66 lines
2.7 KiB
Text
66 lines
2.7 KiB
Text
inductive Term where
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| const : String → Term
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| app : String → List Term → Term
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namespace Term
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mutual
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def numConsts : Term → Nat
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| const _ => 1
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| app _ cs => numConstsLst cs
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def numConstsLst : List Term → Nat
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| [] => 0
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| c :: cs => numConsts c + numConstsLst cs
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end
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mutual
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def replaceConst (a b : String) : Term → Term
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| const c => if a == c then const b else const c
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| app f cs => app f (replaceConstLst a b cs)
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def replaceConstLst (a b : String) : List Term → List Term
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| [] => []
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| c :: cs => replaceConst a b c :: replaceConstLst a b cs
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end
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/--
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info: Term.replaceConst.induct (a b : String) (motive1 : Term → Prop) (motive2 : List Term → Prop)
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(case1 : ∀ (a_1 : String), (a == a_1) = true → motive1 (const a_1))
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(case2 : ∀ (a_1 : String), ¬(a == a_1) = true → motive1 (const a_1))
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(case3 : ∀ (a : String) (cs : List Term), motive2 cs → motive1 (app a cs)) (case4 : motive2 [])
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(case5 : ∀ (c : Term) (cs : List Term), motive1 c → motive2 cs → motive2 (c :: cs)) : ∀ (a : Term), motive1 a
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-/
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#guard_msgs in
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#check replaceConst.induct
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theorem numConsts_replaceConst (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
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apply replaceConst.induct
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(motive1 := fun e => numConsts (replaceConst a b e) = numConsts e)
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(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es)
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case case1 => intro c h; guard_hyp h :ₛ (a == c) = true; simp [replaceConst, numConsts, *]
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case case2 => intro c h; guard_hyp h :ₛ ¬(a == c) = true; simp [replaceConst, numConsts, *]
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case case3 =>
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intros f cs ih
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guard_hyp ih :ₛnumConstsLst (replaceConstLst a b cs) = numConstsLst cs
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simp [replaceConst, numConsts, *]
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case case4 => simp [replaceConstLst, numConstsLst, *]
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case case5 =>
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intro c cs ih₁ ih₂
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guard_hyp ih₁ :ₛ numConsts (replaceConst a b c) = numConsts c
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guard_hyp ih₂ :ₛ numConstsLst (replaceConstLst a b cs) = numConstsLst cs
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simp [replaceConstLst, numConstsLst, *]
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theorem numConsts_replaceConst' (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
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apply replaceConst.induct
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(motive1 := fun e => numConsts (replaceConst a b e) = numConsts e)
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(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es)
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<;> intros <;> simp [replaceConst, numConsts, replaceConstLst, numConstsLst, *]
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theorem numConsts_replaceConst'' (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
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induction e using replaceConst.induct (a := a) (b := b)
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(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es) <;>
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simp [replaceConst, numConsts, replaceConstLst, numConstsLst, *]
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end Term
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