milestone in basic_2 with additions in static_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / cnv_eval.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_eval.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_eval.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_eval.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 "basic_2/rt_computation/cpue_csx.ma".
+include "basic_2/rt_equivalence/cpes_cprs.ma".
+include "basic_2/dynamic/cnv_cpue.ma".
+include "basic_2/dynamic/cnv_cpes.ma".
+include "basic_2/dynamic/cnv_preserve_cpes.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ{p}W.T
+                     | (∀p,W,T. X = ⓛ{p}W.T → ⊥).
+* [ #I | * [ #p * | #I ] #V #T ]
+[3: /3 width=4 by ex1_3_intro, or_introl/ ]
+@or_intror #q #W #U #H destruct
+qed-.
+
+(* main properties with evaluations for rt-transition on terms **************)
+
+theorem cnv_dec (a) (h) (G) (L) (T):
+        Decidable (⦃G,L⦄ ⊢ T ![a,h]).
+#a #h #G #L #T
+@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HT destruct -IH
+  /2 width=1 by cnv_sort, or_introl/
+| #i #HG #HL #HT destruct
+  elim (drops_F_uni L i) [| * * ]
+  [ /3 width=8 by cnv_inv_lref_atom, or_intror/
+  | /3 width=9 by cnv_inv_lref_unit, or_intror/
+  | #I #V #K #HLK
+    elim (IH G K V) -IH [3: /2 width=2 by fqup_lref/ ]
+    [ /3 width=5 by cnv_lref_drops, or_introl/
+    | /4 width=5 by cnv_inv_lref_pair, or_intror/
+    ]
+  ]
+| #l #HG #HL #HT destruct -IH
+  /3 width=6 by cnv_inv_gref, or_intror/
+| #p #I #V #T #HG #HL #HT destruct
+  elim (IH G L V) [| -IH | // ] #HV
+  [ elim (IH G (L.ⓑ{I}V) T) -IH [3: // ] #HT
+    [ /3 width=1 by cnv_bind, or_introl/ ]
+  ]
+  @or_intror #H
+  elim (cnv_inv_bind … H) -H /2 width=1 by/
+| #V #T #HG #HL #HT destruct
+  elim (IH G L V) [| -IH #HV | // ]
+  [ elim (IH G L T) -IH [| #HT #HV | // ]
+    [ cases a #HT #HV
+      [ elim (cnv_fwd_aaa … HT) #A #HA
+        elim (cpme_total_aaa h 1 … HA) #X
+        elim (abst_dec X) [ * ]
+        [ #p #W #U #H #HTU destruct
+          elim (cnv_cpes_dec … 1 0 … HV W) [ #HVW | #HnVW ]
+          [ elim HTU -HTU #HTU #_
+            /3 width=7 by cnv_appl_cpes, or_introl/
+          | @or_intror #H
+            elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #Hm1 #_ #_ #HVW0
+            >Hm1 -m1 [| // ] #HTU0
+            elim (cnv_cpme_cpms_conf … HT … HTU0 … HTU) -T #HU0 #_
+            elim (cpms_inv_abst_bi … HU0) -HU0 #_ #HW0 #_ -p -q -U0
+            /3 width=3 by cpes_cprs_trans/
+          | lapply (cnv_cpme_trans … HT … HTU) -T #H
+            elim (cnv_inv_bind … H) -H #HW #_ //
+          ]
+        | #HnTU #HTX
+          @or_intror #H
+          elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #Hm1 #_ #_ #_
+          >Hm1 -m1 [| // ] #HTU0
+          elim (cnv_cpme_cpms_conf … HT … HTU0 … HTX) -T #HX #_
+          elim (cpms_inv_abst_sn … HX) -HX #V0 #T0 #_ #_ #H destruct -W0 -U0
+          /2 width=4 by/
+        ]
+      | elim (cpue_total_csx h G L T)
+        [| /3 width=2 by cnv_fwd_fsb, fsb_inv_csx/ ] #X
+        elim (abst_dec X) [ * ]
+        [ #p #W #U #H #HTU destruct
+          elim (cnv_cpes_dec … 1 0 … HV W) [ #HVW | #HnVW ]
+          [ elim HTU -HTU #n #HTU #_
+            @or_introl @(cnv_appl_cpes … HV … HT … HVW … HTU) #H destruct
+          | @or_intror #H
+            elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #_ #_ #_ #HVW0 #HTU0
+            elim (cnv_cpue_cpms_conf … HT … HTU0 … HTU) -m1 -T #X * #m2 #HU0X #_ #HUX
+            elim (tueq_inv_bind1 … HUX) -HUX #X0 #_ #H destruct -U
+            elim (cpms_inv_abst_bi … HU0X) -HU0X #_ #HW0 #_ -p -q -m2 -U0 -X0
+            /3 width=3 by cpes_cprs_trans/
+          | lapply (cnv_cpue_trans … HT … HTU) -T #H
+            elim (cnv_inv_bind … H) -H #HW #_ //
+          ]
+        | #HnTU #HTX
+          @or_intror #H
+          elim (cnv_inv_appl_SO_cpes … H) -H #m1 #q #W0 #U0 #_ #_ #_ #_ #HTU0
+          elim (cnv_cpue_cpms_conf … HT … HTU0 … HTX) -m1 -T #X0 * #m2 #HUX0 #_ #HX0
+          elim (cpms_inv_abst_sn … HUX0) -HUX0 #V0 #T0 #_ #_ #H destruct -m2 -W0 -U0
+          elim (tueq_inv_bind2 … HX0) -HX0 #X0 #_ #H destruct
+          /2 width=4 by/
+        ]
+      ]
+    ]
+  ]
+  @or_intror #H
+  elim (cnv_inv_appl_SO … H) -H /2 width=1 by/
+| #U #T #HG #HL #HT destruct
+  elim (IH G L U) [| -IH | // ] #HU
+  [ elim (IH G L T) -IH [3: // ] #HT
+    [ elim (cnv_cpes_dec … 0 1 … HU … HT) #HUT
+      [ /3 width=1 by cnv_cast_cpes, or_introl/ ]
+    ]
+  ]
+  @or_intror #H
+  elim (cnv_inv_cast_cpes … H) -H /2 width=1 by/
+]
+qed-.