319 lines
14 KiB
Text
319 lines
14 KiB
Text
/-
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Copyright (c) 2014 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Floris van Doorn
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An explicit treatment of quotients, without using Lean's built-in quotient types.
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-/
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import logic data.subtype logic.cast algebra.relation data.prod
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import logic.instances
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import .util
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open relation prod inhabited nonempty tactic eq.ops
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open subtype relation.iff_ops
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namespace quotient
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/- definition and basics -/
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-- TODO: make this a structure
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definition is_quotient {A B : Type} (R : A → A → Prop) (abs : A → B) (rep : B → A) : Prop :=
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(∀b, abs (rep b) = b) ∧
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(∀b, R (rep b) (rep b)) ∧
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(∀r s, R r s ↔ (R r r ∧ R s s ∧ abs r = abs s))
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theorem intro {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(H1 : ∀b, abs (rep b) = b) (H2 : ∀b, R (rep b) (rep b))
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(H3 : ∀r s, R r s ↔ (R r r ∧ R s s ∧ abs r = abs s)) : is_quotient R abs rep :=
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and.intro H1 (and.intro H2 H3)
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theorem and_absorb_left {a : Prop} (b : Prop) (Ha : a) : a ∧ b ↔ b :=
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iff.intro (assume Hab, and.elim_right Hab) (assume Hb, and.intro Ha Hb)
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theorem intro_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(H1 : reflexive R) (H2 : ∀b, abs (rep b) = b)
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(H3 : ∀r s, R r s ↔ abs r = abs s) : is_quotient R abs rep :=
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intro
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H2
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(take b, H1 (rep b))
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(take r s,
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have H4 : R r s ↔ R s s ∧ abs r = abs s,
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from
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subst (symm (and_absorb_left _ (H1 s))) (H3 r s),
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subst (symm (and_absorb_left _ (H1 r))) H4)
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theorem abs_rep {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (b : B) : abs (rep b) = b :=
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and.elim_left Q b
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theorem refl_rep {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (b : B) : R (rep b) (rep b) :=
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and.elim_left (and.elim_right Q) b
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theorem R_iff {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (r s : A) : R r s ↔ (R r r ∧ R s s ∧ abs r = abs s) :=
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and.elim_right (and.elim_right Q) r s
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theorem refl_left {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {r s : A} (H : R r s) : R r r :=
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and.elim_left (iff.elim_left (R_iff Q r s) H)
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theorem refl_right {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {r s : A} (H : R r s) : R s s :=
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and.elim_left (and.elim_right (iff.elim_left (R_iff Q r s) H))
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theorem eq_abs {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {r s : A} (H : R r s) : abs r = abs s :=
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and.elim_right (and.elim_right (iff.elim_left (R_iff Q r s) H))
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theorem R_intro {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {r s : A} (H1 : R r r) (H2 : R s s) (H3 : abs r = abs s) : R r s :=
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iff.elim_right (R_iff Q r s) (and.intro H1 (and.intro H2 H3))
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theorem R_intro_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (H1 : reflexive R) {r s : A} (H2 : abs r = abs s) : R r s :=
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iff.elim_right (R_iff Q r s) (and.intro (H1 r) (and.intro (H1 s) H2))
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theorem rep_eq {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {a b : B} (H : R (rep a) (rep b)) : a = b :=
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calc
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a = abs (rep a) : eq.symm (abs_rep Q a)
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... = abs (rep b) : {eq_abs Q H}
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... = b : abs_rep Q b
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theorem R_rep_abs {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {a : A} (H : R a a) : R a (rep (abs a)) :=
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have H3 : abs a = abs (rep (abs a)), from eq.symm (abs_rep Q (abs a)),
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R_intro Q H (refl_rep Q (abs a)) H3
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theorem quotient_imp_symm {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) : symmetric R :=
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take a b : A,
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assume H : R a b,
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have Ha : R a a, from refl_left Q H,
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have Hb : R b b, from refl_right Q H,
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have Hab : abs b = abs a, from eq.symm (eq_abs Q H),
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R_intro Q Hb Ha Hab
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theorem quotient_imp_trans {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) : transitive R :=
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take a b c : A,
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assume Hab : R a b,
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assume Hbc : R b c,
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have Ha : R a a, from refl_left Q Hab,
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have Hc : R c c, from refl_right Q Hbc,
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have Hac : abs a = abs c, from eq.trans (eq_abs Q Hab) (eq_abs Q Hbc),
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R_intro Q Ha Hc Hac
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/- recursion -/
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-- (maybe some are superfluous)
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definition rec {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : B → Type} (f : forall (a : A), C (abs a)) (b : B) : C b :=
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eq.rec_on (abs_rep Q b) (f (rep b))
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theorem comp {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : B → Type} {f : forall (a : A), C (abs a)}
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(H : forall (r s : A) (H' : R r s), eq.rec_on (eq_abs Q H') (f r) = f s)
