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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/notation/relations/ineint_5.ma".
16 include "basic_2/grammar/aarity.ma".
17 include "basic_2/multiple/mr2_mr2.ma".
18 include "basic_2/multiple/lifts_lift_vector.ma".
19 include "basic_2/multiple/drops_drop.ma".
20 include "basic_2/computation/gcp.ma".
22 (* GENERIC COMPUTATION PROPERTIES *******************************************)
24 definition S0 ≝ λC:candidate. ∀G,L2,L1,T1,d,e.
25 C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
27 definition S0s ≝ λC:candidate.
28 ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
29 ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
31 (* Note: this is Girard's CR1 *)
32 definition S1 ≝ λRP,C:candidate.
33 ∀G,L,T. C G L T → RP G L T.
35 (* Note: this is Tait's iii, or Girard's CR4 *)
36 definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
37 ∀G,L,Vs. all … (RP G L) Vs →
38 ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
40 (* Note: this generalizes Tait's ii *)
41 definition S3 ≝ λC:candidate.
43 C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
45 definition S4 ≝ λRP,C:candidate.
46 ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
48 definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
49 ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[0, i+1] V1 ≡ V2 →
50 C G L (ⒶVs.V2) → C G L (ⒶVs.#i).
52 definition S6 ≝ λRP,C:candidate.
53 ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
54 ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
56 definition S7 ≝ λC:candidate.
57 ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
59 (* requirements for the generic reducibility candidate *)
60 record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
71 (* the functional construction for candidates *)
72 definition cfun: candidate → candidate → candidate ≝
73 λC1,C2,G,K,T. ∀V. C1 G K V → C2 G K (ⓐV.T).
75 (* the reducibility candidate associated to an atomic arity *)
76 let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
79 | APair B A ⇒ cfun (acr RP B) (acr RP A)
83 "candidate of reducibility of an atomic arity (abstract)"
84 'InEInt RP G L T A = (acr RP A G L T).
86 (* Basic properties *********************************************************)
88 (* Basic_1: was: sc3_lift1 *)
89 lemma gcr_lifts: ∀C. S0 C → S0s C.
90 #C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
91 [ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
92 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
93 elim (lifts_inv_cons … H) -H /3 width=10 by/
97 axiom rp_lift: ∀RP. S0 RP.
100 axiom rp_lifts: ∀RR,RS,RP. gcr RR RS RP RP →
101 ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
102 RP G L V → RP G L0 V0.
104 #RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
105 @gcr_lifts /width=7 by/
109 (* Basic_1: was only: sns3_lifts1 *)
110 axiom rp_liftsv_all: ∀RR,RS,RP. gcr RR RS RP RP →
111 ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
112 all … (RP G L) Vs → all … (RP G L0) V0s.
114 #RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
115 #T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
119 lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
121 #RR #RS #RP #H1RP #H2RP #A elim A -A /2 width=7 by rp_lift/
122 #B #A #IHB #IHA #G #L2 #L1 #T1 #d #e #IH #T2 #HL21 #HT12 #V #HB
123 @(IHA … HL21) [3: @(lift_flat … HT12) |1: skip |
126 sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
128 lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
129 ∀A. gcr RR RS RP (acr RP A).
130 #RR #RS #RP #H1RP #H2RP #A elim A -A //
131 #B #A #IHB #IHA @mk_gcr
133 elim (cp1 … H1RP G L) #k #HK
135 [ lapply (s2 … IHB G L (◊) … HK) //
136 | #H @(cp2 … H1RP … k) @(s1 … IHA) //
138 | #G #L #Vs #HVs #T #H1T #H2T #V #HB
139 lapply (s1 … IHB … HB) #HV
140 @(s2 … IHA … (V @ Vs))
141 /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
142 | #a #G #L #Vs #U #T #W #HA #V #HB
143 @(s3 … IHA … (V @ Vs)) /2 width=1 by/
144 | #G #L #Vs #HVs #k #V #HB
145 lapply (s1 … IHB … HB) #HV
146 @(s4 … IHA … (V @ Vs)) /3 width=7 by rp_liftsv_all, conj/
147 | #I #G #L #K #Vs #V1 #V2 #i #HLK #HV12 #HA #V #HB
148 @(s5 … IHA … (V @ Vs) … HLK HV12) /2 width=1 by/
149 | #G #L #V1s #V2s #HV12s #a #W #T #HA #HW #V1 #HB
150 elim (lift_total V1 0 1) #V2 #HV12
151 @(s6 … IHA … (V1 @ V1s) (V2 @ V2s)) /2 width=1 by liftv_cons/
152 @HA @(gcr_lift … H1RP H2RP … HB … HV12) /2 width=2 by drop_drop/
153 | #G #L #Vs #T #W #HA #HW #V #HB
154 @(s7 … IHA … (V @ Vs)) /2 width=1 by/
158 lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
159 ∀a,G,L,W,T,B,A. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
160 ∀V. ⦃G, L, V⦄ ϵ[RP] 〚B〛 → ⦃G, L.ⓓⓝW.V, T⦄ ϵ[RP] 〚A〛
162 ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
163 #RR #RS #RP #H1RP #H2RP #a #G #L #W #T #B #A #HW #HA #L0 #V0 #X #des #HL0 #H #HB
164 lapply (acr_gcr … H1RP H2RP A) #HCA
165 lapply (acr_gcr … H1RP H2RP B) #HCB
166 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
167 lapply (gcr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
168 lapply (s3 … HCA … a G L0 (◊)) #H @H -H
169 lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
171 | lapply (s1 … HCB) -HCB #HCB
172 lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
176 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
177 (* Basic_1: removed local theorems 1: sc3_sn3_abst *)