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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 *)
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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/substitution/gr2_gr2.ma".
18 include "basic_2/substitution/lifts_lift_vector.ma".
19 include "basic_2/substitution/ldrops_ldrop.ma".
20 include "basic_2/computation/acp.ma".
22 (* ABSTRACT COMPUTATION PROPERTIES ******************************************)
24 (* Note: this is Girard's CR1 *)
25 definition S1 ≝ λRP,C:relation3 genv lenv term.
26 ∀G,L,T. C G L T → RP G L T.
28 (* Note: this is Tait's iii, or Girard's CR4 *)
29 definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:relation3 genv lenv term.
30 ∀G,L,Vs. all … (RP G L) Vs →
31 ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
33 (* Note: this generalizes Tait's ii *)
34 definition S3 ≝ λC:relation3 genv lenv term.
36 C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
38 definition S4 ≝ λRP,C:relation3 genv lenv term.
39 ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
41 definition S5 ≝ λC:relation3 genv lenv term. ∀I,G,L,K,Vs,V1,V2,i.
42 C G L (ⒶVs.V2) → ⇧[0, i+1] V1 ≡ V2 →
43 ⇩[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
45 definition S6 ≝ λRP,C:relation3 genv lenv term.
46 ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
47 ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
49 definition S7 ≝ λC:relation3 genv lenv term.
50 ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
52 definition S8 ≝ λC:relation3 genv lenv term. ∀G,L2,L1,T1,d,e.
53 C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
55 definition S8s ≝ λC:relation3 genv lenv term.
56 ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
57 ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
59 (* properties of the abstract candidate of reducibility *)
60 record acr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:relation3 genv lenv term) : Prop ≝
71 (* the abstract candidate of reducibility associated to an atomic arity *)
72 let rec aacr (RP:relation3 genv lenv term) (A:aarity) (G:genv) (L:lenv) on A: predicate term ≝
75 | APair B A ⇒ ∀L0,V0,T0,des.
76 aacr RP B G L0 V0 → ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] T ≡ T0 →
77 aacr RP A G L0 (ⓐV0.T0)
81 "candidate of reducibility of an atomic arity (abstract)"
82 'InEInt RP G L T A = (aacr RP A G L T).
84 (* Basic properties *********************************************************)
86 (* Basic_1: was: sc3_lift1 *)
87 lemma acr_lifts: ∀C. S8 C → S8s C.
88 #C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
89 [ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
90 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
91 elim (lifts_inv_cons … H) -H /3 width=10 by/
95 lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
96 ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
97 RP G L V → RP G L0 V0.
98 #RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
99 @acr_lifts /width=7 by/
103 (* Basic_1: was only: sns3_lifts1 *)
104 lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
105 ∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
106 all … (RP G L) Vs → all … (RP G L0) V0s.
107 #RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
108 #T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
112 sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
114 lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
115 ∀A. acr RR RS RP (aacr RP A).
116 #RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
117 #B #A #IHB #IHA @mk_acr normalize
119 elim (cp1 … H1RP G L) #k #HK
120 lapply (H ? (⋆k) ? (⟠) ? ? ?) -H
122 | @(s2 … IHB … (◊)) //
123 | #H @(cp3 … H1RP … k) @(s1 … IHA) //
125 | #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
126 elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
127 lapply (s1 … IHB … HB) #HV0
128 @(s2 … IHA … (V0 @ V0s))
129 /3 width=14 by rp_liftsv_all, acp_lifts, cp2, lifts_simple_dx, conj/
130 | #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #des #HB #HL0 #H
131 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
132 elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
133 elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
134 @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
135 | #G #L #Vs #HVs #k #L0 #V0 #X #des #HB #HL0 #H
136 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
137 >(lifts_inv_sort1 … HY) -Y
138 lapply (s1 … IHB … HB) #HV0
139 @(s4 … IHA … (V0 @ V0s)) /3 width=7 by rp_liftsv_all, conj/
140 | #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
141 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
142 elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
143 elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
144 >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
145 elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
146 elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
147 elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
148 >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
149 @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
150 | #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
151 elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
152 elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
153 elim (lift_total V10 0 1) #V20 #HV120
154 elim (liftv_total 0 1 V10s) #V20s #HV120s
155 @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/
156 @(HA … (des + 1)) /2 width=2 by ldrops_skip/
157 [ @(s8 … IHB … HB … HV120) /2 width=2 by ldrop_drop/
159 elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
160 >(liftv_mono … HV12s … HV10s) -V1s //
162 | #G #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
163 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
164 elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
165 @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
166 | /3 width=7 by ldrops_cons, lifts_cons/
170 lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
171 ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
172 ∀L0,V0,W0,T0,des. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des ] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
173 ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
175 ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
176 #RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
177 lapply (aacr_acr … H1RP H2RP A) #HCA
178 lapply (aacr_acr … H1RP H2RP B) #HCB
179 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
180 lapply (acr_lifts … HL0 … HW0 HW) -HW [ @(s8 … HCB) ] #HW0
182 @(s6 … HCA … (◊) (◊)) //
184 | lapply (s1 … HCB) -HCB #HCB
185 @(cp4 … H1RP) /2 width=1 by/
189 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
190 (* Basic_1: removed local theorems 1: sc3_sn3_abst *)