partial commit of the components before "conversion"

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / acp_cr.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma
index a452a907305da08ff556985eaf047eeddce979ff..a91a9f3180e78b9707b97fe268d08fa5b161f55c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_cr.ma
@@ -33,21 +33,24 @@ definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→p
 definition S3 ≝ λRP,C:lenv→predicate term.
                 ∀a,L,Vs,V,T,W. C L (ⒶVs. ⓓ{a}V. T) → RP L W → C L (ⒶVs. ⓐV. ⓛ{a}W. T).
 
-definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i.
+definition S4 ≝ λRP,C:lenv→predicate term.
+                ∀L,Vs. all … (RP L) Vs → ∀k. C L (ⒶVs.⋆k).
+
+definition S5 ≝ λRP,C:lenv→predicate term. ∀I,L,K,Vs,V1,V2,i.
                 C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 →
-                â\87©[0, i] L â\89¡ K. â\93\93V1 → C L (Ⓐ Vs. #i).
+                â\87©[0, i] L â\89¡ K. â\93\91{I}V1 → C L (Ⓐ Vs. #i).
 
-definition S5 ≝ λRP,C:lenv→predicate term.
+definition S6 ≝ λRP,C:lenv→predicate term.
                 ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
                 ∀a,V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓ{a}V. T).
 
-definition S6 ≝ λRP,C:lenv→predicate term.
+definition S7 ≝ λRP,C:lenv→predicate term.
                 ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓝW. T).
 
-definition S7 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e.
+definition S8 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e.
                 C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2.
 
-definition S7s ≝ λC:lenv→predicate term.
+definition S8s ≝ λC:lenv→predicate term.
                  ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
                  ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
 
@@ -59,7 +62,8 @@ record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate te
   s4: S4 RP C;
   s5: S5 RP C;
   s6: S6 RP C;
-  s7: S7 C
+  s7: S7 RP C;
+  s8: S8 C
 }.
 
 (* the abstract candidate of reducibility associated to an atomic arity *)
@@ -77,7 +81,7 @@ interpretation
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: sc3_lift1 *)
-lemma acr_lifts: ∀C. S7 C → S7s C.
+lemma acr_lifts: ∀C. S8 C → S8s C.
 #C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
 [ #L #T1 #T2 #H #HT1
   <(lifts_inv_nil … H) -H //
@@ -91,7 +95,7 @@ lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
                 RP L V → RP L0 V0.
 #RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV
 @acr_lifts /width=6/
-@(s7 … HRP)
+@(s8 … HRP)
 qed.
 
 (* Basic_1: was only: sns3_lifts1 *)
@@ -111,10 +115,11 @@ lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
 #RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
 #B #A #IHB #IHA @mk_acr normalize
 [ #L #T #H
-  lapply (H ? (⋆0) ? ⟠ ? ? ?) -H
+  elim (cp1 … H1RP L) #k #HK
+  lapply (H ? (⋆k) ? ⟠ ? ? ?) -H
   [1,3: // |2,4: skip
-  | @(s2 … IHB … ◊) // /2 width=2/
-  | #H @(cp3 … H1RP … 0) @(s1 … IHA) //
+  | @(s2 … IHB … ◊) //
+  | #H @(cp3 … H1RP … k) @(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
@@ -125,7 +130,12 @@ lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
   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 #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
+| #L #Vs #HVs #k #L0 #V0 #X #hdes #HB #HL0 #H
+  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  >(lifts_inv_sort1 … HY) -Y
+  lapply (s1 … IHB … HB) #HV0
+  @(s4 … IHA … (V0 @ V0s)) /3 width=6/
+| #I #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
   elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
   elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
   elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
@@ -134,15 +144,15 @@ lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
   elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
   elim (lifts_lift_trans  … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
   >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
-  @(s4 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4/
+  @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4/
 | #L #V1s #V2s #HV12s #a #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/
+  @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /2 width=1/ /2 width=6 by rp_lifts/
   @(HA … (des + 1)) /2 width=1/
-  [ @(s7 … IHB … HB … HV120) /2 width=1/
+  [ @(s8 … IHB … HB … HV120) /2 width=1/
   | @lifts_applv //
     elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
     >(liftv_mono … HV12s … HV10s) -V1s //
@@ -150,7 +160,7 @@ lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
 | #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/
+  @(s7 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /3 width=4/
 | /3 width=7/
 ]
 qed.
@@ -167,7 +177,7 @@ lapply (aacr_acr … H1RP H2RP B) #HCB
 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/
+@(s6 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
 qed.
 
 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)