1b412b6cc0
The previous `cases` tactic would only use the revert/intro idiom for `cases h` when `h` is a hypothesis
50 lines
1.3 KiB
Text
50 lines
1.3 KiB
Text
/- "Proving in the Large" chapter of CPDT -/
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inductive exp : Type
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| Const (n : nat) : exp
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| Plus (e1 e2 : exp) : exp
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| Mult (e1 e2 : exp) : exp
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open exp
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def exp_eval : exp → nat
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| (Const n) := n
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| (Plus e1 e2) := exp_eval e1 + exp_eval e2
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| (Mult e1 e2) := exp_eval e1 * exp_eval e2
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def times (k : nat) : exp → exp
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| (Const n) := Const (k * n)
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| (Plus e1 e2) := Plus (times e1) (times e2)
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| (Mult e1 e2) := Mult (times e1) e2
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attribute [simp] exp_eval times mul_add
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theorem eval_times (k e) : exp_eval (times k e) = k * exp_eval e :=
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by induction e; simp_using_hs
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/- CPDT: induction e; crush. -/
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def reassoc : exp → exp
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| (Const n) := (Const n)
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| (Plus e1 e2) :=
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let e1' := reassoc e1 in
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let e2' := reassoc e2 in
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match e2' with
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| (Plus e21 e22) := Plus (Plus e1' e21) e22
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| _ := Plus e1' e2'
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end
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| (Mult e1 e2) :=
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let e1' := reassoc e1 in
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let e2' := reassoc e2 in
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match e2' with
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| (Mult e21 e22) := Mult (Mult e1' e21) e22
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| _ := Mult e1' e2'
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end
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attribute [simp] reassoc
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lemma rewr : ∀ a b c d : nat, b * c = d → a * b * c = a * d :=
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by intros; simp_using_hs
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theorem reassoc_correct (e) : exp_eval (reassoc e) = exp_eval e :=
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by induction e; simp; try {cases (reassoc e2); dsimp at ih_2; dsimp; simp_using_hs}
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