1. DiscreteGameTheory
2. Examples

import DiscreteGameTheory.Base namespace Base open SignGame def intSign (a b : Int) : Sign := if a > b then .pos else if a = b then .zero else .neg lemma intSign_refl (a : Int) : intSign a a = .zero := bya:ℤ⊢ intSign a a = Sign.zero simp [intSign]All goals completed! 🐙 lemma intSign_antisym (a b : Int) : intSign a b = (intSign b a).flip := bya:ℤb:ℤ⊢ intSign a b = (intSign b a).flip simp only [intSign, Sign.flip]a:ℤb:ℤ⊢ (if a > b then Sign.pos else if a = b then Sign.zero else Sign.neg) = match if b > a then Sign.pos else if b = a then Sign.zero else Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zero; split_ifsposa:ℤb:ℤh✝¹:a > bh✝:b > a⊢ Falseposa:ℤb:ℤh✝²:a > bh✝¹:¬b > ah✝:b = a⊢ Falsenega:ℤb:ℤh✝²:a > bh✝¹:¬b > ah✝:¬b = a⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroposa:ℤb:ℤh✝²:¬a > bh✝¹:a = bh✝:b > a⊢ Falseposa:ℤb:ℤh✝³:¬a > bh✝²:a = bh✝¹:¬b > ah✝:b = a⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeronega:ℤb:ℤh✝³:¬a > bh✝²:a = bh✝¹:¬b > ah✝:¬b = a⊢ Falseposa:ℤb:ℤh✝²:¬a > bh✝¹:¬a = bh✝:b > a⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroposa:ℤb:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:¬b > ah✝:b = a⊢ Falsenega:ℤb:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:¬b > ah✝:¬b = a⊢ False <;>posa:ℤb:ℤh✝¹:a > bh✝:b > a⊢ Falseposa:ℤb:ℤh✝²:a > bh✝¹:¬b > ah✝:b = a⊢ Falsenega:ℤb:ℤh✝²:a > bh✝¹:¬b > ah✝:¬b = a⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroposa:ℤb:ℤh✝²:¬a > bh✝¹:a = bh✝:b > a⊢ Falseposa:ℤb:ℤh✝³:¬a > bh✝²:a = bh✝¹:¬b > ah✝:b = a⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeronega:ℤb:ℤh✝³:¬a > bh✝²:a = bh✝¹:¬b > ah✝:¬b = a⊢ Falseposa:ℤb:ℤh✝²:¬a > bh✝¹:¬a = bh✝:b > a⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroposa:ℤb:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:¬b > ah✝:b = a⊢ Falsenega:ℤb:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:¬b > ah✝:¬b = a⊢ False first | rflnega:ℤb:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:¬b > ah✝:¬b = a⊢ False | omegaAll goals completed! 🐙 lemma intSign_trans (a b c : Int) : (intSign a b).nonneg → (intSign b c).nonneg → (intSign a c).nonneg := bya:ℤb:ℤc:ℤ⊢ (intSign a b).nonneg → (intSign b c).nonneg → (intSign a c).nonneg simp only [intSign, Sign.nonneg]a:ℤb:ℤc:ℤ⊢ (match if a > b then Sign.pos else if a = b then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match if b > c then Sign.pos else if b = c then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match if a > c then Sign.pos else if a = c then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False; split_ifsposa:ℤb:ℤc:ℤh✝²:a > bh✝¹:b > ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝³:a > bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:¬a > bh✝²:a = bh✝¹:b > ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:b > ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False <;>posa:ℤb:ℤc:ℤh✝²:a > bh✝¹:b > ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝³:a > bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:a > bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:a > bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:¬a > bh✝²:a = bh✝¹:b > ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:a = bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝³:¬a > bh✝²:¬a = bh✝¹:b > ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:b > ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:b > ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:¬b > ch✝¹:b = ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁴:¬a > bh✝³:¬a = bh✝²:¬b > ch✝¹:¬b = ch✝:a > c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.pos with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseposa:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.zero with | Sign.pos => True | Sign.zero => True | Sign.neg => Falsenega:ℤb:ℤc:ℤh✝⁵:¬a > bh✝⁴:¬a = bh✝³:¬b > ch✝²:¬b = ch✝¹:¬a > ch✝:¬a = c⊢ (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → (match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False) → match Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False simp_allAll goals completed! 🐙; omegaAll goals completed! 🐙

Construct a 2-player Bool game from payoff matrices. Uses ofPayoffs to get axioms for free.

