1 (**************************************************************************)
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 *)
13 (**************************************************************************)
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 (* Note: this is Girard's CR1 *)
25 definition S1 ≝ λRP,C:candidate.
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:candidate.
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:candidate.
36 C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
38 definition S4 ≝ λRP,C:candidate.
39 ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
41 definition S5 ≝ λC:candidate. ∀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:candidate.
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:candidate.
50 ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
52 (* requirements for the generic reducibility candidate *)
53 record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
63 (* the functional construction for candidates *)
64 definition cfun: candidate → candidate → candidate ≝
65 λC1,C2,G,K,T. ∀L,W,U,cs.
66 ⬇*[Ⓕ, cs] L ≡ K → ⬆*[cs] T ≡ U → C1 G L W → C2 G L (ⓐW.U).
68 (* the reducibility candidate associated to an atomic arity *)
69 let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
72 | APair B A ⇒ cfun (acr RP B) (acr RP A)
76 "candidate of reducibility of an atomic arity (abstract)"
77 'InEInt RP G L T A = (acr RP A G L T).
79 (* Basic properties *********************************************************)
81 (* Basic 1: was: sc3_lift *)
82 lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G) (Ⓕ).
83 #RR #RS #RP #H #A elim A -A
84 /3 width=8 by cp2, drops_cons, lifts_cons/
87 (* Basic_1: was: sc3_lift1 *)
88 lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftables1 (acr RP A G) (Ⓕ).
89 #RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/
93 sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
95 lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
96 ∀A. gcr RR RS RP (acr RP A).
97 #RR #RS #RP #H1RP #H2RP #A elim A -A //
98 #B #A #IHB #IHA @mk_gcr
100 elim (cp1 … H1RP G L) #k #HK
101 lapply (H L (⋆k) T (◊) ? ? ?) -H //
102 [ lapply (s2 … IHB G L (◊) … HK) //
103 | /3 width=6 by s1, cp3/
105 | #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
106 elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
107 lapply (s1 … IHB … HB) #HV0
108 @(s2 … IHA … (V0 @ V0s))
109 /3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
110 | #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
111 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
112 elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
113 elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
114 @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
115 | #G #L #Vs #HVs #k #L0 #V0 #X #cs #HL0 #H #HB
116 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
117 >(lifts_inv_sort1 … HY) -Y
118 lapply (s1 … IHB … HB) #HV0
119 @(s4 … IHA … (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/
120 | #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
121 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
122 elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
123 elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
124 >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
125 elim (drops_inv_skip2 … Hcs0 … H) -H -cs0 #L2 #W1 #cs0 #Hcs0 #HLK #HVW1 #H destruct
126 elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
127 elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
128 >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
129 @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
130 | #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
131 elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
132 elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
133 elim (lift_total V10 0 1) #V20 #HV120
134 elim (liftv_total 0 1 V10s) #V20s #HV120s
135 @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
136 @(HA … (cs + 1)) /2 width=2 by drops_skip/
138 elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
139 >(liftv_mono … HV12s … HV10s) -V1s //
140 | @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
142 | #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
143 elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
144 elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
145 @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
149 lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
150 ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
151 ∀L0,V0,W0,T0,cs. ⬇*[Ⓕ, cs] L0 ≡ L → ⬆*[cs] W ≡ W0 → ⬆*[cs + 1] T ≡ T0 →
152 ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
154 ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
155 #RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #cs #HL0 #H #HB
156 lapply (acr_gcr … H1RP H2RP A) #HCA
157 lapply (acr_gcr … H1RP H2RP B) #HCB
158 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
159 lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
160 lapply (s3 … HCA … a G L0 (◊)) #H @H -H
161 lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
163 | lapply (s1 … HCB) -HCB #HCB
164 lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
168 (* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
169 (* Basic_1: removed local theorems 1: sc3_sn3_abst *)