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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_2A/notation/relations/ineint_5.ma".
16 include "basic_2A/grammar/aarity.ma".
17 include "basic_2A/multiple/lifts_lift_vector.ma".
18 include "basic_2A/multiple/drops_drop.ma".
19 include "basic_2A/computation/gcp.ma".
21 (* GENERIC COMPUTATION PROPERTIES *******************************************)
23 (* Note: this is Girard's CR1 *)
24 definition S1 ≝ λRP,C:candidate.
25 ∀G,L,T. C G L T → RP G L T.
27 (* Note: this is Tait's iii, or Girard's CR4 *)
28 definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
29 ∀G,L,Vs. all … (RP G L) Vs →
30 ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
32 (* Note: this generalizes Tait's ii *)
33 definition S3 ≝ λC:candidate.
35 C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
37 definition S4 ≝ λRP,C:candidate.
38 ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
40 definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
41 C G L (ⒶVs.V2) → ⬆[0, i+1] V1 ≡ V2 →
42 ⬇[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
44 definition S6 ≝ λRP,C:candidate.
45 ∀G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
46 ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
48 definition S7 ≝ λC:candidate.
49 ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
51 (* requirements for the generic reducibility candidate *)
52 record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
62 (* the functional construction for candidates *)
63 definition cfun: candidate → candidate → candidate ≝
64 λC1,C2,G,K,T. ∀L,W,U,cs.
65 ⬇*[Ⓕ, cs] L ≡ K → ⬆*[cs] T ≡ U → C1 G L W → C2 G L (ⓐW.U).
67 (* the reducibility candidate associated to an atomic arity *)
68 let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
71 | APair B A ⇒ cfun (acr RP B) (acr RP A)
75 "candidate of reducibility of an atomic arity (abstract)"
76 'InEInt RP G L T A = (acr RP A G L T).
78 (* Basic properties *********************************************************)
80 (* Basic 1: was: sc3_lift *)
81 lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G) (Ⓕ).
82 #RR #RS #RP #H #A elim A -A
83 /3 width=8 by cp2, drops_cons, lifts_cons/
86 (* Basic_1: was: sc3_lift1 *)
87 lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftables1 (acr RP A G) (Ⓕ).
88 #RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/
92 sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
94 lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
95 ∀A. gcr RR RS RP (acr RP A).
96 #RR #RS #RP #H1RP #H2RP #A elim A -A //
97 #B #A #IHB #IHA @mk_gcr
99 elim (cp1 … H1RP G L) #k #HK
100 lapply (H L (⋆k) T (𝐞) ? ? ?) -H //
101 [ lapply (s2 … IHB G L (ⓔ) … HK) //
102 | /3 width=6 by s1, cp3/
104 | #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
105 elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
106 lapply (s1 … IHB … HB) #HV0
107 @(s2 … IHA … (V0 ⨮ V0s))
108 /3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
109 | #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
110 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
111 elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
112 elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
113 @(s3 … IHA … (V0 ⨮ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
114 | #G #L #Vs #HVs #k #L0 #V0 #X #cs #HL0 #H #HB
115 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
116 >(lifts_inv_sort1 … HY) -Y
117 lapply (s1 … IHB … HB) #HV0
118 @(s4 … IHA … (V0 ⨮ V0s)) /3 width=7 by gcp2_lifts_all, conj/
119 | #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
120 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
121 elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
122 elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
123 >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
124 elim (drops_inv_skip2 … Hcs0 … H) -H -cs0 #L2 #W1 #cs0 #Hcs0 #HLK #HVW1 #H destruct
125 elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
126 elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
127 >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
128 @(s5 … IHA … (V0 ⨮ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
129 | #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
130 elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
131 elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
132 elim (lift_total V10 0 1) #V20 #HV120
133 elim (liftv_total 0 1 V10s) #V20s #HV120s
134 @(s6 … IHA … (V10 ⨮ V10s) (V20 ⨮ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
135 @(HA … (cs + 1)) /2 width=2 by drops_skip/
137 elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
138 >(liftv_mono … HV12s … HV10s) -V1s //
139 | @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
141 | #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
142 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
143 elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
144 @(s7 … IHA … (V0 ⨮ V0s)) /3 width=5 by lifts_applv/
148 lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
149 ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
150 ∀L0,V0,W0,T0,cs. ⬇*[Ⓕ, cs] L0 ≡ L → ⬆*[cs] W ≡ W0 → ⬆*[cs + 1] T ≡ T0 →
151 ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
153 ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
154 #RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #cs #HL0 #H #HB
155 lapply (acr_gcr … H1RP H2RP A) #HCA
156 lapply (acr_gcr … H1RP H2RP B) #HCB
157 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
158 lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
159 lapply (s3 … HCA … a G L0 (ⓔ)) #H @H -H
160 lapply (s6 … HCA G L0 (ⓔ) (ⓔ) ?) // #H @H -H
162 | lapply (s1 … HCB) -HCB #HCB
163 lapply (s7 … H2RP G L0 (ⓔ)) /3 width=1 by/
167 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
168 (* Basic_1: removed local theorems 1: sc3_sn3_abst *)