def game2x2 (u_TT u_TF u_FT u_FF : Int) (v_TT v_TF v_FT v_FF : Int) : SignGame (Fin 2) (fun _ : Fin 2 => Bool) := SignGame.ofPayoffs (fun i s => if (i : ℕ) = 0 then if s 0 then (if s 1 then u_TT else u_TF) else (if s 1 then u_FT else u_FF) else if s 0 then (if s 1 then v_TT else v_TF) else (if s 1 then v_FT else v_FF)) -- Prisoner's Dilemma: C = true, D = false def pd := game2x2 3 0 5 1 3 5 0 1 theorem pd_nash_DD : pd.IsPureNash (fun _ => false) := by⊢ pd.IsPureNash fun x => false intro i vi:Fin 2v:Bool⊢ (pd.signAction i (fun x => false) ((fun x => false) i) v).nonneg; fin_cases i«0»v:Bool⊢ (pd.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (pd.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (pd.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (pd.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (pd.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 theorem pd_not_nash_CC : ¬pd.IsPureNash (fun _ => true) := by⊢ ¬pd.IsPureNash fun x => true intro hh:pd.IsPureNash fun x => true⊢ False; have := h 0 falseh:pd.IsPureNash fun x => truethis:(pd.signAction 0 (fun x => true) ((fun x => true) 0) false).nonneg := h 0 false⊢ False; revert thish:pd.IsPureNash fun x => true⊢ (pd.signAction 0 (fun x => true) ((fun x => true) 0) false).nonneg → False; decideAll goals completed! 🐙 -- Matching Pennies: H = true, T = false def mp := game2x2 1 0 0 1 0 1 1 0 theorem mp_no_pureNash : ∀ p : PureProfile (Fin 2) (fun _ : Fin 2 => Bool), ¬mp.IsPureNash p := by⊢ ∀ (p : PureProfile (Fin 2) fun x => Bool), ¬mp.IsPureNash p intro p hpp:PureProfile (Fin 2) fun x => Boolhp:mp.IsPureNash p⊢ False have h0t := hp 0 truep:PureProfile (Fin 2) fun x => Boolhp:mp.IsPureNash ph0t:(mp.signAction 0 p (p 0) true).nonneg := hp 0 true⊢ False; have h0f := hp 0 falsep:PureProfile (Fin 2) fun x => Boolhp:mp.IsPureNash ph0t:(mp.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp.signAction 0 p (p 0) false).nonneg := hp 0 false⊢ False have h1t := hp 1 truep:PureProfile (Fin 2) fun x => Boolhp:mp.IsPureNash ph0t:(mp.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp.signAction 0 p (p 0) false).nonneg := hp 0 falseh1t:(mp.signAction 1 p (p 1) true).nonneg := hp 1 true⊢ False; have h1f := hp 1 falsep:PureProfile (Fin 2) fun x => Boolhp:mp.IsPureNash ph0t:(mp.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp.signAction 0 p (p 0) false).nonneg := hp 0 falseh1t:(mp.signAction 1 p (p 1) true).nonneg := hp 1 trueh1f:(mp.signAction 1 p (p 1) false).nonneg := hp 1 false⊢ False simp only [mp, game2x2, SignGame.ofPayoffs, IsPureNash, SignGame.signAction, Sign.nonneg, Function.update] at *p:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False⊢ False cases h0 : p 0falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = false⊢ Falsetruep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = true⊢ False <;>falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = false⊢ Falsetruep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = true⊢ False cases h1 : p 1true.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = false⊢ Falsetrue.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = true⊢ False <;>false.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = falseh1:p 1 = false⊢ Falsefalse.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = falseh1:p 1 = true⊢ Falsetrue.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = false⊢ Falsetrue.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) > if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.pos else if (if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ p i else p 0) = true then if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ p i else p 1) = true then 1 else 0) = if ↑i = 0 then if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 0 = i then ⋯ ▸ v else p 0) = true then if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 0 else 1 else if (if h : 1 = i then ⋯ ▸ v else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : True then true else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) > if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑0 = 0 then if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : True then p 0 else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ p 0 else p 1) = true then 1 else 0) = if ↑0 = 0 then if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : True then false else p 0) = true then if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 0 else 1 else if (if h : 1 = 0 then ⋯ ▸ false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 1 else 0 else if (if h : True then true else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ true else p 0) = true then if (if h : True then true else p 1) = true then 0 else 1 else if (if h : True then true else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) > if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.pos else if (if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 1 else 0 else if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ p 1 else p 0) = true then if (if h : True then p 1 else p 1) = true then 0 else 1 else if (if h : True then p 1 else p 1) = true then 1 else 0) = if ↑1 = 0 then if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 1 else 0 else if (if h : True then false else p 1) = true then 0 else 1 else if (if h : 0 = 1 then ⋯ ▸ false else p 0) = true then if (if h : True then false else p 1) = true then 0 else 1 else if (if h : True then false else p 1) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = true⊢ False simp_allAll goals completed! 🐙 -- Stag Hunt: S = true, H = false def sh := game2x2 4 0 3 3 4 3 0 3 theorem sh_nash_SS : sh.IsPureNash (fun _ => true) := by⊢ sh.IsPureNash fun x => true intro i vi:Fin 2v:Bool⊢ (sh.signAction i (fun x => true) ((fun x => true) i) v).nonneg; fin_cases i«0»v:Bool⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 theorem sh_nash_HH : sh.IsPureNash (fun _ => false) := by⊢ sh.IsPureNash fun x => false intro i vi:Fin 2v:Bool⊢ (sh.signAction i (fun x => false) ((fun x => false) i) v).nonneg; fin_cases i«0»v:Bool⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (sh.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (sh.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 -- Battle of the Sexes: O = true, F = false def bos := game2x2 3 0 0 2 2 0 0 3 theorem bos_nash_OO : bos.IsPureNash (fun _ => true) := by⊢ bos.IsPureNash fun x => true intro i vi:Fin 2v:Bool⊢ (bos.signAction i (fun x => true) ((fun x => true) i) v).nonneg; fin_cases i«0»v:Bool⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 theorem bos_nash_FF : bos.IsPureNash (fun _ => false) := by⊢ bos.IsPureNash fun x => false intro i vi:Fin 2v:Bool⊢ (bos.signAction i (fun x => false) ((fun x => false) i) v).nonneg; fin_cases i«0»v:Bool⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (bos.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (bos.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙

A 3-player coordination game: each player gets payoff 1 if all agree, 0 otherwise.

