70 lines
1.7 KiB
Text
70 lines
1.7 KiB
Text
namespace imp
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open tactic
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@[reducible]
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def uname := string
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inductive aexp
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| val : nat → aexp
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| var : uname → aexp
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| plus : aexp → aexp → aexp
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| times : aexp → aexp → aexp
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instance : decidable_eq aexp :=
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by mk_dec_eq_instance
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@[reducible]
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def value := nat
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def state := uname → value
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open aexp
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def aval : aexp → state → value
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| (val n) s := n
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| (var x) s := s x
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| (plus a₁ a₂) s := aval a₁ s + aval a₂ s
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| (times a₁ a₂) s := aval a₁ s * aval a₂ s
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example : aval (plus (val 3) (var "x")) (λ x, 0) = 3 :=
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rfl
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def updt (s : state) (x : uname) (v : value) : state :=
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λ y, if x = y then v else s y
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def asimp_const : aexp → aexp
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| (val n) := val n
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| (var x) := var x
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| (plus a₁ a₂) :=
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match asimp_const a₁, asimp_const a₂ with
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| val n₁, val n₂ := val (n₁ + n₂)
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| b₁, b₂ := plus b₁ b₂
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end
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| (times a₁ a₂) :=
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match asimp_const a₁, asimp_const a₂ with
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| val n₁, val n₂ := val (n₁ * n₂)
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| b₁, b₂ := times b₁ b₂
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end
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example : asimp_const (plus (plus (val 2) (val 3)) (var "x")) = plus (val 5) (var "x") :=
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rfl
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attribute [ematch] asimp_const aval
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-- set_option trace.smt.ematch true
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meta def not_done : tactic unit := fail_if_success now
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lemma aval_asimp_const (a : aexp) (s : state) : aval (asimp_const a) s = aval a s :=
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begin [smt]
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induction a,
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all_goals {destruct (asimp_const a_1), all_goals {destruct (asimp_const a_2), eblast}}
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end
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lemma ex2 (a : aexp) (s : state) : aval (asimp_const a) s = aval a s :=
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begin [smt]
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induction a,
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all_goals {destruct (asimp_const a_1), all_goals {destruct (asimp_const a_2), eblast_using [asimp_const, aval]}}
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end
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end imp
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