106 lines
3 KiB
Text
106 lines
3 KiB
Text
opaque g : Nat → Nat
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set_option trace.Meta.debug true
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@[grind] def f (a : Nat) :=
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match a with
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| 0 => 10
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| x+1 => g (f x)
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set_option grind.debug true
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set_option grind.debug.proofs true
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set_option trace.grind.ematch.instance true
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set_option trace.grind.assert true
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/--
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trace: [grind.assert] f (y + 1) = a
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[grind.assert] ¬a = g (f y)
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[grind.ematch.instance] f.eq_2: f y.succ = g (f y)
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[grind.assert] f (y + 1) = g (f y)
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-/
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#guard_msgs (trace) in
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example : f (y + 1) = a → a = g (f y):= by
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grind
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@[grind] def app (xs ys : List α) :=
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match xs with
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| [] => ys
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| x::xs => x :: app xs ys
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/--
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trace: [grind.assert] app [1, 2] ys = xs
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[grind.assert] ¬xs = 1 :: 2 :: ys
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[grind.ematch.instance] app.eq_2: app [1, 2] ys = 1 :: app [2] ys
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[grind.assert] app [1, 2] ys = 1 :: app [2] ys
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[grind.ematch.instance] app.eq_2: app [2] ys = 2 :: app [] ys
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[grind.assert] app [2] ys = 2 :: app [] ys
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[grind.ematch.instance] app.eq_1: app [] ys = ys
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[grind.assert] app [] ys = ys
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-/
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#guard_msgs (trace) in
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example : app [1, 2] ys = xs → xs = 1::2::ys := by
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grind
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opaque p : Nat → Nat → Prop
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opaque q : Nat → Prop
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@[grind =] theorem pq : p x x ↔ q x := by sorry
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/--
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trace: [grind.assert] p a a
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[grind.assert] ¬q a
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[grind.ematch.instance] pq: p a a ↔ q a
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[grind.assert] p a a = q a
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-/
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#guard_msgs (trace) in
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example : p a a → q a := by
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grind
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opaque appV (xs : Vector α n) (ys : Vector α m) : Vector α (n + m) :=
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Vector.append xs ys
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@[grind =]
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theorem appV_assoc (a : Vector α n) (b : Vector α m) (c : Vector α n') :
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appV a (appV b c) ≍ appV (appV a b) c := sorry
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/--
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trace: [grind.assert] x1 = appV a_2 b
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[grind.assert] x2 = appV x1 c
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[grind.assert] x3 = appV b c
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[grind.assert] x4 = appV a_2 x3
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[grind.assert] ¬x2 ≍ x4
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[grind.ematch.instance] appV_assoc: appV a_2 (appV b c) ≍ appV (appV a_2 b) c
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[grind.assert] appV a_2 (appV b c) ≍ appV (appV a_2 b) c
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-/
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#guard_msgs (trace) in
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example : x1 = appV a b → x2 = appV x1 c → x3 = appV b c → x4 = appV a x3 → x2 ≍ x4 := by
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grind
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/--
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info: appV_assoc': [@appV #6 #5 (@HAdd.hAdd `[Nat] `[Nat] `[Nat] `[instHAdd] #4 #3) #2 (@appV _ #4 #3 #1 #0)]
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-/
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#guard_msgs (info) in
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@[grind? =]
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theorem appV_assoc' (a : Vector α n) (b : Vector α m) (c : Vector α n') :
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appV a (appV b c) ≍ appV (appV a b) c := sorry
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example (p : Prop) (h₁ h₂ : Decidable p) : h₁ = h₂ := by
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grind
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example (p q : Prop) (h₁ : Decidable p) (h₂ : Decidable (p ∧ q)) : (p ↔ q) → h₁ ≍ h₂ := by
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grind
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example (a₁ a₂ : α) (b₁ b₂ : β) : a₁ ≍ b₁ → a₂ ≍ b₂ → (a₁ = a₂) = (b₁ = b₂) := by
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grind
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example (a₁ a₂ : α) (b₁ b₂ : β) : a₁ ≍ b₁ → a₂ ≍ b₂ → (a₁ = a₂) = (b₂ = b₁) := by
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grind
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example (a₁ a₂ : α) (b₁ b₂ : β) : a₁ ≍ b₁ → a₂ ≍ b₂ → (a₁ = a₂) ≍ (b₁ = b₂) := by
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grind
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example (a₁ a₂ : α) (b₁ b₂ : β) : a₁ ≍ b₁ → a₂ ≍ b₂ → (a₁ = a₂) ≍ (b₂ = b₁) := by
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grind
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