(* *)
(**************************************************************************)
-include "Basic_2/static/aaa.ma".
-include "Basic_2/computation/lsubc.ma".
+include "Basic_2/unfold/lifts_lifts.ma".
+include "Basic_2/unfold/ldrops_ldrops.ma".
+include "Basic_2/static/aaa_lifts.ma".
+include "Basic_2/static/aaa_aaa.ma".
+include "Basic_2/computation/lsubc_ldrops.ma".
(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
(* Main propertis ***********************************************************)
-axiom aacr_aaa_csubc: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,T,A. L1 ⊢ T ÷ A →
- ∀L2. L2 [RP] ⊑ L1 → ⦃L2, T⦄ [RP] ϵ 〚A〛.
-(*
+(* Basic_1: was: sc3_arity_csubc *)
+theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP.
+ acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,T,A. L1 ⊢ T ÷ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
+ ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 [RP] ⊑ L0 →
+ ⦃L2, T0⦄ [RP] ϵ 〚A〛.
#RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A
-[ #L #k #L2 #HL2
- lapply (aacr_acr … H1RP H2RP 𝕒) #HAtom
+[ #L #k #L0 #des #HL0 #X #H #L2 #HL20
+ >(lifts_inv_sort1 … H) -H
+ lapply (aacr_acr … H1RP H2RP ⓪) #HAtom
@(s2 … HAtom … ◊) // /2 width=2/
-| * #L #K #V #B #i #HLK #_ #IHB #L2 #HL2
- [
- | lapply (aacr_acr … H1RP H2RP B) #HB
- @(s2 … HB … ◊) //
- @(cp2 … H1RP)
-| #L #V #T #B #A #_ #_ #IHB #IHA #L2 #HL2
- lapply (aacr_acr … H1RP H2RP A) #HA
+| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20
lapply (aacr_acr … H1RP H2RP B) #HB
- lapply (s1 … HB) -HB #HB
- @(s5 … HA … ◊ ◊) // /3 width=1/
-| #L #W #T #B #A #_ #_ #IHB #IHA #L2 #HL2
+ elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b
+ elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1
+ >(at_mono … Hi1 … Hi0) -i1
+ elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct
+ elim (lsubc_ldrop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H
+ elim (lsubc_inv_pair2 … H) -H *
+ [ #K2 #HK20 #H destruct
+ generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2
+ [ elim (lift_total V0 0 (i0 +1)) #V #HV0
+ elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
+ @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *)
+ | @(s2 … HB … ◊) // /2 width=3/
+ ]
+ | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
+ #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b
+ lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
+ >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B
+ elim (lift_total V2 0 (i0 +1)) #V #HV2
+ @(s4 … HB … ◊ … HV2 HLK2)
+ @(s7 … HB … HKV2B) //
+ ]
+| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (aacr_acr … H1RP H2RP A) #HA
lapply (aacr_acr … H1RP H2RP B) #HB
lapply (s1 … HB) -HB #HB
- @(aacr_abst … H1RP H2RP) /3 width=1/ -HB /4 width=3/
-| /3 width=1/
-| #L #V #T #A #_ #_ #IH1A #IH2A #L2 #HL2
+ @(s5 … HA … ◊ ◊) // /3 width=5/
+| #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
+ elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+ @(aacr_abst … H1RP H2RP)
+ [ lapply (aacr_acr … H1RP H2RP B) #HB
+ @(s1 … HB) /2 width=5/
+ | -IHB
+ #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
+ elim (lifts_total des3 W0) #W2 #HW02
+ elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
+ lapply (aaa_lifts … L2 W2 … (des @ des3) … HLWB) -HLWB /2 width=3/ #HLW2B
+ @(IHA (L2. ⓛW2) … (des + 1 @ des3 + 1)) -IHA
+ /2 width=3/ /3 width=5/
+ ]
+| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ /3 width=10/
+| #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
lapply (aacr_acr … H1RP H2RP A) #HA
lapply (s1 … HA) #H
- @(s6 … HA … ◊) /2 width=1/ /3 width=1/
+ @(s6 … HA … ◊) /2 width=5/ /3 width=5/
]
-*)
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L,T,A. L ⊢ T ÷ A → ⦃L, T⦄ [RP] ϵ 〚A〛.
+/2 width=8/ qed.
+
lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
∀L,T,A. L ⊢ T ÷ A → RP L T.
#RR #RS #RP #H1RP #H2RP #L #T #A #HT