the support for reducibility candidates evolves ,,,,

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / computation / acp_cr.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/computation/acp_cr.ma b/matita/matita/contribs/lambda_delta/Basic_2/computation/acp_cr.ma
index c45bc25914fbbef41fd70ef470ab1ce12b78a601..f954ca15b7da21885c0c66a8f3047cd2b320d08f 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/computation/acp_cr.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/computation/acp_cr.ma
@@ -13,8 +13,7 @@
 (**************************************************************************)
 
 include "Basic_2/grammar/aarity.ma".
-include "Basic_2/grammar/term_simple.ma".
-include "Basic_2/substitution/lift_vector.ma".
+include "Basic_2/unfold/lifts_vector.ma".
 include "Basic_2/computation/acp.ma".
 
 (* ABSTRACT COMPUTATION PROPERTIES ******************************************)
@@ -42,6 +41,10 @@ definition S6 ≝ λRP,C:lenv→predicate term.
 definition S7 ≝ λC:lenv→predicate term. ∀L1,L2,T1,T2,d,e.
                 C L1 T1 → ⇓[d, e] L2 ≡ L1 → ⇑[d, e] T1 ≡ T2 → C L2 T2.
 
+definition S7s ≝ λC:lenv→predicate term.
+                 ∀L1,L2,des. ⇓[des] L2 ≡ L1 →
+                 ∀T1,T2. ⇑[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
+
 (* properties of the abstract candidate of reducibility *)
 record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝
 { s1: S1 RP C;
@@ -56,7 +59,8 @@ record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate te
 let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
 λT. match A with
 [ AAtom     ⇒ RP L T
-| APair B A ⇒ ∀V. aacr RP B L V → aacr RP A L (𝕔{Appl} V. T)
+| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇓[des] L0 ≡ L → ⇑[des] T ≡ T0 →
+              aacr RP A L0 (𝕔{Appl} V0. T0)
 ].
 
 interpretation
@@ -65,37 +69,80 @@ interpretation
 
 (* Basic properties *********************************************************)
 
+lemma acr_lifts: ∀C. S7 C → S7s C.
+#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
+[ #L #T1 #T2 #H #HT1
+  <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+  elim (lifts_inv_cons … H) -H /3 width=9/
+]
+qed.
+
+lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+                ∀des,L0,L,V,V0. ⇓[des] L0 ≡ L → ⇑[des] V ≡ V0 →
+                RP L V → RP L0 V0.
+#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV 
+@acr_lifts /width=6/
+@(s7 … HRP)
+qed.
+
+lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+                     ∀des,L0,L,Vs,V0s. ⇑[des] Vs ≡ V0s →  ⇓[des] L0 ≡ L →
+                     all … (RP L) Vs → all … (RP L0) V0s.
+#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s
+@conj /2 width=1/ /2 width=6 by rp_lifts/
+qed.
+
 axiom aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
                 ∀A. acr RR RS RP (aacr RP A).
 (*
 #RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
 #B #A #IHB #IHA @mk_acr normalize
 [ #L #T #H
-  lapply (H (⋆0) ?) -H [ @(s2 … IHB … ◊) // /2 width=2/ ] #H
-  @(cp3 … H1RP … 0) @(s1 … IHA) //
-| #L #Vs #HVs #T #H1T #H2T #V #HB
-  lapply (s1 … IHB … HB) #HV
-  @(s2 … IHA … (V :: Vs)) // /2 width=1/
-| #L #Vs #V #T #W #HA #HW #V0 #HB
-  @(s3 … IHA … (V0 :: Vs)) // /2 width=1/
-| #L #V1s #V2s #HV12s #V #T #HA #HV #V1 #HB
-  elim (lift_total V1 0 1) #V2 #HV12
-  @(s5 … IHA … (V1 :: V1s) (V2 :: V2s)) // /2 width=1/
-  @HA @(s7 … IHB … HB … HV12) /2 width=1/
-| #L #Vs #T #W #HA #HW #V0 #HB
-  @(s6 … IHA … (V0 :: Vs)) // /2 width=1/
-| #L1 #L2 #T1 #T2 #d #e #HA #HL21 #HT12 #V2 #HB
-  @(s7 … IHA … HL21) [2: @HA [2: 
+  lapply (H ? (⋆0) ? ⟠ ? ? ?) -H 
+  [1,3: // |2,4: skip
+  | @(s2 … IHB … ◊) // /2 width=2/
+  | #H @(cp3 … H1RP … 0) @(s1 … IHA) //
+  ]
+| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
+  elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+  lapply (s1 … IHB … HB) #HV0
+  @(s2 … IHA … (V0 :: V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/
+| #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
+  elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
+  @(s3 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /4 width=5/
+| #L #V1s #V2s #HV12s #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
+  elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+  elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
+  elim (lift_total V10 0 1) #V20 #HV120
+  elim (liftv_total 0 1 V10s) #V20s #HV120s
+  @(s5 … IHA … (V10 :: V10s) (V20 :: V20s)) /2 width=1/ /2 width=6 by rp_lifts/
+  @(HA … (ss des)) /2 width=1/
+  [ @(s7 … IHB … HB … HV120) /2 width=1/
+  | @liftsv_applv //
+  ]
+| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
+  @(s6 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /3 width=4/ 
+| /3 width=7/
 ]
 qed.
 *)
 lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
-                 ∀L,W,T,A,B. RP L W →
-                 (∀V. ⦃L, V⦄ [RP] ϵ 〚B〛 → ⦃L. 𝕓{Abbr}V, T⦄ [RP] ϵ 〚A〛) →
-                              ⦃L, 𝕓{Abst}W. T⦄ [RP] ϵ 〚𝕔B. A〛.
-#RR #RS #RP #H1RP #H2RP #L #W #T #A #B #HW #HA #V #HB
+                 ∀L,W,T,A,B. RP L W → (
+                    ∀L0,V0,T0,des. ⇓[des] L0 ≡ L → ⇑[ss des] T ≡ T0 →
+                                   ⦃L0, V0⦄ [RP] ϵ 〚B〛→ ⦃L0. 𝕓{Abbr} V0, T0⦄ [RP] ϵ 〚A〛
+                 ) →
+                 ⦃L, 𝕓{Abst} W. T⦄ [RP] ϵ 〚𝕔 B. A〛.
+#RR #RS #RP #H1RP #H2RP #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
 lapply (aacr_acr … H1RP H2RP A) #HCA
 lapply (aacr_acr … H1RP H2RP B) #HCB
-lapply (s1 … HCB) -HCB #HCB
-@(s3 … HCA … ◊) // @(s5 … HCA … ◊ ◊) // /2 width=1/
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (s1 … HCB) -HCB #HCB 
+@(s3 … HCA … ◊) /2 width=6 by rp_lifts/
+@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
 qed.