ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_eq.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_eq.ma
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-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "ground_2/xoa/ex_3_2.ma".
-include "ground_2/notation/relations/ideq_2.ma".
-include "ground_2/relocation/rtmap.ma".
-
-(* RELOCATION MAP ***********************************************************)
-
-coinductive eq: relation rtmap ≝
-| eq_push: ∀f1,f2,g1,g2. eq f1 f2 → ⫯f1 = g1 → ⫯f2 = g2 → eq g1 g2
-| eq_next: ∀f1,f2,g1,g2. eq f1 f2 → ↑f1 = g1 → ↑f2 = g2 → eq g1 g2
-.
-
-interpretation "extensional equivalence (rtmap)"
-   'IdEq f1 f2 = (eq f1 f2).
-
-definition eq_repl (R:relation …) ≝
-                   ∀f1,f2. f1 ≡ f2 → R f1 f2.
-
-definition eq_repl_back (R:predicate …) ≝
-                        ∀f1. R f1 → ∀f2. f1 ≡ f2 → R f2.
-
-definition eq_repl_fwd (R:predicate …) ≝
-                       ∀f1. R f1 → ∀f2. f2 ≡ f1 → R f2.
-
-(* Basic properties *********************************************************)
-
-corec lemma eq_refl: reflexive … eq.
-#f cases (pn_split f) *
-#g #Hg [ @(eq_push … Hg Hg) | @(eq_next … Hg Hg) ] -Hg //
-qed.
-
-corec lemma eq_sym: symmetric … eq.
-#f1 #f2 * -f1 -f2
-#f1 #f2 #g1 #g2 #Hf #H1 #H2
-[ @(eq_push … H2 H1) | @(eq_next … H2 H1) ] -g2 -g1 /2 width=1 by/
-qed-.
-
-lemma eq_repl_sym: ∀R. eq_repl_back R → eq_repl_fwd R.
-/3 width=3 by eq_sym/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma eq_inv_px: ∀g1,g2. g1 ≡ g2 → ∀f1. ⫯f1 = g1 →
-                 ∃∃f2. f1 ≡ f2 & ⫯f2 = g2.
-#g1 #g2 * -g1 -g2
-#f1 #f2 #g1 #g2 #Hf * * -g1 -g2
-#x1 #H
-[ lapply (injective_push … H) -H /2 width=3 by ex2_intro/
-| elim (discr_push_next … H)
-]
-qed-.
-
-lemma eq_inv_nx: ∀g1,g2. g1 ≡ g2 → ∀f1. ↑f1 = g1 →
-                 ∃∃f2. f1 ≡ f2 & ↑f2 = g2.
-#g1 #g2 * -g1 -g2
-#f1 #f2 #g1 #g2 #Hf * * -g1 -g2
-#x1 #H
-[ elim (discr_next_push … H)
-| lapply (injective_next … H) -H /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma eq_inv_xp: ∀g1,g2. g1 ≡ g2 → ∀f2. ⫯f2 = g2 →
-                 ∃∃f1. f1 ≡ f2 & ⫯f1 = g1.
-#g1 #g2 * -g1 -g2
-#f1 #f2 #g1 #g2 #Hf * * -g1 -g2
-#x2 #H
-[ lapply (injective_push … H) -H /2 width=3 by ex2_intro/
-| elim (discr_push_next … H)
-]
-qed-.
-
-lemma eq_inv_xn: ∀g1,g2. g1 ≡ g2 → ∀f2. ↑f2 = g2 →
-                 ∃∃f1. f1 ≡ f2 & ↑f1 = g1.
-#g1 #g2 * -g1 -g2
-#f1 #f2 #g1 #g2 #Hf * * -g1 -g2
-#x2 #H
-[ elim (discr_next_push … H)
-| lapply (injective_next … H) -H /2 width=3 by ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma eq_inv_pp: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ≡ f2.
-#g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_px … H … H1) -g1
-#x2 #Hx2 * -g2
-#H lapply (injective_push … H) -H //
-qed-.
-
-lemma eq_inv_nn: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ≡ f2.
-#g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_nx … H … H1) -g1
-#x2 #Hx2 * -g2
-#H lapply (injective_next … H) -H //
-qed-.
-
-lemma eq_inv_pn: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 → ⊥.
-#g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_px … H … H1) -g1
-#x2 #Hx2 * -g2
-#H elim (discr_next_push … H)
-qed-.
-
-lemma eq_inv_np: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 → ⊥.
-#g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_nx … H … H1) -g1
-#x2 #Hx2 * -g2
-#H elim (discr_push_next … H)
-qed-.
-
-lemma eq_inv_gen: ∀f1,f2. f1 ≡ f2 →
-                  (∃∃g1,g2. g1 ≡ g2 & ⫯g1 = f1 & ⫯g2 = f2) ∨
-                  ∃∃g1,g2. g1 ≡ g2 & ↑g1 = f1 & ↑g2 = f2.
-#f1 elim (pn_split f1) * #g1 #H1 #f2 #Hf
-[ elim (eq_inv_px … Hf … H1) -Hf /3 width=5 by or_introl, ex3_2_intro/
-| elim (eq_inv_nx … Hf … H1) -Hf /3 width=5 by or_intror, ex3_2_intro/
-]
-qed-.
-
-(* Main properties **********************************************************)
-
-corec theorem eq_trans: Transitive … eq.
-#f1 #f * -f1 -f
-#f1 #f #g1 #g #Hf1 #H1 #H #f2 #Hf2
-[ cases (eq_inv_px … Hf2 … H) | cases (eq_inv_nx … Hf2 … H) ] -g
-/3 width=5 by eq_push, eq_next/
-qed-.
-
-theorem eq_canc_sn: ∀f2. eq_repl_back (λf. f ≡ f2).
-/3 width=3 by eq_trans, eq_sym/ qed-.
-
-theorem eq_canc_dx: ∀f1. eq_repl_fwd (λf. f1 ≡ f).
-/3 width=3 by eq_trans, eq_sym/ qed-.