def coordGame3 : SignGame (Fin 3) (fun _ : Fin 3 => Bool) where sign _i p q := let allAgree (s : ∀ _ : Fin 3, Bool) : Bool := s 0 == s 1 && s 1 == s 2 let payP : Int := if allAgree p then 1 else 0 let payQ : Int := if allAgree q then 1 else 0 intSign payP payQ sign_refl _i p := by_i:Fin 3p:PureProfile (Fin 3) fun x => Bool⊢ (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree p = true then 1 else 0; intSign payP payQ) = Sign.zero simp [intSign]All goals completed! 🐙 sign_antisym _i p q := by_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Bool⊢ (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree q = true then 1 else 0; intSign payP payQ) = (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree q = true then 1 else 0; let payQ := if allAgree p = true then 1 else 0; intSign payP payQ).flip simp only [intSign, Sign.flip]_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Bool⊢ (if (if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0) > if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0 then Sign.pos else if (if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0) = if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0 then Sign.zero else Sign.neg) = match if (if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0) > if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0 then Sign.pos else if (if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0) = if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0 then Sign.zero else Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zero; split_ifspos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(q 0 == q 1 && q 1 == q 2) = trueh✝:1 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬1 > 1h✝:1 = 1⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬1 > 1h✝:¬1 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:1 > 0h✝:0 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:1 > 0h✝¹:¬0 > 1h✝:0 = 1⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:1 > 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬1 > 0h✝¹:1 = 0h✝:0 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:1 = 0h✝¹:¬0 > 1h✝:0 = 1⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:1 = 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬1 > 0h✝¹:¬1 = 0h✝:0 > 1⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:¬1 = 0h✝¹:¬0 > 1h✝:0 = 1⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:¬1 = 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:0 > 1h✝:1 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:0 > 1h✝¹:¬1 > 0h✝:1 = 0⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:0 > 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬0 > 1h✝¹:0 = 1h✝:1 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:0 = 1h✝¹:¬1 > 0h✝:1 = 0⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:0 = 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬0 > 1h✝¹:¬0 = 1h✝:1 > 0⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:¬0 = 1h✝¹:¬1 > 0h✝:1 = 0⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:¬0 = 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(q 0 == q 1 && q 1 == q 2) = trueh✝:0 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬0 > 0h✝:0 = 0⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬0 > 0h✝:¬0 = 0⊢ False <;>pos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(q 0 == q 1 && q 1 == q 2) = trueh✝:1 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬1 > 1h✝:1 = 1⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬1 > 1h✝:¬1 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:1 > 0h✝:0 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:1 > 0h✝¹:¬0 > 1h✝:0 = 1⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:1 > 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬1 > 0h✝¹:1 = 0h✝:0 > 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:1 = 0h✝¹:¬0 > 1h✝:0 = 1⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:1 = 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:(p 0 == p 1 && p 1 == p 2) = trueh✝³:¬(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬1 > 0h✝¹:¬1 = 0h✝:0 > 1⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:¬1 = 0h✝¹:¬0 > 1h✝:0 = 1⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:¬(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬1 > 0h✝²:¬1 = 0h✝¹:¬0 > 1h✝:¬0 = 1⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:(q 0 == q 1 && q 1 == q 2) = trueh✝¹:0 > 1h✝:1 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:0 > 1h✝¹:¬1 > 0h✝:1 = 0⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:0 > 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Sign.pos = match Sign.neg with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬0 > 1h✝¹:0 = 1h✝:1 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:0 = 1h✝¹:¬1 > 0h✝:1 = 0⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:0 = 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁴:¬(p 0 == p 1 && p 1 == p 2) = trueh✝³:(q 0 == q 1 && q 1 == q 2) = trueh✝²:¬0 > 1h✝¹:¬0 = 1h✝:1 > 0⊢ Sign.neg = match Sign.pos with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeropos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:¬0 = 1h✝¹:¬1 > 0h✝:1 = 0⊢ Falseneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝⁵:¬(p 0 == p 1 && p 1 == p 2) = trueh✝⁴:(q 0 == q 1 && q 1 == q 2) = trueh✝³:¬0 > 1h✝²:¬0 = 1h✝¹:¬1 > 0h✝:¬1 = 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(q 0 == q 1 && q 1 == q 2) = trueh✝:0 > 0⊢ Falsepos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬0 > 0h✝:0 = 0⊢ Sign.zero = match Sign.zero with | Sign.pos => Sign.neg | Sign.neg => Sign.pos | Sign.zero => Sign.zeroneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬0 > 0h✝:¬0 = 0⊢ False first | rflneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolh✝³:¬(p 0 == p 1 && p 1 == p 2) = trueh✝²:¬(q 0 == q 1 && q 1 == q 2) = trueh✝¹:¬0 > 0h✝:¬0 = 0⊢ False | omegaAll goals completed! 