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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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15 include "basic_2/grammar/aarity.ma".
16 include "basic_2/substitution/gr2_gr2.ma".
17 include "basic_2/substitution/lifts_lift_vector.ma".
18 include "basic_2/substitution/ldrops_ldrop.ma".
19 include "basic_2/computation/acp.ma".
21 (* ABSTRACT COMPUTATION PROPERTIES ******************************************)
23 (* Note: this is Girard's CR1 *)
24 definition S1 ≝ λRP,C:lenv→predicate term.
27 (* Note: this is Tait's iii, or Girard's CR4 *)
28 definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term.
29 ∀L,Vs. all … (RP L) Vs →
30 ∀T. 𝐒⦃T⦄ → NF … (RR L) RS T → C L (ⒶVs.T).
32 (* Note: this generalizes Tait's ii *)
33 definition S3 ≝ λC:lenv→predicate term.
34 ∀a,L,Vs,V,T,W. C L (ⒶVs.ⓓ{a}ⓝW.V.T) → C L (ⒶVs.ⓐV.ⓛ{a}W.T).
36 definition S4 ≝ λRP,C:lenv→predicate term.
37 ∀L,Vs. all … (RP L) Vs → ∀k. C L (ⒶVs.⋆k).
39 definition S5 ≝ λC:lenv→predicate term. ∀I,L,K,Vs,V1,V2,i.
40 C L (ⒶVs.V2) → ⇧[0, i + 1] V1 ≡ V2 →
41 ⇩[0, i] L ≡ K.ⓑ{I}V1 → C L (Ⓐ Vs.#i).
43 definition S6 ≝ λRP,C:lenv→predicate term.
44 ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
45 ∀a,V,T. C (L.ⓓV) (ⒶV2s.T) → RP L V → C L (ⒶV1s.ⓓ{a}V.T).
47 definition S7 ≝ λC:lenv→predicate term.
48 ∀L,Vs,T,W. C L (ⒶVs.T) → C L (ⒶVs.W) → C L (ⒶVs.ⓝW.T).
50 definition S8 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e.
51 C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2.
53 definition S8s ≝ λC:lenv→predicate term.
54 ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
55 ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
57 (* properties of the abstract candidate of reducibility *)
58 record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝
69 (* the abstract candidate of reducibility associated to an atomic arity *)
70 let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
73 | APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
78 "candidate of reducibility of an atomic arity (abstract)"
79 'InEInt RP L T A = (aacr RP A L T).
81 (* Basic properties *********************************************************)
83 (* Basic_1: was: sc3_lift1 *)
84 lemma acr_lifts: ∀C. S8 C → S8s C.
85 #C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
87 <(lifts_inv_nil … H) -H //
88 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
89 elim (lifts_inv_cons … H) -H /3 width=9/
93 lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
94 ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 →
96 #RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV
101 (* Basic_1: was only: sns3_lifts1 *)
102 lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
103 ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[des] L0 ≡ L →
104 all … (RP L) Vs → all … (RP L0) V0s.
105 #RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
106 #T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s
107 @conj /2 width=1/ /2 width=6 by rp_lifts/
111 sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
113 lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
114 ∀A. acr RR RS RP (aacr RP A).
115 #RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
116 #B #A #IHB #IHA @mk_acr normalize
118 elim (cp1 … H1RP L) #k #HK
119 lapply (H ? (⋆k) ? ⟠ ? ? ?) -H
122 | #H @(cp3 … H1RP … k) @(s1 … IHA) //
124 | #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
125 elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
126 lapply (s1 … IHB … HB) #HV0
127 @(s2 … IHA … (V0 @ V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/
128 | #a #L #Vs #U #T #W #HA #L0 #V0 #X #des #HB #HL0 #H
129 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
130 elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
131 elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
132 @(s3 … IHA … (V0 @ V0s)) /5 width=5/
133 | #L #Vs #HVs #k #L0 #V0 #X #hdes #HB #HL0 #H
134 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
135 >(lifts_inv_sort1 … HY) -Y
136 lapply (s1 … IHB … HB) #HV0
137 @(s4 … IHA … (V0 @ V0s)) /3 width=6/
138 | #I #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
139 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
140 elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
141 elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
142 >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
143 elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
144 elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
145 elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
146 >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
147 @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4/
148 | #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
149 elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
150 elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
151 elim (lift_total V10 0 1) #V20 #HV120
152 elim (liftv_total 0 1 V10s) #V20s #HV120s
153 @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /2 width=1/ /3 width=6 by rp_lifts/
154 @(HA … (des + 1)) /2 width=1/
155 [ @(s8 … IHB … HB … HV120) /2 width=1/
157 elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
158 >(liftv_mono … HV12s … HV10s) -V1s //
160 | #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
161 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
162 elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
163 @(s7 … IHA … (V0 @ V0s)) /3 width=4/
168 lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
169 ∀a,L,W,T,A,B. ⦃L, W⦄ ϵ[RP] 〚B〛 → (
170 ∀L0,V0,W0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des ] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
171 ⦃L0, V0⦄ ϵ[RP] 〚B〛 → ⦃L0, W0⦄ ϵ[RP] 〚B〛 → ⦃L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
173 ⦃L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
174 #RR #RS #RP #H1RP #H2RP #a #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
175 lapply (aacr_acr … H1RP H2RP A) #HCA
176 lapply (aacr_acr … H1RP H2RP B) #HCB
177 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
178 lapply (acr_lifts … HL0 … HW0 HW) -HW [ @(s8 … HCB) ] #HW0
182 | lapply (s1 … HCB) -HCB #HCB
183 @(cp4 … H1RP) /2 width=1/
187 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
188 (* Basic_1: removed local theorems 1: sc3_sn3_abst *)