2c87905d77
This PR implements E-matching for the (WIP) `grind` tactic. We still need to finalize and internalize the new instances.
97 lines
3.2 KiB
Text
97 lines
3.2 KiB
Text
set_option warningAsError false
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set_option grind.debug true
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set_option grind.debug.proofs true
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example (a b : Nat) (f : Nat → Nat) : (h₁ : a = b) → f a = f b := by
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grind
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example (a b : Nat) (f : Nat → Nat) : (h₁ : a = b) → (h₂ : f a ≠ f b) → False := by
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grind
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example (a b : Nat) (f : Nat → Nat) : a = b → f (f a) ≠ f (f b) → False := by
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grind
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example (a b c : Nat) (f : Nat → Nat) : a = b → c = b → f (f a) ≠ f (f c) → False := by
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grind
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example (a b c : Nat) (f : Nat → Nat → Nat) : a = b → c = b → f (f a b) a ≠ f (f c c) c → False := by
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grind
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example (a b c : Nat) (f : Nat → Nat → Nat) : a = b → c = b → f (f a b) a = f (f c c) c := by
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grind
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example (a b c d : Nat) : a = b → b = c → c = d → a = d := by
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grind
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infix:50 "===" => HEq
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example (a b c d : Nat) : a === b → b = c → c === d → a === d := by
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grind
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example (a b c d : Nat) : a = b → b = c → c === d → a === d := by
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grind
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opaque f {α : Type} : α → α → α := fun a _ => a
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opaque g : Nat → Nat
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example (a b c : Nat) : a = b → g a === g b := by
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grind
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example (a b c : Nat) : a = b → c = b → f (f a b) (g c) = f (f c a) (g b) := by
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grind
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example (a b c d e x y : Nat) : a = b → a = x → b = y → c = d → c = e → c = b → a = e := by
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grind
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namespace Ex1
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opaque f (a b : Nat) : a > b → Nat
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opaque g : Nat → Nat
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example (a₁ a₂ b₁ b₂ c d : Nat)
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(H₁ : a₁ > b₁)
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(H₂ : a₂ > b₂) :
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a₁ = c → a₂ = c →
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b₁ = d → d = b₂ →
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g (g (f a₁ b₁ H₁)) = g (g (f a₂ b₂ H₂)) := by
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grind
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end Ex1
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namespace Ex2
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def f (α : Type) (a : α) : α := a
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example (a : α) (b : β) : (h₁ : α = β) → (h₂ : HEq a b) → HEq (f α a) (f β b) := by
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grind
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end Ex2
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example (f g : (α : Type) → α → α) (a : α) (b : β) : (h₁ : α = β) → (h₂ : HEq a b) → (h₃ : f = g) → HEq (f α a) (g β b) := by
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grind
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set_option trace.grind.debug.proof true in
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example (f : {α : Type} → α → Nat → Bool → Nat) (a b : Nat) : f a 0 true = v₁ → f b 0 true = v₂ → a = b → v₁ = v₂ := by
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grind
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set_option trace.grind.debug.proof true in
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example (f : {α : Type} → α → Nat → Bool → Nat) (a b : Nat) : f a b x = v₁ → f a b y = v₂ → x = y → v₁ = v₂ := by
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grind
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set_option trace.grind.debug.proof true in
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theorem ex1 (f : {α : Type} → α → Nat → Bool → Nat) (a b c : Nat) : f a b x = v₁ → f c b y = v₂ → a = c → x = y → v₁ = v₂ := by
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grind
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#print ex1
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example (n1 n2 n3 : Nat) (v1 w1 : Vector Nat n1) (w1' : Vector Nat n3) (v2 w2 : Vector Nat n2) :
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n1 = n3 → v1 = w1 → HEq w1 w1' → v2 = w2 → HEq (v1 ++ v2) (w1' ++ w2) := by
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grind
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example (n1 n2 n3 : Nat) (v1 w1 : Vector Nat n1) (w1' : Vector Nat n3) (v2 w2 : Vector Nat n2) :
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HEq n1 n3 → v1 = w1 → HEq w1 w1' → HEq v2 w2 → HEq (v1 ++ v2) (w1' ++ w2) := by
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grind
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theorem ex2 (n1 n2 n3 : Nat) (v1 w1 v : Vector Nat n1) (w1' : Vector Nat n3) (v2 w2 w : Vector Nat n2) :
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HEq n1 n3 → v1 = w1 → HEq w1 w1' → HEq v2 w2 → HEq (w1' ++ w2) (v ++ w) → HEq (v1 ++ v2) (v ++ w) := by
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grind
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#print ex2
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