🐙 sign_trans _i p q r := by_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Bool⊢ (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree q = true then 1 else 0; intSign payP payQ).nonneg → (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree q = true then 1 else 0; let payQ := if allAgree r = true then 1 else 0; intSign payP payQ).nonneg → (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree r = true then 1 else 0; intSign payP payQ).nonneg intro hpq hqr_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolhpq:(let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree q = true then 1 else 0; intSign payP payQ).nonneghqr:(let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree q = true then 1 else 0; let payQ := if allAgree r = true then 1 else 0; intSign payP payQ).nonneg⊢ (let allAgree := fun s => s 0 == s 1 && s 1 == s 2; let payP := if allAgree p = true then 1 else 0; let payQ := if allAgree r = true then 1 else 0; intSign payP payQ).nonneg; simp only [] at *_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolhpq:(intSign (if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0) (if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0)).nonneghqr:(intSign (if (q 0 == q 1 && q 1 == q 2) = true then 1 else 0) (if (r 0 == r 1 && r 1 == r 2) = true then 1 else 0)).nonneg⊢ (intSign (if (p 0 == p 1 && p 1 == p 2) = true then 1 else 0) (if (r 0 == r 1 && r 1 == r 2) = true then 1 else 0)).nonneg split_ifs at *pos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 1).nonneghqr:(intSign 1 1).nonneg⊢ (intSign 1 1).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 0).nonneghqr:(intSign 0 1).nonneg⊢ (intSign 1 1).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 1).nonneghqr:(intSign 1 0).nonneg⊢ (intSign 1 0).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 0).nonneghqr:(intSign 0 0).nonneg⊢ (intSign 1 0).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 1).nonneghqr:(intSign 1 1).nonneg⊢ (intSign 0 1).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 0).nonneghqr:(intSign 0 1).nonneg⊢ (intSign 0 1).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 1).nonneghqr:(intSign 1 0).nonneg⊢ (intSign 0 0).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 0).nonneghqr:(intSign 0 0).nonneg⊢ (intSign 0 0).nonneg <;>pos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 1).nonneghqr:(intSign 1 1).nonneg⊢ (intSign 1 1).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 0).nonneghqr:(intSign 0 1).nonneg⊢ (intSign 1 1).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 1).nonneghqr:(intSign 1 0).nonneg⊢ (intSign 1 0).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 1 0).nonneghqr:(intSign 0 0).nonneg⊢ (intSign 1 0).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 1).nonneghqr:(intSign 1 1).nonneg⊢ (intSign 0 1).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 0).nonneghqr:(intSign 0 1).nonneg⊢ (intSign 0 1).nonnegpos_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 1).nonneghqr:(intSign 1 0).nonneg⊢ (intSign 0 0).nonnegneg_i:Fin 3p:PureProfile (Fin 3) fun x => Boolq:PureProfile (Fin 3) fun x => Boolr:PureProfile (Fin 3) fun x => Boolh✝²:¬(p 0 == p 1 && p 1 == p 2) = trueh✝¹:¬(r 0 == r 1 && r 1 == r 2) = trueh✝:¬(q 0 == q 1 && q 1 == q 2) = truehpq:(intSign 0 0).nonneghqr:(intSign 0 0).nonneg⊢ (intSign 0 0).nonneg exact intSign_trans _ _ _ hpq hqrAll goals completed! 🐙 theorem coordGame3_nash_allTrue : coordGame3.IsPureNash (fun _ => true) := by⊢ coordGame3.IsPureNash fun x => true intro i vi:Fin 3v:Bool⊢ (coordGame3.signAction i (fun x => true) ((fun x => true) i) v).nonneg; fin_cases i«0»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg«2»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg«2»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) v).nonneg cases v«2».false⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) false).nonneg«2».true⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) true).nonneg <;>«0».false⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg«2».false⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) false).nonneg«2».true⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => true) ((fun x => true) ((fun i => i) ⟨2, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 theorem coordGame3_nash_allFalse : coordGame3.IsPureNash (fun _ => false) := by⊢ coordGame3.IsPureNash fun x => false intro i vi:Fin 3v:Bool⊢ (coordGame3.signAction i (fun x => false) ((fun x => false) i) v).nonneg; fin_cases i«0»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg«2»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg«2»v:Bool⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) v).nonneg cases v«2».false⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) false).nonneg«2».true⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) true).nonneg <;>«0».false⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (coordGame3.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (coordGame3.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg«2».false⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) false).nonneg«2».true⊢ (coordGame3.signAction ((fun i => i) ⟨2, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨2, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 theorem coordGame3_not_nash_mixed : ¬coordGame3.IsPureNash (fun i : Fin 3 => if i = 0 then true else false) := by⊢ ¬coordGame3.IsPureNash fun i => if i = 0 then true else false intro hh:coordGame3.IsPureNash fun i => if i = 0 then true else false⊢ False; have := h 0 falseh:coordGame3.IsPureNash fun i => if i = 0 then true else falsethis:(coordGame3.signAction 0 (fun i => if i = 0 then true else false) ((fun i => if i = 0 then true else false) 0) false).nonneg := h 0 false⊢ False; revert thish:coordGame3.IsPureNash fun i => if i = 0 then true else false⊢ (coordGame3.signAction 0 (fun i => if i = 0 then true else false) ((fun i => if i = 0 then true else false) 0) false).nonneg → False; decideAll goals completed! 🐙

