syntactic components detached from basic_2 become static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / relocation / seq.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/seq.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/seq.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 "static_2/notation/relations/ideqsn_3.ma".
+include "static_2/syntax/ceq_ext.ma".
+include "static_2/relocation/sex.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR SELECTED LOCAL ENVIRONMENTS ********************)
+
+(* Basic_2A1: includes: lreq_atom lreq_zero lreq_pair lreq_succ *)
+definition seq: relation3 rtmap lenv lenv ≝ sex ceq_ext cfull.
+
+interpretation
+  "syntactic equivalence on selected entries (local environment)"
+  'IdEqSn f L1 L2 = (seq f L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma seq_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ≡[f] L2).
+/2 width=3 by sex_eq_repl_back/ qed-.
+
+lemma seq_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ≡[f] L2).
+/2 width=3 by sex_eq_repl_fwd/ qed-.
+
+lemma sle_seq_trans: ∀f2,L1,L2. L1 ≡[f2] L2 →
+                     ∀f1. f1 ⊆ f2 → L1 ≡[f1] L2.
+/2 width=3 by sle_sex_trans/ qed-.
+
+(* Basic_2A1: includes: lreq_refl *)
+lemma seq_refl: ∀f. reflexive … (seq f).
+/2 width=1 by sex_refl/ qed.
+
+(* Basic_2A1: includes: lreq_sym *)
+lemma seq_sym: ∀f. symmetric … (seq f).
+/3 width=2 by sex_sym, cext2_sym/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: includes: lreq_inv_atom1 *)
+lemma seq_inv_atom1: ∀f,Y. ⋆ ≡[f] Y → Y = ⋆.
+/2 width=4 by sex_inv_atom1/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair1 *)
+lemma seq_inv_next1: ∀g,J,K1,Y. K1.ⓘ{J} ≡[↑g] Y →
+                     ∃∃K2. K1 ≡[g] K2 & Y = K2.ⓘ{J}.
+#g #J #K1 #Y #H
+elim (sex_inv_next1 … H) -H #Z #K2 #HK12 #H1 #H2 destruct
+<(ceq_ext_inv_eq … H1) -Z /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero1 lreq_inv_succ1 *)
+lemma seq_inv_push1: ∀g,J1,K1,Y. K1.ⓘ{J1} ≡[⫯g] Y →
+                     ∃∃J2,K2. K1 ≡[g] K2 & Y = K2.ⓘ{J2}.
+#g #J1 #K1 #Y #H elim (sex_inv_push1 … H) -H /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_atom2 *)
+lemma seq_inv_atom2: ∀f,X. X ≡[f] ⋆ → X = ⋆.
+/2 width=4 by sex_inv_atom2/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair2 *)
+lemma seq_inv_next2: ∀g,J,X,K2. X ≡[↑g] K2.ⓘ{J} →
+                     ∃∃K1. K1 ≡[g] K2 & X = K1.ⓘ{J}.
+#g #J #X #K2 #H
+elim (sex_inv_next2 … H) -H #Z #K1 #HK12 #H1 #H2 destruct
+<(ceq_ext_inv_eq … H1) -J /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero2 lreq_inv_succ2 *)
+lemma seq_inv_push2: ∀g,J2,X,K2. X ≡[⫯g] K2.ⓘ{J2} →
+                     ∃∃J1,K1. K1 ≡[g] K2 & X = K1.ⓘ{J1}.
+#g #J2 #X #K2 #H elim (sex_inv_push2 … H) -H /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair *)
+lemma seq_inv_next: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[↑f] L2.ⓘ{I2} →
+                    ∧∧ L1 ≡[f] L2 & I1 = I2.
+#f #I1 #I2 #L1 #L2 #H elim (sex_inv_next … H) -H
+/3 width=3 by ceq_ext_inv_eq, conj/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_succ *)
+lemma seq_inv_push: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[⫯f] L2.ⓘ{I2} → L1 ≡[f] L2.
+#f #I1 #I2 #L1 #L2 #H elim (sex_inv_push … H) -H /2 width=1 by conj/
+qed-.
+
+lemma seq_inv_tl: ∀f,I,L1,L2. L1 ≡[⫱f] L2 → L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+/2 width=1 by sex_inv_tl/ qed-.
+
+(* Basic_2A1: removed theorems 5:
+              lreq_pair_lt lreq_succ_lt lreq_pair_O_Y lreq_O2 lreq_inv_O_Y
+*)