38 lines
1.1 KiB
Text
38 lines
1.1 KiB
Text
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import algebra.e_closure
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open eq
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namespace relation
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section
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parameters {A : Type}
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(R : A → A → Type)
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local abbreviation T := e_closure R
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variables ⦃a a' : A⦄ {s : R a a'} {r : T a a}
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parameter {R}
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theorem ap_ap_e_closure_elim_h₁ {B C D : Type} {f : A → B}
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{g : B → C} (h : C → D)
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(e : Π⦃a a' : A⦄, R a a' → f a = f a')
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{e' : Π⦃a a' : A⦄, R a a' → g (f a) = g (f a')}
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(p : Π⦃a a' : A⦄ (s : R a a'), ap g (e s) = e' s) (t : T a a')
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: square (ap (ap h) (ap_e_closure_elim_h e p t))
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(ap_e_closure_elim_h e (λa a' s, ap_compose h g (e s)) t)
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(ap_compose h g (e_closure.elim e t))⁻¹
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(ap_e_closure_elim_h e' (λa a' s, (ap (ap h) (p s))⁻¹) t) :=
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begin
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induction t,
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apply sorry,
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apply sorry,
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{
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rewrite [↑e_closure.elim, ↑ap_e_closure_elim_h, ap_con (ap h)],
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refine (transpose !ap_compose_inv)⁻¹ᵛ ⬝h _,
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rewrite [con_inv,inv_inv,-inv2_inv],
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exact !ap_inv2 ⬝v square_inv2 v_0
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},
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apply sorry
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end
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end
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end relation
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