The Prisoner's Dilemma has a unique pure Nash equilibrium at (D,D). This illustrates the central tension of PD: mutual cooperation is Pareto-better but not an equilibrium.

theorem pd_unique_pureNash : ∀ p : PureProfile (Fin 2) (fun _ : Fin 2 => Bool), pd.IsPureNash p ↔ p = (fun _ => false) := by⊢ ∀ (p : PureProfile (Fin 2) fun x => Bool), pd.IsPureNash p ↔ p = fun x => false intro pp:PureProfile (Fin 2) fun x => Bool⊢ pd.IsPureNash p ↔ p = fun x => false constructormpp:PureProfile (Fin 2) fun x => Bool⊢ pd.IsPureNash p → p = fun x => falsemprp:PureProfile (Fin 2) fun x => Bool⊢ (p = fun x => false) → pd.IsPureNash p ·mpp:PureProfile (Fin 2) fun x => Bool⊢ pd.IsPureNash p → p = fun x => false intro hpmpp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash p⊢ p = fun x => false by_contra hmpp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => false⊢ False have : ∃ i, p i = true := by⊢ ∀ (p : PureProfile (Fin 2) fun x => Bool), pd.IsPureNash p ↔ p = fun x => false by_contra hallp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehall:¬∃ i, p i = true⊢ False; push_neg at hallp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehall:∀ (i : Fin 2), p i ≠ true⊢ False apply hp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehall:∀ (i : Fin 2), p i ≠ true⊢ p = fun x => false; ext ihp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehall:∀ (i : Fin 2), p i ≠ truei:Fin 2⊢ p i = false; exact Bool.eq_false_iff.mpr (hall i)All goals completed! 🐙 obtain ⟨i, hi⟩ := thismpp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsei:Fin 2hi:p i = true⊢ False have hpi := hp i falsempp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsei:Fin 2hi:p i = truehpi:(pd.signAction i p (p i) false).nonneg := hp i false⊢ False; have hpi' := hp i truempp:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsei:Fin 2hi:p i = truehpi:(pd.signAction i p (p i) false).nonneg := hp i falsehpi':(pd.signAction i p (p i) true).nonneg := hp i true⊢ False fin_cases imp.«0»p:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨0, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨0, ⋯⟩) p (p ((fun i => i) ⟨0, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨0, ⋯⟩) p (p ((fun i => i) ⟨0, ⋯⟩)) true).nonneg⊢ Falsemp.«1»p:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonneg⊢ False <;>mp.«0»p:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨0, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨0, ⋯⟩) p (p ((fun i => i) ⟨0, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨0, ⋯⟩) p (p ((fun i => i) ⟨0, ⋯⟩)) true).nonneg⊢ Falsemp.«1»p:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonneg⊢ False (cases h0 : p 0mp.«1».falsep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = false⊢ Falsemp.«1».truep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = true⊢ False <;>mp.«1».falsep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = false⊢ Falsemp.«1».truep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = true⊢ False cases h1 : p 1mp.«1».true.falsep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = trueh1:p 1 = false⊢ Falsemp.«1».true.truep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = trueh1:p 1 = true⊢ False <;>mp.«1».false.falsep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = falseh1:p 1 = false⊢ Falsemp.«1».false.truep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = falseh1:p 1 = true⊢ Falsemp.«1».true.falsep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = trueh1:p 1 = false⊢ Falsemp.«1».true.truep:PureProfile (Fin 2) fun x => Boolhp:pd.IsPureNash ph:¬p = fun x => falsehi:p ((fun i => i) ⟨1, ⋯⟩) = truehpi:(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) false).nonneghpi':(pd.signAction ((fun i => i) ⟨1, ⋯⟩) p (p ((fun i => i) ⟨1, ⋯⟩)) true).nonnegh0:p 0 = trueh1:p 1 = true⊢ False simp_all [pd, game2x2, SignGame.ofPayoffs, SignGame.signAction, Sign.nonneg, Function.update]All goals completed! 🐙) ·mprp:PureProfile (Fin 2) fun x => Bool⊢ (p = fun x => false) → pd.IsPureNash p rintro rflmpr⊢ pd.IsPureNash fun x => false; exact pd_nash_DDAll goals completed! 🐙

Matching Pennies has no pure Nash equilibrium (mp_no_pureNash), but the fully mixed profile where both players play {H, T} is a Nash equilibrium in the sign-game sense: no face strictly dominates the full face.

