Merge branch 'matita-lablgtk3' of ssh://matita.cs.unibo.it:/srv/git/helm into matita...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / cnr_tdeq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnr_tdeq.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/relocation/lifts_tdeq.ma".
+include "basic_2/rt_transition/cpr_drops_basic.ma".
+include "basic_2/rt_transition/cnr_simple.ma".
+include "basic_2/rt_transition/cnr_drops.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
+
+(* Properties with context-free sort-irrelevant equivalence for terms *******)
+
+(* Basic_1: was: nf2_dec *)
+(* Basic_2A1: uses: cnr_dec *)
+lemma cnr_dec_tdeq (h) (G) (L):
+      ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄
+            | ∃∃T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥).
+#h #G #L #T1
+@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
+[ #s #HG #HL #HT destruct -IH
+  /3 width=4 by cnr_sort, or_introl/
+| #i #HG #HL #HT destruct -IH
+  elim (drops_F_uni L i)
+  [ /3 width=6 by cnr_lref_atom, or_introl/
+  | * * [ #I | * #V ] #K #HLK
+    [ /3 width=7 by cnr_lref_unit, or_introl/
+    | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+      @or_intror @(ex2_intro … W) [ /2 width=6 by cpm_delta_drops/ ] #H
+      lapply (tdeq_inv_lref1 … H) -H #H destruct
+      /2 width=5 by lifts_inv_lref2_uni_lt/
+    | /3 width=7 by cnr_lref_abst, or_introl/
+    ]
+  ]
+| #l #HG #HL #HT destruct -IH
+  /3 width=4 by cnr_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
+  [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
+    elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
+    [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
+      @or_intror @(ex2_intro … X1) [ /2 width=3 by cpm_zeta/ ] #H
+      /2 width=7 by tdeq_lifts_inv_pair_sn/
+    | @or_intror @(ex2_intro … (+ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    ]
+  | elim (IH G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * | // ] -IH
+    [ #HT1 #HV1 /3 width=6 by cnr_abbr_neg, or_introl/
+    | #T2 #HT12 #HnT12 #_
+      @or_intror @(ex2_intro … (-ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    | #V2 #HV12 #HnV12
+      @or_intror @(ex2_intro … (-ⓓV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    ]
+  | elim (IH G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * | // ] -IH
+    [ #HT1 #HV1 /3 width=6 by cnr_abst, or_introl/
+    | #T2 #HT12 #HnT12 #_
+      @or_intror @(ex2_intro … (ⓛ{p}V1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    | #V2 #HV12 #HnV12
+      @or_intror @(ex2_intro … (ⓛ{p}V2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    ]
+  ]
+| * #V1 #T1 #HG #HL #HT destruct [| -IH ]
+  [ elim (IH G L V1) [ elim (IH G L T1) [| * | // ] | * | // ] -IH
+    [ #HT1 #HV1
+      elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
+      [ /3 width=6 by cnr_appl_simple, or_introl/
+      | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+        @or_intror @(ex2_intro … (ⓓ{p}W1.ⓐX1.U1)) [ /2 width=3 by cpm_theta/ ] #H
+        elim (tdeq_inv_pair … H) -H #H destruct
+      | @or_intror @(ex2_intro … (ⓓ{p}ⓝW1.V1.U1)) [ /2 width=1 by cpm_beta/ ] #H
+        elim (tdeq_inv_pair … H) -H #H destruct
+      ]
+    | #T2 #HT12 #HnT12 #_
+      @or_intror @(ex2_intro … (ⓐV1.T2)) [ /2 width=1 by cpm_appl/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    | #V2 #HV12 #HnV12
+      @or_intror @(ex2_intro … (ⓐV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+      elim (tdeq_inv_pair … H) -H /2 width=1 by/
+    ]
+  | @or_intror @(ex2_intro … T1) [ /2 width=1 by cpm_eps/ ] #H
+    /2 width=4 by tdeq_inv_pair_xy_y/
+  ]
+]
+qed-.