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refactor(library/init/sigma): simplify lex.accessible proof using 'cases' tactic

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Leonardo de Moura 2014-12-12 12:36:51 -08:00
parent 29aaa21f2a
commit b900e9171d
1 changed files with 5 additions and 37 deletions
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42
library/init/sigma.lean
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@ -37,52 +37,20 @@ namespace sigma
parameters {Ra : A → A → Prop} {Rb : Π a : A, B a → B a → Prop}
infix `≺`:50 := lex Ra Rb
set_option pp.beta true
variables {C : Πa, B a → Type} {D : Πa b, C a b → Type}
variables {a a' : A}
{b : B a} {b' : B a'}
{c : C a b} {c' : C a' b'}
{d : D a b c} {d' : D a' b' c'}
private theorem hcongr_arg2 (f : Πa b, C a b) (Ha : a = a') (Hb : b == b') : f a b == f a' b' :=
hcongr (hcongr_arg f Ha) (hcongr_arg C Ha) Hb
private theorem hcongr_arg3 (f : Πa b c, D a b c) (Ha : a = a') (Hb : b == b') (Hc : c == c')
: f a b c == f a' b' c' :=
hcongr (hcongr_arg2 f Ha Hb) (hcongr_arg2 D Ha Hb) Hc
definition lex.accessible {a} (aca : acc Ra a) (acb : ∀a, well_founded (Rb a)) : ∀ (b : B a), acc (lex Ra Rb) ⟨a, b⟩ :=
acc.rec_on aca
(λxa aca (iHa : ∀y, Ra y xa → ∀b : B y, acc (lex Ra Rb) ⟨y, b⟩),
λb : B xa, acc.rec_on (acb xa b)
(λxb acb
(iHb : ∀y, Rb xa y xb → acc (lex Ra Rb) ⟨xa, y⟩),
(iHb : ∀ (y : B xa), Rb xa y xb → acc (lex Ra Rb) ⟨xa, y⟩),
acc.intro ⟨xa, xb⟩ (λp (lt : p ≺ ⟨xa, xb⟩),
have aux : xa = xa → xb == xb → acc (lex Ra Rb) p, from
@lex.rec_on A B Ra Rb (λp₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex Ra Rb) p₁)
p ⟨xa, xb⟩ lt
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
show acc (lex Ra Rb) ⟨a₁, b₁⟩, from
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
iHa a₁ Ra₁ b₁)
(λa b₁ b₂ (H : Rb a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
-- TODO(Leo): cleanup this proof
show acc (lex Ra Rb) ⟨a, b₁⟩, from
let b₁' : B xa := eq.rec_on eq₂ b₁ in
have aux₁ : b₁ == b₁', from
heq.symm (eq_rec_heq eq₂ b₁),
have aux₂ : Rb a b₁ b₂ = Rb xa b₁' xb, from
heq.to_eq (hcongr_arg3 Rb eq₂ aux₁ eq₃),
have aux₃ : Rb xa b₁' xb, from
eq.rec_on aux₂ H,
have aux₄ : acc (lex Ra Rb) ⟨xa, b₁'⟩, from
iHb b₁' aux₃,
have aux₅ : ∀ (b₁ b₂ : B a) (H₁ : a = a) (H₂ : b₁ == b₂), acc (lex Ra Rb) ⟨a, b₁⟩ → acc (lex Ra Rb) ⟨a, b₂⟩, from
λ b₁ b₂ H₁ H₂ Ha, eq.rec_on (heq.to_eq H₂) Ha,
have aux₆ : ∀ (b₁ : B xa) (b₂ : B a) (H₁ : a = xa) (H₂ : b₁ == b₂), acc (lex Ra Rb) ⟨xa, b₁⟩ → acc (lex Ra Rb) ⟨a, b₂⟩, from
eq.rec_on eq₂ aux₅,
aux₆ b₁' b₁ eq₂ (heq.symm aux₁) aux₄),
(λ (a₁ : A) (b₁ : B a₁) (a₂ : A) (b₂ : B a₂) (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
begin cases eq₂, exact (iHa a₁ H b₁) end)
(λ (a : A) (b₁ b₂ : B a) (H : Rb a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
begin cases eq₂, cases eq₃, exact (iHb b₁ H) end),
aux rfl !heq.refl)))
-- The lexicographical order of well founded relations is well-founded