theorem mp_mixed_nash : mp.IsNash (fun _ : Fin 2 => Face.full (V := Bool)) := by⊢ mp.IsNash fun x => Face.full intro i Ai:Fin 2A:Face ((fun x => Bool) i)⊢ ¬mp.StrictDom i (fun x => Face.full) A ⟨hfwd, _⟩i:Fin 2A:Face ((fun x => Bool) i)hfwd:mp.Dominates i (fun x => Face.full) A ((fun x => Face.full) i)right✝:¬mp.Dominates i (fun x => Face.full) ((fun x => Face.full) i) A⊢ False obtain ⟨a, ha⟩ := A.2i:Fin 2A:Face ((fun x => Bool) i)hfwd:mp.Dominates i (fun x => Face.full) A ((fun x => Face.full) i)right✝:¬mp.Dominates i (fun x => Face.full) ((fun x => Face.full) i) Aa:Boolha:a ∈ ↑A⊢ False have hcon : ∀ (p : ∀ _ : Fin 2, Bool), ConsistentAt (fun _ : Fin 2 => Face.full (V := Bool)) i p := fun _ _ _ => Finset.mem_univ _i:Fin 2A:Face ((fun x => Bool) i)hfwd:mp.Dominates i (fun x => Face.full) A ((fun x => Face.full) i)right✝:¬mp.Dominates i (fun x => Face.full) ((fun x => Face.full) i) Aa:Boolha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) i p := fun x x_1 x_2 => Finset.mem_univ (x x_1)⊢ False have h1 := hfwd a ha (fun _ => true) (hcon _) (!a) (Finset.mem_univ _)i:Fin 2A:Face ((fun x => Bool) i)hfwd:mp.Dominates i (fun x => Face.full) A ((fun x => Face.full) i)right✝:¬mp.Dominates i (fun x => Face.full) ((fun x => Face.full) i) Aa:Boolha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) i p := fun x x_1 x_2 => Finset.mem_univ (x x_1)h1:(mp.signAction i (fun x => true) a !a).nonneg := hfwd a ha (fun x => true) (hcon fun x => true) (!a) (Finset.mem_univ !a)⊢ False have h2 := hfwd a ha (fun _ => false) (hcon _) (!a) (Finset.mem_univ _)i:Fin 2A:Face ((fun x => Bool) i)hfwd:mp.Dominates i (fun x => Face.full) A ((fun x => Face.full) i)right✝:¬mp.Dominates i (fun x => Face.full) ((fun x => Face.full) i) Aa:Boolha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) i p := fun x x_1 x_2 => Finset.mem_univ (x x_1)h1:(mp.signAction i (fun x => true) a !a).nonneg := hfwd a ha (fun x => true) (hcon fun x => true) (!a) (Finset.mem_univ !a)h2:(mp.signAction i (fun x => false) a !a).nonneg := hfwd a ha (fun x => false) (hcon fun x => false) (!a) (Finset.mem_univ !a)⊢ False fin_cases i«0»a:BoolA:Face ((fun x => Bool) ((fun i => i) ⟨0, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩)) Aha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩) ph1:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) a !a).nonnegh2:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) a !a).nonneg⊢ False«1»a:BoolA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Aha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) ph1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) a !a).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) a !a).nonneg⊢ False <;>«0»a:BoolA:Face ((fun x => Bool) ((fun i => i) ⟨0, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩)) Aha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩) ph1:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) a !a).nonnegh2:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) a !a).nonneg⊢ False«1»a:BoolA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Aha:a ∈ ↑Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) ph1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) a !a).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) a !a).nonneg⊢ False cases a«1».falseA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) pha:false ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) false !false).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) false !false).nonneg⊢ False«1».trueA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) pha:true ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) true !true).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) true !true).nonneg⊢ False <;>«0».falseA:Face ((fun x => Bool) ((fun i => i) ⟨0, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩) pha:false ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) false !false).nonnegh2:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) false !false).nonneg⊢ False«0».trueA:Face ((fun x => Bool) ((fun i => i) ⟨0, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨0, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨0, ⋯⟩) pha:true ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => true) true !true).nonnegh2:(mp.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) true !true).nonneg⊢ False«1».falseA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) pha:false ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) false !false).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) false !false).nonneg⊢ False«1».trueA:Face ((fun x => Bool) ((fun i => i) ⟨1, ⋯⟩))hfwd:mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) A ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩))right✝:¬mp.Dominates ((fun i => i) ⟨1, ⋯⟩) (fun x => Face.full) ((fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩)) Ahcon:∀ (p : Fin 2 → Bool), ConsistentAt (fun x => Face.full) ((fun i => i) ⟨1, ⋯⟩) pha:true ∈ ↑Ah1:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => true) true !true).nonnegh2:(mp.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) true !true).nonneg⊢ False simp_all [mp, game2x2, SignGame.ofPayoffs, SignGame.signAction, Sign.nonneg, Function.update]All goals completed! 🐙

The Dominates ordering is genuinely partial on mixed profiles: in Matching Pennies, when the opponent plays the full face {H,T}, neither {H} nor {T} dominates the other for player 0. This shows that sign-game dominance is not a total order on faces.

theorem mp_partial_order : let σ : Profile (Fin 2) (fun _ : Fin 2 => Bool) := fun _ => Face.full ¬mp.Dominates 0 σ (Face.vertex true) (Face.vertex false) ∧ ¬mp.Dominates 0 σ (Face.vertex false) (Face.vertex true) := by⊢ let σ := fun x => Face.full; ¬mp.Dominates 0 σ (Face.vertex true) (Face.vertex false) ∧ ¬mp.Dominates 0 σ (Face.vertex false) (Face.vertex true) refine ⟨fun h => ?_, fun h => ?_⟩refine_1h:mp.Dominates 0 (fun x => Face.full) (Face.vertex true) (Face.vertex false)⊢ Falserefine_2h:mp.Dominates 0 (fun x => Face.full) (Face.vertex false) (Face.vertex true)⊢ False ·refine_1h:mp.Dominates 0 (fun x => Face.full) (Face.vertex true) (Face.vertex false)⊢ False have := h true (Finset.mem_singleton_self _) (fun _ => false) (fun _ _ => Finset.mem_univ _ : ConsistentAt _ (0 : Fin 2) _) false (Finset.mem_singleton_self _)refine_1h:mp.Dominates 0 (fun x => Face.full) (Face.vertex true) (Face.vertex false)this:(mp.signAction 0 (fun x => false) true false).nonneg := h true (Finset.mem_singleton_self true) (fun x => false) (fun x x_1 => Finset.mem_univ false) false (Finset.mem_singleton_self false)⊢ False simp_all [mp, game2x2, SignGame.ofPayoffs, SignGame.signAction, Sign.nonneg, Function.update]All goals completed! 🐙 ·refine_2h:mp.Dominates 0 (fun x => Face.full) (Face.vertex false) (Face.vertex true)⊢ False have := h false (Finset.mem_singleton_self _) (fun _ => true) (fun _ _ => Finset.mem_univ _ : ConsistentAt _ (0 : Fin 2) _) true (Finset.mem_singleton_self _)refine_2h:mp.Dominates 0 (fun x => Face.full) (Face.vertex false) (Face.vertex true)this:(mp.signAction 0 (fun x => true) false true).nonneg := h false (Finset.mem_singleton_self false) (fun x => true) (fun x x_1 => Finset.mem_univ true) true (Finset.mem_singleton_self true)⊢ False simp_all [mp, game2x2, SignGame.ofPayoffs, SignGame.signAction, Sign.nonneg, Function.update]All goals completed! 🐙

Build a symmetric 2-player game from a ranking function. rank me opp gives the ordinal payoff when I play me and my opponent plays opp. Symmetry means both players share the same ranking function.