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{a : A} (Ha : R a a) : rec Q f (abs a) = f a :=
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assert H2 : R a (rep (abs a)), from
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R_rep_abs Q Ha,
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assert Heq : abs (rep (abs a)) = abs a, from
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abs_rep Q (abs a),
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calc
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rec Q f (abs a) = eq.rec_on Heq (f (rep (abs a))) : rfl
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... = eq.rec_on Heq (eq.rec_on (Heq⁻¹) (f a)) : {(H _ _ H2)⁻¹}
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... = f a : eq.rec_on_compose (eq_abs Q H2) _ _
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definition rec_constant {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : Type} (f : A → C) (b : B) : C :=
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f (rep b)
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theorem comp_constant {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : Type} {f : A → C}
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(H : forall (r s : A) (H' : R r s), f r = f s)
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{a : A} (Ha : R a a) : rec_constant Q f (abs a) = f a :=
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have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
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calc
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rec_constant Q f (abs a) = f (rep (abs a)) : rfl
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... = f a : {(H _ _ H2)⁻¹}
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definition rec_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : Type} (f : A → A → C) (b c : B) : C :=
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f (rep b) (rep c)
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theorem comp_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {C : Type} {f : A → A → C}
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(H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), f a b = f a' b')
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{a b : A} (Ha : R a a) (Hb : R b b) : rec_binary Q f (abs a) (abs b) = f a b :=
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have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
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have H3 : R b (rep (abs b)), from R_rep_abs Q Hb,
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calc
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rec_binary Q f (abs a) (abs b) = f (rep (abs a)) (rep (abs b)) : rfl
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... = f a b : {(H _ _ _ _ H2 H3)⁻¹}
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theorem comp_binary_refl {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (Hrefl : reflexive R) {C : Type} {f : A → A → C}
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(H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), f a b = f a' b')
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(a b : A) : rec_binary Q f (abs a) (abs b) = f a b :=
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comp_binary Q H (Hrefl a) (Hrefl b)
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definition quotient_map {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (f : A → A) (b : B) : B :=
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abs (f (rep b))
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theorem comp_quotient_map {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {f : A → A}
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(H : forall (a a' : A) (Ha : R a a'), R (f a) (f a'))
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{a : A} (Ha : R a a) : quotient_map Q f (abs a) = abs (f a) :=
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have H2 : R a (rep (abs a)), from R_rep_abs Q Ha,
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have H3 : R (f a) (f (rep (abs a))), from H _ _ H2,
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have H4 : abs (f a) = abs (f (rep (abs a))), from eq_abs Q H3,
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H4⁻¹
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definition quotient_map_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) (f : A → A → A) (b c : B) : B :=
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abs (f (rep b) (rep c))
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theorem comp_quotient_map_binary {A B : Type} {R : A → A → Prop} {abs : A → B} {rep : B → A}
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(Q : is_quotient R abs rep) {f : A → A → A}
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(H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), R (f a b) (f a' b'))
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{a b : A} (Ha : R a a) (Hb : R b b) : quotient_map_binary Q f (abs a) (abs b) = abs (f a b) :=
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have Ha2 : R a (rep (abs a)), from R_rep_abs Q Ha,
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have Hb2 : R b (rep (abs b)), from R_rep_abs Q Hb,
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have H2 : R (f a b) (f (rep (abs a)) (rep (abs b))), from H _ _ _ _ Ha2 Hb2,
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(eq_abs Q H2)⁻¹
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theorem comp_quotient_map_binary_refl {A B : Type} {R : A → A → Prop} (Hrefl : reflexive R)
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{abs : A → B} {rep : B → A} (Q : is_quotient R abs rep) {f : A → A → A}
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(H : forall (a a' b b' : A) (Ha : R a a') (Hb : R b b'), R (f a b) (f a' b'))
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(a b : A) : quotient_map_binary Q f (abs a) (abs b) = abs (f a b) :=
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comp_quotient_map_binary Q H (Hrefl a) (Hrefl b)
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/- image -/
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definition image {A B : Type} (f : A → B) := subtype (fun b, ∃a, f a = b)
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theorem image_inhabited {A B : Type} (f : A → B) (H : inhabited A) : inhabited (image f) :=
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inhabited.mk (tag (f (default A)) (exists.intro (default A) rfl))
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theorem image_inhabited2 {A B : Type} (f : A → B) (a : A) : inhabited (image f) :=
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image_inhabited f (inhabited.mk a)
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definition fun_image {A B : Type} (f : A → B) (a : A) : image f :=
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tag (f a) (exists.intro a rfl)
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theorem fun_image_def {A B : Type} (f : A → B) (a : A) :
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fun_image f a = tag (f a) (exists.intro a rfl) := rfl
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theorem elt_of_fun_image {A B : Type} (f : A → B) (a : A) : elt_of (fun_image f a) = f a :=
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elt_of.tag _ _
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theorem image_elt_of {A B : Type} {f : A → B} (u : image f) : ∃a, f a = elt_of u :=
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has_property u
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theorem fun_image_surj {A B : Type} {f : A → B} (u : image f) : ∃a, fun_image f a = u :=
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subtype.destruct u
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(take (b : B) (H : ∃a, f a = b),
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obtain a (H': f a = b), from H,
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(exists.intro a (tag_eq H')))
