cf36ac986d
This PR optimizes the construction on congruence proofs in `simp`. It uses some of the ideas used in `Sym.simp`.
77 lines
2 KiB
Text
77 lines
2 KiB
Text
x : Nat
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h : f (f x) = x
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⊢ (have y := x * x;
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if True then 1 else y + 1) =
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1
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theorem ex0 : ∀ (x : Nat),
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f (f x) = x →
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(have y := 0 + x * x;
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if f (f x) = x then 1 else y + 1) =
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1 :=
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fun x h =>
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Eq.mpr
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(id
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(congrFun'
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(congrArg Eq
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(id
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(id
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(have_congr' (Nat.zero_add (x * x)) fun y =>
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ite_congr (Eq.trans (congrFun' (congrArg Eq h) x) (eq_self x)) (fun a => Eq.refl 1) fun a =>
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Eq.refl (y + 1)))))
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1))
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(of_eq_true (Eq.trans (congrFun' (congrArg Eq (ite_cond_eq_true 1 (x * x + 1) (Eq.refl True))) 1) (eq_self 1)))
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x : Nat
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h : f (f x) = x
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⊢ (have y := x * x;
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if True then 1 else y + 1) =
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1
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theorem ex1 : ∀ (x : Nat),
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f (f x) = x →
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(have y := x * x;
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if f (f x) = x then 1 else y + 1) =
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1 :=
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fun x h =>
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Eq.mpr
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(id
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(congrFun'
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(congrArg Eq
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(id
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(id
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(have_body_congr' (x * x) fun y =>
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ite_congr (Eq.trans (congrFun' (congrArg Eq h) x) (eq_self x)) (fun a => Eq.refl 1) fun a =>
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Eq.refl (y + 1)))))
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1))
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(of_eq_true (Eq.trans (congrFun' (congrArg Eq (ite_cond_eq_true 1 (x * x + 1) (Eq.refl True))) 1) (eq_self 1)))
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x z : Nat
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h : f (f x) = x
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h' : z = x
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⊢ (have y := x;
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y) =
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z
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theorem ex2 : ∀ (x z : Nat),
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f (f x) = x →
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z = x →
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(have y := f (f x);
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y) =
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z :=
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fun x z h h' =>
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Eq.mpr (id (congrFun' (congrArg Eq (id (id (have_val_congr' h)))) z))
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(of_eq_true (Eq.trans (congrArg (Eq x) h') (eq_self x)))
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x z : Nat
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⊢ (let α := Nat;
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fun x => 0 + x) =
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id
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p : Prop
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h : p
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⊢ (have n := 10;
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fun x => True) =
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fun z => p
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theorem ex4 : ∀ (p : Prop),
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p →
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(have n := 10;
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fun x => x = x) =
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fun z => p :=
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fun p h =>
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Eq.mpr
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(id (congrFun' (congrArg Eq (id (id (have_body_congr_dep' 10 fun n => funext fun x => eq_self x)))) fun z => p))
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(of_eq_true (Eq.trans (congrArg (Eq fun x => True) (funext fun z => eq_true h)) (eq_self fun x => True)))
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