X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Fcprs.ma;h=640f4dbfce2192b8a6a3feb6ef0dd923f875aeaf;hb=32bdf7f107be22a121fab8225c5fae4eb6b33633;hp=1b3bcf66638c0246bb4c8137a78fc70ade583171;hpb=ef49e0e7f5f298c299afdd3cbfdc2404ecb93879;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
index 1b3bcf666..640f4dbfc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
@@ -12,155 +12,119 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/predstar_4.ma".
 include "basic_2/reduction/cnr.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
 
 (* Basic_1: includes: pr1_pr0 *)
-definition cprs: lenv → relation term ≝ LTC … cpr.
+definition cprs: relation4 genv lenv term term ≝
+                 λG. LTC … (cpr G).
 
 interpretation "context-sensitive parallel computation (term)"
-   'PRedStar L T1 T2 = (cprs L T1 T2).
+   'PRedStar G L T1 T2 = (cprs G L T1 T2).
 
 (* Basic eliminators ********************************************************)
 
-lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
-                ∀T2. L ⊢ T1 ➡* T2 → R T2.
-#L #T1 #R #HT1 #IHT1 #T2 #HT12
+lemma cprs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2.
+#G #L #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
-lemma cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) →
-                   ∀T1. L ⊢ T1 ➡* T2 → R T1.
-#L #T2 #R #HT2 #IHT2 #T1 #HT12
+lemma cprs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1.
+#G #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: pr3_pr2 *)
-lemma cpr_cprs: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ➡* T2.
+lemma cpr_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 /2 width=1/ qed.
 
-lemma cpss_cprs: ∀L,T1,T2. L ⊢ T1 ▶* T2 → L ⊢ T1 ➡* T2.
-/3 width=1/ qed.
-
 (* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
+lemma cprs_refl: ∀G,L,T. ⦃G, L⦄ ⊢ T ➡* T.
 /2 width=1/ qed.
 
-lemma cprs_strap1: ∀L,T1,T,T2.
-                   L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2.
+lemma cprs_strap1: ∀G,L,T1,T,T2.
+                   ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 normalize /2 width=3/ qed.
 
 (* Basic_1: was: pr3_step *)
-lemma cprs_strap2: ∀L,T1,T,T2.
-                   L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+lemma cprs_strap2: ∀G,L,T1,T,T2.
+                   ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 normalize /2 width=3/ qed.
 
-lemma cprs_cpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ▶* T2 → L ⊢ T1 ➡* T2.
-/3 width=3/ qed-.
-
-lemma cpss_cprs_trans: ∀L,T1,T. L ⊢ T1 ▶* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/3 width=3/ qed-.
-
-lemma cprs_lsubr_trans: lsubr_trans … cprs.
-/3 width=3 by cpr_lsubr_trans, TC_lsubr_trans/
+lemma lsubr_cprs_trans: ∀G. lsub_trans … (cprs G) lsubr.
+/3 width=5 by lsubr_cpr_trans, LTC_lsub_trans/
 qed-.
 
 (* Basic_1: was: pr3_pr1 *)
-lemma tprs_cprs: ∀L,T1,T2. ⋆ ⊢ T1 ➡* T2 → L ⊢ T1 ➡* T2.
-#L #T1 #T2 #H @(cprs_lsubr_trans … H) -H //
+lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
+#G #L #T1 #T2 #H @(lsubr_cprs_trans … H) -H //
 qed.
 
-lemma cprs_ext_bind_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡* T2 →
-                        ∀a,I. L ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2
-/3 width=1/ /3 width=6/
-qed.
-
-lemma cprs_bind_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 →
-                    ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
-#L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
+lemma cprs_bind_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡* T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
 /3 width=1/ /3 width=3/
 qed.
 
 (* Basic_1: was only: pr3_thin_dx *)
-lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 →
-                    L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
+lemma cprs_flat_dx: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
 #T #T2 #_ #HT2 #IHT1
 @(cprs_strap1 … IHT1) -V1 -T1 /2 width=1/
 qed.
 
-lemma cprs_flat_sn: ∀I,L,T1,T2. L ⊢ T1 ➡ T2 → ∀V1,V2. L ⊢ V1 ➡* V2 →
-                    L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2 /3 width=1/
+lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2 /3 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_strap1 … IHV1) -V1 -T1 /2 width=1/
 qed.
 
