lean2/tests/lean/run/tree3.lean
Leonardo de Moura 6bc89f0916 feat(library/definitional): define ibelow and below
These are helper definitions for brec_on and binduction_on
2014-11-12 16:38:46 -08:00

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import logic data.prod
open eq.ops prod tactic
inductive tree (A : Type) :=
leaf : A → tree A,
node : tree A → tree A → tree A
inductive one.{l} : Type.{max 1 l} :=
star : one
set_option pp.universes true
namespace tree
namespace manual
section
universe variables l₁ l₂
variable {A : Type.{l₁}}
variable (C : tree A → Type.{l₂})
definition below (t : tree A) : Type :=
rec_on t (λ a, one.{l₂}) (λ t₁ t₂ r₁ r₂, C t₁ × C t₂ × r₁ × r₂)
end
section
universe variables l₁ l₂
variable {A : Type.{l₁}}
variable {C : tree A → Type.{l₂}}
definition below_rec_on (t : tree A) (H : Π (n : tree A), below C n → C n) : C t
:= have general : C t × below C t, from
rec_on t
(λa, (H (leaf a) one.star, one.star))
(λ (l r : tree A) (Hl : C l × below C l) (Hr : C r × below C r),
have b : below C (node l r), from
(pr₁ Hl, pr₁ Hr, pr₂ Hl, pr₂ Hr),
have c : C (node l r), from
H (node l r) b,
(c, b)),
pr₁ general
end
end manual
check no_confusion
theorem leaf_ne_tree {A : Type} (a : A) (l r : tree A) : leaf a ≠ node l r :=
assume h : leaf a = node l r,
no_confusion h
constant A : Type₁
constants l₁ l₂ r₁ r₂ : tree A
axiom node_eq : node l₁ r₁ = node l₂ r₂
check no_confusion node_eq
definition tst : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂ := no_confusion node_eq
check tst (λ e₁ e₂, e₁)
theorem node.inj1 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
assume h,
have trivial : (l₁ = l₂ → r₁ = r₂ → l₁ = l₂) → l₁ = l₂, from no_confusion h,
trivial (λ e₁ e₂, e₁)
theorem node.inj2 {A : Type} (l₁ l₂ r₁ r₂ : tree A) : node l₁ r₁ = node l₂ r₂ → l₁ = l₂ :=
begin
intro h,
apply (no_confusion h),
intros, assumption
end
theorem node.inj3 {A : Type} {l₁ l₂ r₁ r₂ : tree A} (h : node l₁ r₁ = node l₂ r₂) : l₁ = l₂ :=
begin
apply (no_confusion h),
assume (e₁ : l₁ = l₂) (e₂ : r₁ = r₂), e₁
end
theorem node.inj4 {A : Type} {l₁ l₂ r₁ r₂ : tree A} (h : node l₁ r₁ = node l₂ r₂) : l₁ = l₂ :=
begin
apply (no_confusion h),
assume e₁ e₂, e₁
end
end tree