def symGame2x2 (rank : Bool → Bool → ℕ) : SignGame (Fin 2) (fun _ : Fin 2 => Bool) := SignGame.ofPayoffs (fun i s => (rank (s i) (s (1 - i)) : Int))

Build an asymmetric 2-player game from per-player ranking functions.

def game2x2_rank (rank₀ rank₁ : Bool → Bool → ℕ) : SignGame (Fin 2) (fun _ : Fin 2 => Bool) := SignGame.ofPayoffs (fun i s => if (i : ℕ) = 0 then (rank₀ (s 0) (s 1) : Int) else (rank₁ (s 1) (s 0) : Int)) -- Prisoner's Dilemma via ranking: C = true, D = false def pd_rank (me opp : Bool) : ℕ := match me, opp with | true, false => 0 -- I cooperate, they defect (worst) | false, false => 1 -- both defect | true, true => 2 -- both cooperate | false, true => 3 -- I defect, they cooperate (best def pd' := symGame2x2 pd_rank lemma pd'_nash_DD : pd'.IsPureNash (fun _ => false) := by⊢ pd'.IsPureNash fun x => false intro i vi:Fin 2v:Bool⊢ (pd'.signAction i (fun x => false) ((fun x => false) i) v).nonneg; fin_cases i«0»v:Bool⊢ (pd'.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg <;>«0»v:Bool⊢ (pd'.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) v).nonneg«1»v:Bool⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) v).nonneg cases v«1».false⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg <;>«0».false⊢ (pd'.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) false).nonneg«0».true⊢ (pd'.signAction ((fun i => i) ⟨0, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨0, ⋯⟩)) true).nonneg«1».false⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) false).nonneg«1».true⊢ (pd'.signAction ((fun i => i) ⟨1, ⋯⟩) (fun x => false) ((fun x => false) ((fun i => i) ⟨1, ⋯⟩)) true).nonneg decideAll goals completed! 🐙 lemma pd'_not_nash_CC : ¬pd'.IsPureNash (fun _ => true) := by⊢ ¬pd'.IsPureNash fun x => true intro hh:pd'.IsPureNash fun x => true⊢ False; have := h 0 falseh:pd'.IsPureNash fun x => truethis:(pd'.signAction 0 (fun x => true) ((fun x => true) 0) false).nonneg := h 0 false⊢ False; revert thish:pd'.IsPureNash fun x => true⊢ (pd'.signAction 0 (fun x => true) ((fun x => true) 0) false).nonneg → False; decideAll goals completed! 🐙

Two ranking functions that differ by an affine transform produce the same sign function, illustrating ordinal invariance.

