4c9601e60f
This PR completes support for injective functions in grind. See
examples:
```lean
/-! Add some injectivity theorems. -/
def double (x : Nat) := 2*x
@[grind inj] theorem double_inj : Function.Injective double := by
grind [Function.Injective, double]
structure InjFn (α : Type) (β : Type) where
f : α → β
h : Function.Injective f
instance : CoeFun (InjFn α β) (fun _ => α → β) where
coe s := s.f
@[grind inj] theorem fn_inj (F : InjFn α β) : Function.Injective (F : α → β) := by
grind [Function.Injective, cases InjFn]
def toList (a : α) : List α := [a]
@[grind inj] theorem toList_inj : Function.Injective (toList : α → List α) := by
grind [Function.Injective, toList]
/-! Examples -/
example (x y : Nat) : toList (double x) = toList (double y) → x = y := by
grind
example (f : InjFn (List Nat) α) (x y z : Nat)
: f (toList (double x)) = f (toList y) →
y = double z →
x = z := by
grind
```
64 lines
1.9 KiB
Text
64 lines
1.9 KiB
Text
reset_grind_attrs%
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set_option warn.sorry false
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set_option trace.grind.inj true
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def succ (x : Nat) := x+1
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@[grind inj] theorem succ_inj : Function.Injective succ := by
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grind [Function.Injective, succ]
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def double (x : Nat) := 2*x
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@[grind inj] theorem double_inj : Function.Injective double := by
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grind [Function.Injective, double]
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@[grind inj] theorem mul_2_inj : Function.Injective (2 * ·) := by
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grind [Function.Injective]
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def Array.IsId (as : Array Nat) : Prop :=
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∀ i : Fin as.size, as[i] = i
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@[grind inj] theorem array_inj {as : Array Nat} (h : as.IsId) : Function.Injective (as[·]? : Fin as.size → Option Nat) := by
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intro a b; simp; have := h a; have := h b; simp_all; grind
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structure InjFn (α : Type) (β : Type) where
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f : α → β
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h : Function.Injective f
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instance : CoeFun (InjFn α β) (fun _ => α → β) where
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coe s := s.f
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@[grind inj] theorem fn_inj (F : InjFn α β) : Function.Injective (F : α → β) := by
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grind [Function.Injective, cases InjFn]
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def toList (a : α) : List α := [a]
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@[grind inj] theorem toList_inj : Function.Injective (toList : α → List α) := by
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grind [Function.Injective, toList]
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example (x y : Nat) : succ (double x) = succ (double y) → x = y := by
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grind
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example (x y : Nat) : toList x = toList y → x = y := by
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grind
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example (x y : Nat) : toList (succ (double x)) = toList (succ (double y)) → x = y := by
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grind
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example (as : Array Nat) (h : as.IsId) (i j : Fin as.size) : as[i]? = as[j]? → i = j := by
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grind
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example (F : InjFn α β) : F x = F y → x = y := by
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grind
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example (F : InjFn α Nat) : toList (succ (F x)) = toList (succ (F y)) → x = y := by
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grind
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example (F : InjFn α Nat) : toList (succ (F x)) = a → a = toList (succ (F y)) → x = y := by
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grind
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opaque p : Nat → Nat → Prop
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@[grind =] theorem peq : p x y = (x = double (succ y)) := sorry
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example (x y : Nat) : (double (succ x)) = a → p a y → x = y := by
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grind
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