93 lines
3.3 KiB
Text
93 lines
3.3 KiB
Text
theorem tst1 {α : Type} {p : Prop} (xs : List α) (h₁ : (a : α) → (as : List α) → xs = a :: as → p) (h₂ : xs = [] → p) : p :=
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by match (generalizing := false) h : xs with
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| [] => exact h₂ h
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| z::zs => apply h₁ z zs; assumption
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theorem tst1' {α : Type} {p : Prop} (xs : List α) (h₁ : (a : α) → (as : List α) → xs = a :: as → p) (h₂ : xs = [] → p) : p :=
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by match xs with
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| [] => exact h₂ rfl
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| z::zs => exact h₁ z zs rfl
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theorem tst2 {α : Type} {p : Prop} (xs : List α) (h₁ : (a : α) → (as : List α) → xs = a :: as → p) (h₂ : xs = [] → p) : p :=
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by match (generalizing := false) h:xs with
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| [] => ?nilCase
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| z::zs => ?consCase;
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case consCase => exact h₁ z zs h;
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case nilCase => exact h₂ h
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def tst3 {α β γ : Type} (h : α × β × γ) : β × α × γ :=
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by {
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match h with
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| (a, b, c) => exact (b, a, c)
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}
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theorem tst4 {α : Type} {p : Prop} (xs : List α) (h₁ : (a : α) → (as : List α) → xs = a :: as → p) (h₂ : xs = [] → p) : p := by
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match (generalizing := false) h : xs with
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| [] => _
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| z::zs => _
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case match_2 => exact h₁ z zs h
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exact h₂ h
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theorem tst5 {p q r} (h : p ∨ q ∨ r) : r ∨ q ∨ p:= by
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match h with
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| Or.inl h => exact Or.inr (Or.inr h)
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| Or.inr (Or.inl h) => ?c1
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| Or.inr (Or.inr h) => ?c2
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case c2 =>
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apply Or.inl
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assumption
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case c1 =>
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apply Or.inr
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apply Or.inl
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assumption
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theorem tst6 {p q r} (h : p ∨ q ∨ r) : r ∨ q ∨ p:= by
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match h with
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| Or.inl h => exact Or.inr (Or.inr h)
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| Or.inr (Or.inl h) => ?c1
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| Or.inr (Or.inr h) =>
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apply Or.inl
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assumption
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case c1 => apply Or.inr; apply Or.inl; assumption
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theorem tst7 {p q r} (h : p ∨ q ∨ r) : r ∨ q ∨ p:=
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by match h with
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| Or.inl h =>
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exact Or.inr (Or.inr h)
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| Or.inr (Or.inl h) =>
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apply Or.inr;
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apply Or.inl;
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assumption
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| Or.inr (Or.inr h) =>
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apply Or.inl;
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assumption
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inductive ListLast.{u} {α : Type u} : List α → Type u
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| empty : ListLast []
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| nonEmpty : (as : List α) → (a : α) → ListLast (as ++ [a])
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axiom last {α} (xs : List α) : ListLast xs
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axiom back {α} [Inhabited α] (xs : List α) : α
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axiom popBack {α} : List α → List α
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axiom backEq {α} [Inhabited α] : (xs : List α) → (x : α) → back (xs ++ [x]) = x
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axiom popBackEq {α} : (xs : List α) → (x : α) → popBack (xs ++ [x]) = xs
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theorem tst8 {α} [Inhabited α] (xs : List α) : xs ≠ [] → xs = popBack xs ++ [back xs] :=
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match (generalizing := false) xs, h:last xs with
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| _, ListLast.empty => fun h => absurd rfl h
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| _, ListLast.nonEmpty ys y => fun _ => sorry
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theorem tst9 {α} [Inhabited α] (xs : List α) : xs ≠ [] → xs = popBack xs ++ [back xs] := by
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match (generalizing := false) xs, h:last xs with
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| _, ListLast.empty => intro h; exact absurd rfl h
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| _, ListLast.nonEmpty ys y => intro; rw [popBackEq, backEq]
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theorem tst8' {α} [Inhabited α] (xs : List α) : xs ≠ [] → xs = popBack xs ++ [back xs] :=
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match xs, last xs with
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| _, ListLast.empty => fun h => absurd rfl h
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| _, ListLast.nonEmpty ys y => fun _ => sorry
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theorem tst8'' {α} [Inhabited α] (xs : List α) (h : xs ≠ []) : xs = popBack xs ++ [back xs] :=
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match xs, last xs with
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| _, ListLast.empty => absurd rfl h
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| _, ListLast.nonEmpty ys y => sorry
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