(* *)
(**************************************************************************)
-include "Basic_2/unfold/gr2_gr2.ma".
include "Basic_2/unfold/lifts_lifts.ma".
include "Basic_2/unfold/ldrops_ldrops.ma".
-include "Basic_2/static/aaa.ma".
-include "Basic_2/computation/lsubc.ma".
-
-(* NOTE: The constant (0) can not be generalized *)
-axiom lsubc_ldrop_trans: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 [RP] ⊑ K2.
-
-axiom ldrops_lsubc_trans: ∀RP,L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
- ∃∃L2. L1 [RP] ⊑ L2 & ⇩*[des] L2 ≡ K2.
+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_lifts: ∀RR,RS,RP.
+(* 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 #L0 #des #HL0 #X #H #L2 #HL20
>(lifts_inv_sort1 … H) -H
- lapply (aacr_acr … H1RP H2RP 𝕒) #HAtom
+ lapply (aacr_acr … H1RP H2RP ⓪) #HAtom
@(s2 … HAtom … ◊) // /2 width=2/
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L0 #des #HL01 #X #H #L2 #HL20
+| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20
lapply (aacr_acr … H1RP H2RP B) #HB
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_trans … HL20 … HL0) -HL0 #X #HLK2 #H
- elim (lift_total V0 0 (i0 +1)) #V #HV0
+ 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
- [ @(s4 … HB … ◊ … HV0 HLK2)
- @(IHB … HL20) [2: /2 width=6/ | skip ]
- | skip
- ]
-(⇧*[des0]V1≡V0) → (⇧[O,i0+1]V0≡V) → (@[i]des≡i0) → (des+1▭i+1≡des0+1) →
-⇧*[{O,i+1}::des]V1≡V)
-
- Theorem lift1_free: (hds:?; i:?; t:?)
- (lift1 hds (lift (S i) (0) t)) =
- (lift (S (trans hds i)) (0) (lift1 (ptrans hds i) t)).
-
-
-
-
+ [ 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/
]
- | #K2 #V2 #A2 #HV2 #HV0 #HK20 #H1 #H2 destruct
+ | -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 B) #HB
lapply (s1 … HB) -HB #HB
@(s5 … HA … ◊ ◊) // /3 width=5/
-| #L #W #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
+| #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/
- | #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
+ | -IHB
+ #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
elim (lifts_total des3 W0) #W2 #HW02
- elim (ldrops_lsubc_trans … HL32 … HL02) -L2 #L2 #HL32 #HL20
- @(IHA (L2. 𝕓{Abst} W2) … (des + 1 @ des3 + 1))
- /2 width=3/ /3 width=5/ /4 width=6/
+ 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
lapply (s1 … HA) #H
@(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
lapply (aacr_acr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=8/
+@(s1 … HA) /2 width=4/
qed.