7fa1a8b114
This PR eliminates uses of `intros x y z` (with arguments) and updates the `intros` docstring to suggest that `intro x y z` should be used instead. The `intros` tactic is historical, and can be traced all the way back to Lean 2, when `intro` could only introduce a single hypothesis. Since 2020, the `intro` tactic has superceded it. The `intros` tactic (without arguments) is currently still useful.
71 lines
3 KiB
Text
71 lines
3 KiB
Text
inductive Term where
|
|
| const : String → Term
|
|
| app : String → List Term → Term
|
|
|
|
namespace Term
|
|
|
|
mutual
|
|
def numConsts : Term → Nat
|
|
| const _ => 1
|
|
| app _ cs => numConstsLst cs
|
|
|
|
def numConstsLst : List Term → Nat
|
|
| [] => 0
|
|
| c :: cs => numConsts c + numConstsLst cs
|
|
end
|
|
|
|
mutual
|
|
-- Since #4733 this function can be compiled using structural recursion,
|
|
-- but then the construction of the functional induction principle falls over
|
|
-- TODO: Fix funind, and then omit the `termination_by` here (or test both variants)
|
|
def replaceConst (a b : String) : Term → Term
|
|
| const c => if a == c then const b else const c
|
|
| app f cs => app f (replaceConstLst a b cs)
|
|
termination_by t => sizeOf t
|
|
|
|
def replaceConstLst (a b : String) : List Term → List Term
|
|
| [] => []
|
|
| c :: cs => replaceConst a b c :: replaceConstLst a b cs
|
|
termination_by ts => sizeOf ts
|
|
end
|
|
|
|
|
|
/--
|
|
info: Term.replaceConst.induct (a : String) (motive1 : Term → Prop) (motive2 : List Term → Prop)
|
|
(case1 : ∀ (a_1 : String), (a == a_1) = true → motive1 (const a_1))
|
|
(case2 : ∀ (a_1 : String), ¬(a == a_1) = true → motive1 (const a_1))
|
|
(case3 : ∀ (a : String) (cs : List Term), motive2 cs → motive1 (app a cs)) (case4 : motive2 [])
|
|
(case5 : ∀ (c : Term) (cs : List Term), motive1 c → motive2 cs → motive2 (c :: cs)) (a✝ : Term) : motive1 a✝
|
|
-/
|
|
#guard_msgs in
|
|
#check replaceConst.induct
|
|
|
|
theorem numConsts_replaceConst (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
|
|
apply replaceConst.induct
|
|
(motive1 := fun e => numConsts (replaceConst a b e) = numConsts e)
|
|
(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es)
|
|
case case1 => intro c h; guard_hyp h :ₛ (a == c) = true; simp [replaceConst, numConsts, *]
|
|
case case2 => intro c h; guard_hyp h :ₛ ¬(a == c) = true; simp [replaceConst, numConsts, *]
|
|
case case3 =>
|
|
intro f cs ih
|
|
guard_hyp ih :ₛnumConstsLst (replaceConstLst a b cs) = numConstsLst cs
|
|
simp [replaceConst, numConsts, *]
|
|
case case4 => simp [replaceConstLst, numConstsLst, *]
|
|
case case5 =>
|
|
intro c cs ih₁ ih₂
|
|
guard_hyp ih₁ :ₛ numConsts (replaceConst a b c) = numConsts c
|
|
guard_hyp ih₂ :ₛ numConstsLst (replaceConstLst a b cs) = numConstsLst cs
|
|
simp [replaceConstLst, numConstsLst, *]
|
|
|
|
theorem numConsts_replaceConst' (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
|
|
apply replaceConst.induct
|
|
(motive1 := fun e => numConsts (replaceConst a b e) = numConsts e)
|
|
(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es)
|
|
<;> intros <;> simp [replaceConst, numConsts, replaceConstLst, numConstsLst, *]
|
|
|
|
theorem numConsts_replaceConst'' (a b : String) (e : Term) : numConsts (replaceConst a b e) = numConsts e := by
|
|
induction e using replaceConst.induct (a := a)
|
|
(motive2 := fun es => numConstsLst (replaceConstLst a b es) = numConstsLst es) <;>
|
|
simp [replaceConst, numConsts, replaceConstLst, numConstsLst, *]
|
|
|
|
end Term
|