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theorem image_tag {A B : Type} {f : A → B} (u : image f) : ∃a H, tag (f a) H = u :=
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obtain a (H : fun_image f a = u), from fun_image_surj u,
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exists.intro a (exists.intro (exists.intro a rfl) H)
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open eq.ops
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theorem fun_image_eq {A B : Type} (f : A → B) (a a' : A)
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: (f a = f a') ↔ (fun_image f a = fun_image f a') :=
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iff.intro
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(assume H : f a = f a', tag_eq H)
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(assume H : fun_image f a = fun_image f a',
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(congr_arg elt_of H ▸ elt_of_fun_image f a) ▸ elt_of_fun_image f a')
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theorem idempotent_image_elt_of {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f)
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: fun_image f (elt_of u) = u :=
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obtain (a : A) (Ha : fun_image f a = u), from fun_image_surj u,
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calc
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fun_image f (elt_of u) = fun_image f (elt_of (fun_image f a)) : {Ha⁻¹}
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... = fun_image f (f a) : {elt_of_fun_image f a}
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... = fun_image f a : {iff.elim_left (fun_image_eq f (f a) a) (H a)}
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... = u : Ha
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theorem idempotent_image_fix {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f)
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: f (elt_of u) = elt_of u :=
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obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
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calc
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f (elt_of u) = f (f a) : {Ha⁻¹}
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... = f a : H a
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... = elt_of u : Ha
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/- construct quotient from representative map -/
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theorem representative_map_idempotent {A : Type} {R : A → A → Prop} {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b ↔ R a a ∧ R b b ∧ f a = f b) (a : A) :
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f (f a) = f a :=
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(and.elim_right (and.elim_right (iff.elim_left (H2 a (f a)) (H1 a))))⁻¹
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theorem representative_map_idempotent_equiv {A : Type} {R : A → A → Prop} {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b) (a : A) :
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f (f a) = f a :=
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(H2 a (f a) (H1 a))⁻¹
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theorem representative_map_refl_rep {A : Type} {R : A → A → Prop} {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b ↔ R a a ∧ R b b ∧ f a = f b) (a : A) :
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R (f a) (f a) :=
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representative_map_idempotent H1 H2 a ▸ H1 (f a)
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theorem representative_map_image_fix {A : Type} {R : A → A → Prop} {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a a', R a a' ↔ R a a ∧ R a' a' ∧ f a = f a') (b : image f) :
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f (elt_of b) = elt_of b :=
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idempotent_image_fix (representative_map_idempotent H1 H2) b
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theorem representative_map_to_quotient {A : Type} {R : A → A → Prop} {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a a', R a a' ↔ R a a ∧ R a' a' ∧ f a = f a') :
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is_quotient R (fun_image f) elt_of :=
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let abs := fun_image f in
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intro
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(take u : image f,
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obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
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have H : elt_of (abs (elt_of u)) = elt_of u, from
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calc
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elt_of (abs (elt_of u)) = f (elt_of u) : elt_of_fun_image _ _
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... = f (f a) : {Ha⁻¹}
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... = f a : representative_map_idempotent H1 H2 a
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... = elt_of u : Ha,
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show abs (elt_of u) = u, from subtype.equal H)
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(take u : image f,
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show R (elt_of u) (elt_of u), from
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obtain (a : A) (Ha : f a = elt_of u), from image_elt_of u,
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Ha ▸ (@representative_map_refl_rep A R f H1 H2 a))
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(take a a',
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subst (fun_image_eq f a a') (H2 a a'))
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theorem representative_map_equiv_inj {A : Type} {R : A → A → Prop}
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(equiv : is_equivalence R) {f : A → A}
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(H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b) {a b : A} (H3 : f a = f b) :
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R a b :=
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have symmR : relation.symmetric R, from rel_symm R,
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have transR : relation.transitive R, from rel_trans R,
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show R a b, from
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have H2 : R a (f b), from H3 ▸ (H1 a),
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have H3 : R (f b) b, from symmR (H1 b),
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transR H2 H3
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theorem representative_map_to_quotient_equiv {A : Type} {R : A → A → Prop}
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(equiv : is_equivalence R) {f : A → A} (H1 : ∀a, R a (f a)) (H2 : ∀a b, R a b → f a = f b) :
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@is_quotient A (image f) R (fun_image f) elt_of :=
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representative_map_to_quotient
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H1
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(take a b,
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have reflR : reflexive R, from rel_refl R,
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have H3 : f a = f b → R a b, from representative_map_equiv_inj equiv H1 H2,
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have H4 : R a b ↔ f a = f b, from iff.intro (H2 a b) H3,
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have H5 : R a b ↔ R b b ∧ f a = f b,
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from subst (symm (and_absorb_left _ (reflR b))) H4,
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subst (symm (and_absorb_left _ (reflR a))) H5)
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end quotient
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