def pd_rank_alt (me opp : Bool) : ℕ := match me, opp with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40 theorem pd_same_game : (symGame2x2 pd_rank).sign = (symGame2x2 pd_rank_alt).sign := by⊢ (symGame2x2 pd_rank).sign = (symGame2x2 pd_rank_alt).sign ext i p qh.h.hi:Fin 2p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (symGame2x2 pd_rank).sign i p q = (symGame2x2 pd_rank_alt).sign i p q simp only [symGame2x2, SignGame.ofPayoffs, pd_rank, pd_rank_alt]h.h.hi:Fin 2p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (if ↑(match p i, p (1 - i) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q i, q (1 - i) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p i, p (1 - i) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q i, q (1 - i) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p i, p (1 - i) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q i, q (1 - i) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p i, p (1 - i) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q i, q (1 - i) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg fin_cases ih.h.h.«0»p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1»p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg <;>h.h.h.«0»p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨0, ⋯⟩), p (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨0, ⋯⟩), q (1 - (fun i => i) ⟨0, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1»p:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Bool⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg ( cases h0 : p 0h.h.h.«1».falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg <;>h.h.h.«1».falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg cases h1 : p 1h.h.h.«1».true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg <;>h.h.h.«1».false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg cases h2 : q 0h.h.h.«1».true.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg <;>h.h.h.«1».false.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg cases h3 : q 1h.h.h.«1».true.true.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = trueh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = trueh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg <;>h.h.h.«1».false.false.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = falseh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.false.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = falseh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.false.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = trueh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.false.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = falseh2:q 0 = trueh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = falseh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = falseh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = trueh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».false.true.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = falseh1:p 1 = trueh2:q 0 = trueh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = falseh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = falseh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = trueh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.false.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = falseh2:q 0 = trueh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.false.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = falseh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.false.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = falseh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.true.falsep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = trueh3:q 1 = false⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.negh.h.h.«1».true.true.true.truep:PureProfile (Fin 2) fun x => Boolq:PureProfile (Fin 2) fun x => Boolh0:p 0 = trueh1:p 1 = trueh2:q 0 = trueh3:q 1 = true⊢ (if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 0 | false, false => 1 | true, true => 2 | false, true => 3) then Sign.zero else Sign.neg) = if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) > ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.pos else if ↑(match p ((fun i => i) ⟨1, ⋯⟩), p (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) = ↑(match q ((fun i => i) ⟨1, ⋯⟩), q (1 - (fun i => i) ⟨1, ⋯⟩) with | true, false => 10 | false, false => 20 | true, true => 30 | false, true => 40) then Sign.zero else Sign.neg simp_allAll goals completed! 🐙) -- Matching Pennies via ranking: P0 wants to match, P1 wants to differ def mp' := game2x2_rank (fun me opp => match me, opp with -- P0 wants to match | true, true => 1 | false, false => 1 | _, _ => 0) (fun me opp => match me, opp with -- P1 wants to differ | true, false => 1 | false, true => 1 | _, _ => 0) theorem mp'_no_pureNash : ∀ p, ¬mp'.IsPureNash p := by⊢ ∀ (p : PureProfile (Fin 2) fun x => Bool), ¬mp'.IsPureNash p intro p hpp:PureProfile (Fin 2) fun x => Boolhp:mp'.IsPureNash p⊢ False have h0t := hp 0 truep:PureProfile (Fin 2) fun x => Boolhp:mp'.IsPureNash ph0t:(mp'.signAction 0 p (p 0) true).nonneg := hp 0 true⊢ False; have h0f := hp 0 falsep:PureProfile (Fin 2) fun x => Boolhp:mp'.IsPureNash ph0t:(mp'.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp'.signAction 0 p (p 0) false).nonneg := hp 0 false⊢ False have h1t := hp 1 truep:PureProfile (Fin 2) fun x => Boolhp:mp'.IsPureNash ph0t:(mp'.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp'.signAction 0 p (p 0) false).nonneg := hp 0 falseh1t:(mp'.signAction 1 p (p 1) true).nonneg := hp 1 true⊢ False; have h1f := hp 1 falsep:PureProfile (Fin 2) fun x => Boolhp:mp'.IsPureNash ph0t:(mp'.signAction 0 p (p 0) true).nonneg := hp 0 trueh0f:(mp'.signAction 0 p (p 0) false).nonneg := hp 0 falseh1t:(mp'.signAction 1 p (p 1) true).nonneg := hp 1 trueh1f:(mp'.signAction 1 p (p 1) false).nonneg := hp 1 false⊢ False simp only [mp', game2x2_rank, SignGame.ofPayoffs, SignGame.signAction, IsPureNash, Sign.nonneg, Function.update] at *p:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => False⊢ False cases h0 : p 0falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = false⊢ Falsetruep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = true⊢ False <;>falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = false⊢ Falsetruep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = true⊢ False cases h1 : p 1true.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = false⊢ Falsetrue.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = true⊢ False <;>false.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = falseh1:p 1 = false⊢ Falsefalse.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = falseh1:p 1 = true⊢ Falsetrue.falsep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = false⊢ Falsetrue.truep:PureProfile (Fin 2) fun x => Boolhp:∀ (i : Fin 2) (v : Bool), match if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ p i else p 0, if h : 1 = i then ⋯ ▸ p i else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ p i else p 1, if h : 0 = i then ⋯ ▸ p i else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑i = 0 then ↑(match if h : 0 = i then ⋯ ▸ v else p 0, if h : 1 = i then ⋯ ▸ v else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = i then ⋯ ▸ v else p 1, if h : 0 = i then ⋯ ▸ v else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0t:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then true else p 0, if h : 1 = 0 then ⋯ ▸ true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ true else p 1, if h : True then true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0f:match if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑0 = 0 then ↑(match if h : True then p 0 else p 0, if h : 1 = 0 then ⋯ ▸ p 0 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ p 0 else p 1, if h : True then p 0 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑0 = 0 then ↑(match if h : True then false else p 0, if h : 1 = 0 then ⋯ ▸ false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : 1 = 0 then ⋯ ▸ false else p 1, if h : True then false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1t:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ true else p 0, if h : True then true else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then true else p 1, if h : 0 = 1 then ⋯ ▸ true else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh1f:match if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) > if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.pos else if (if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ p 1 else p 0, if h : True then p 1 else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then p 1 else p 1, if h : 0 = 1 then ⋯ ▸ p 1 else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0)) = if ↑1 = 0 then ↑(match if h : 0 = 1 then ⋯ ▸ false else p 0, if h : True then false else p 1 with | true, true => 1 | false, false => 1 | x, x_1 => 0) else ↑(match if h : True then false else p 1, if h : 0 = 1 then ⋯ ▸ false else p 0 with | true, false => 1 | false, true => 1 | x, x_1 => 0) then Sign.zero else Sign.neg with | Sign.pos => True | Sign.zero => True | Sign.neg => Falseh0:p 0 = trueh1:p 1 = true⊢ False simp_allAll goals completed! 🐙

Unlike nash_exists, which is a pure existence theorem, findNash is a def returning a concrete Nash profile. It runs the descent algorithm as a terminating program, so we can #eval it to see the equilibrium.

Prisoner's Dilemma has a unique pure Nash equilibrium at (D, D). Running the descent from the fully mixed profile {C, D}² converges there:

#eval (pd.findNash.1 0).1   -- {false}  (D for player 0)
#eval (pd.findNash.1 1).1   -- {false}  (D for player 1)

Matching Pennies has no pure Nash equilibrium, but the fully mixed profile is an ordinal Nash equilibrium — no proper subface strictly dominates the full face. The descent recognises this immediately and returns the starting profile:

#eval (mp.findNash.1 0).1   -- {false, true}
#eval (mp.findNash.1 1).1   -- {false, true}

The descent finds the unique pure Nash of the Prisoner's Dilemma. Since the whole algorithm is computable, decide discharges the claim by running it.

theorem pd_findNash_is_DD : (pd.findNash.1 0).1 = {false} ∧ (pd.findNash.1 1).1 = {false} := by⊢ ↑(↑pd.findNash 0) = {false} ∧ ↑(↑pd.findNash 1) = {false} native_decideAll goals completed! 🐙

The descent leaves Matching Pennies at the fully mixed profile — the ordinal Nash equilibrium.

theorem mp_findNash_is_full : (mp.findNash.1 0).1 = {false, true} ∧ (mp.findNash.1 1).1 = {false, true} := by⊢ ↑(↑mp.findNash 0) = {false, true} ∧ ↑(↑mp.findNash 1) = {false, true} native_decideAll goals completed! 🐙 end Base