-lemma cprs_zeta: ∀L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
-                 L.ⓓV ⊢ T1 ➡* T → L ⊢ +ⓓV.T1 ➡* T2.
-#L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
+lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* T2.
+#G #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
 qed.
 
-lemma cprs_tau: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ∀V. L ⊢ ⓝV.T1 ➡* T2.
-#L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
+lemma cprs_tau: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
+#G #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
 qed.
 
-lemma cprs_beta_dx: ∀a,L,V1,V2,W,T1,T2.
-                    L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 →
-                    L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2.
-#a #L #V1 #V2 #W #T1 #T2 #HV12 * -T2 /3 width=1/
-/4 width=6 by cprs_strap1, cprs_bind_dx, cprs_flat_dx, cpr_beta/ (**) (* auto too slow without trace *)
+lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
+                    ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2 /3 width=1/
+/4 width=7 by cprs_strap1, cprs_bind_dx, cprs_flat_dx, cpr_beta/ (**) (* auto too slow without trace *)
 qed.
 
-lemma cprs_theta_dx: ∀a,L,V1,V,V2,W1,W2,T1,T2.
-                     L ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
-                     L ⊢ W1 ➡ W2 → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
-#a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
+lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
+                     ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡* T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
 /4 width=9 by cprs_strap1, cprs_bind_dx, cprs_flat_dx, cpr_theta/ (**) (* auto too slow without trace *)
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
 (* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k.
-#L #U2 #k #H @(cprs_ind … H) -U2 //
+lemma cprs_inv_sort1: ∀G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡* U2 → U2 = ⋆k.
+#G #L #U2 #k #H @(cprs_ind … H) -U2 //
 #U2 #U #_ #HU2 #IHU2 destruct
 >(cpr_inv_sort1 … HU2) -HU2 //
 qed-.
 
-(* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 →
-                      ∨∨ ∃∃V2,T2.       L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
-                                        U2 = ⓐV2. T2
-                       | ∃∃a,V2,W,T.    L ⊢ V1 ➡* V2 &
-                                        L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ➡* U2
-                       | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
-                                        L ⊢ T1 ➡* ⓓ{a}V. T & L ⊢ ⓓ{a}V. ⓐV2. T ➡* U2.
-#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
-#U #U2 #_ #HU2 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
-  elim (cpr_inv_appl1 … HU2) -HU2 *
-  [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
-  | #a #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct
-    lapply (cprs_strap1 … HV10 … HV02) -V0 /5 width=7/
-  | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
-    @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
-  ]
-| /4 width=9/
-| /4 width=11/
-]
-qed-.
-
 (* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓝW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
-                      ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
-#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
+#G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
 #U2 #U #_ #HU2 * /3 width=3/ *
 #W #T #HW1 #HT1 #H destruct
 elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
@@ -168,30 +132,12 @@ elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
 qed-.
 
 (* Basic_1: was: nf2_pr3_unfold *)
-lemma cprs_inv_cnr1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U.
-#L #T #U #H @(cprs_ind_dx … H) -T //
+lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → T = U.
+#G #L #T #U #H @(cprs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
 qed-.
 
-(* Basic forward lemmas *****************************************************)
-
-(* Basic_1: was: pr3_gen_abst *)
-lemma cprs_fwd_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1. T1 ➡* U2 → ∀I,W.
-                      ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓑ{I} W ⊢ T1 ➡* T2 &
-                               U2 = ⓛ{a}V2. T2.
-#a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
-#U #U2 #_ #HU2 #IHU1 #I #W
-elim (IHU1 I W) -IHU1 #V #T #HV1 #HT1 #H destruct
-elim (cpr_fwd_abst1 … HU2 I W) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
-qed-.
-
-lemma cprs_fwd_abst: ∀a,L,V1,V2,T1,T2. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2 → ∀I,W.
-                     L ⊢ V1 ➡* V2 ∧ L. ⓑ{I} W ⊢ T1 ➡* T2.
-#a #L #V1 #V2 #T1 #T2 #H #I #W
-elim (cprs_fwd_abst1 … H I W) -H #V #T #HV1 #HT1 #H destruct /2 width=1/
-qed-.
-
 (* Basic_1: removed theorems 13:
    pr1_head_1 pr1_head_2 pr1_comp
    clear_pr3_trans pr3_cflat pr3_gen_bind