(* *)
(**************************************************************************)
+include "basic_2/notation/relations/ineint_5.ma".
include "basic_2/grammar/aarity.ma".
-include "basic_2/substitution/gr2_gr2.ma".
-include "basic_2/substitution/lifts_lift_vector.ma".
-include "basic_2/substitution/ldrops_ldrop.ma".
+include "basic_2/multiple/gr2_gr2.ma".
+include "basic_2/multiple/lifts_lift_vector.ma".
+include "basic_2/multiple/ldrops_ldrop.ma".
include "basic_2/computation/acp.ma".
(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
-(* Note: this is Girard's CR1 *)
-definition S1 ≝ λRP,C:lenv→predicate term.
- ∀L,T. C L T → RP L T.
+definition S0 ≝ λC:relation3 genv lenv term. ∀G,L2,L1,T1,d,e.
+ C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
-(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term.
- ∀L,Vs. all … (RP L) Vs →
- ∀T. 𝐒⦃T⦄ → NF … (RR L) RS T → C L (ⒶVs.T).
+definition S0s ≝ λC:relation3 genv lenv term.
+ ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
-(* Note: this is Tait's ii *)
-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).
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:relation3 genv lenv term.
+ ∀G,L,T. C G L T → RP G L T.
-definition S4 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs. all … (RP L) Vs → ∀k. C L (ⒶVs.⋆k).
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:relation3 genv lenv term.
+ ∀G,L,Vs. all … (RP G L) Vs →
+ ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
-definition S5 ≝ λRP,C:lenv→predicate term. ∀I,L,K,Vs,V1,V2,i.
- C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 →
- ⇩[0, i] L ≡ K. ⓑ{I}V1 → C L (Ⓐ Vs. #i).
+(* Note: this generalizes Tait's ii *)
+definition S3 ≝ λC:relation3 genv lenv term.
+ ∀a,G,L,Vs,V,T,W.
+ C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-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 S4 ≝ λRP,C:relation3 genv lenv term.
+ ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
-definition S7 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓝW. T).
+definition S5 ≝ λC:relation3 genv lenv term. ∀I,G,L,K,Vs,V1,V2,i.
+ C G L (ⒶVs.V2) → ⇧[0, i+1] V1 ≡ V2 →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
-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 S6 ≝ λRP,C:relation3 genv lenv term.
+ ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
-definition S8s ≝ λC:lenv→predicate term.
- ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
+definition S7 ≝ λC:relation3 genv lenv term.
+ ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
(* 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;
+record acr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:relation3 genv lenv term) : Prop ≝
+{ s0: S0 C;
+ s1: S1 RP C;
s2: S2 RR RS RP C;
- s3: S3 RP C;
+ s3: S3 C;
s4: S4 RP C;
- s5: S5 RP C;
+ s5: S5 C;
s6: S6 RP C;
- s7: S7 RP C;
- s8: S8 C
+ s7: S7 C
}.
(* the abstract candidate of reducibility associated to an atomic arity *)
-let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
+let rec aacr (RP:relation3 genv lenv term) (A:aarity) (G:genv) (L:lenv) on A: predicate term ≝
λT. match A with
-[ AAtom ⇒ RP L T
-| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
- aacr RP A L0 (ⓐV0. T0)
+[ AAtom ⇒ RP G L T
+| APair B A ⇒ ∀L0,V0,T0,des.
+ aacr RP B G L0 V0 → ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] T ≡ T0 →
+ aacr RP A G L0 (ⓐV0.T0)
].
interpretation
"candidate of reducibility of an atomic arity (abstract)"
- 'InEInt RP L T A = (aacr RP A L T).
+ 'InEInt RP G L T A = (aacr RP A G L T).
(* Basic properties *********************************************************)
(* Basic_1: was: sc3_lift1 *)
-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 //
+lemma acr_lifts: ∀C. S0 C → S0s C.
+#C #HC #G #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/
+ elim (lifts_inv_cons … H) -H /3 width=10 by/
]
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/
-@(s8 … HRP)
+lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
+ ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
+ RP G L V → RP G L0 V0.
+#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
+@acr_lifts /width=7 by/
+@(s0 … HRP)
qed.
(* Basic_1: was only: sns3_lifts1 *)
-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/
+lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
+ ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
+ all … (RP G L) Vs → all … (RP G L0) V0s.
+#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
qed.
(* Basic_1: was:
sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
*)
-lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G 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
- elim (cp1 … H1RP L) #k #HK
- lapply (H ? (⋆k) ? ⟠ ? ? ?) -H
+[ /3 width=7 by ldrops_cons, lifts_cons/
+| #G #L #T #H
+ elim (cp1 … H1RP G L) #k #HK
+ lapply (H ? (⋆k) ? (⟠) ? ? ?) -H
[1,3: // |2,4: skip
- | @(s2 … IHB … ◊) //
- | #H @(cp3 … H1RP … k) @(s1 … IHA) //
+ | @(s2 … IHB … (◊)) //
+ | #H @(cp2 … H1RP … k) @(s1 … IHA) //
]
-| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
+| #G #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/
-| #a #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+ @(s2 … IHA … (V0 @ V0s))
+ /3 width=14 by rp_liftsv_all, acp_lifts, cp0, lifts_simple_dx, conj/
+| #a #G #L #Vs #U #T #W #HA #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 #Vs #HVs #k #L0 #V0 #X #hdes #HB #HL0 #H
+ @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #k #L0 #V0 #X #des #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
+ @(s4 … IHA … (V0 @ V0s)) /3 width=7 by rp_liftsv_all, conj/
+| #I #G #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
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
- @(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
+ @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
+| #G #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
- @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /2 width=1/ /2 width=6 by rp_lifts/
- @(HA … (des + 1)) /2 width=1/
- [ @(s8 … IHB … HB … HV120) /2 width=1/
+ @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/
+ @(HA … (des + 1)) /2 width=2 by ldrops_skip/
+ [ @(s0 … IHB … HB … HV120) /2 width=2 by ldrop_drop/
| @lifts_applv //
elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
>(liftv_mono … HV12s … HV10s) -V1s //
]
-| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+| #G #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
- @(s7 … IHA … (V0 @ V0s)) /2 width=6 by rp_lifts/ /3 width=4/
-| /3 width=7/
+ @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
]
qed.
-lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀a,L,W,T,A,B. RP L W → (
- ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 →
- ⦃L0, V0⦄ ϵ[RP] 〚B〛 → ⦃L0. ⓓV0, T0⦄ ϵ[RP] 〚A〛
+lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
+ ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+ ∀L0,V0,W0,T0,des. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des ] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
+ ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
) →
- ⦃L, ⓛ{a}W. T⦄ ϵ[RP] 〚②B. A〛.
-#RR #RS #RP #H1RP #H2RP #a #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
+ ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #a #G #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
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/
-@(s6 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
+lapply (acr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
+@(s3 … HCA … (◊))
+@(s6 … HCA … (◊) (◊)) //
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+ @(cp3 … H1RP) /2 width=1 by/
+]
qed.
(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
projects / helm.git / blobdiff