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 "ground_2/xoa/ex_3_5.ma".
16 include "ground_2/xoa/ex_5_7.ma".
17 include "basic_2/rt_transition/cpm_lsubr.ma".
18 include "basic_2/rt_computation/cpms_drops.ma".
19 include "basic_2/rt_computation/cprs.ma".
21 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
23 (* Main properties **********************************************************)
25 (* Basic_2A1: includes: cprs_bind *)
26 theorem cpms_bind (n) (h) (G) (L):
27 ∀I,V1,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[n,h] T2 →
28 ∀V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
29 ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[n,h] ⓑ[p,I]V2.T2.
30 #n #h #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
31 [ /2 width=1 by cpms_bind_dx/
32 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
33 /3 width=3 by cpr_pair_sn, cpms_step_dx/
37 theorem cpms_appl (n) (h) (G) (L):
38 ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
39 ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
40 ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
41 #n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
42 [ /2 width=1 by cpms_appl_dx/
43 | #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
44 /3 width=3 by cpr_pair_sn, cpms_step_dx/
48 (* Basic_2A1: includes: cprs_beta_rc *)
49 theorem cpms_beta_rc (n) (h) (G) (L):
50 ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
51 ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
52 ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
53 ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
54 #n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
55 [ /2 width=1 by cpms_beta_dx/
56 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
57 /4 width=3 by cpr_pair_sn, cpms_step_dx/
61 (* Basic_2A1: includes: cprs_beta *)
62 theorem cpms_beta (n) (h) (G) (L):
63 ∀W1,T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
64 ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
65 ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h] V2 →
66 ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
67 #n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
68 [ /2 width=1 by cpms_beta_rc/
69 | #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
70 /4 width=5 by cpms_step_dx, cpr_pair_sn, cpm_cast/
74 (* Basic_2A1: includes: cprs_theta_rc *)
75 theorem cpms_theta_rc (n) (h) (G) (L):
76 ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V → ∀V2. ⇧*[1] V ≘ V2 →
77 ∀W1,T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
78 ∀W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
79 ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
80 #n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
81 [ /2 width=3 by cpms_theta_dx/
82 | #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
83 /3 width=3 by cpr_pair_sn, cpms_step_dx/
87 (* Basic_2A1: includes: cprs_theta *)
88 theorem cpms_theta (n) (h) (G) (L):
89 ∀V,V2. ⇧*[1] V ≘ V2 → ∀W1,W2. ❪G,L❫ ⊢ W1 ➡*[h] W2 →
90 ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
91 ∀V1. ❪G,L❫ ⊢ V1 ➡*[h] V →
92 ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
93 #n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
94 [ /2 width=3 by cpms_theta_rc/
95 | #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
96 /3 width=3 by cpr_pair_sn, cpms_step_sn/
100 (* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *)
101 theorem cpms_trans (h) (G) (L):
102 ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T →
103 ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
104 /2 width=3 by ltc_trans/ qed-.
106 (* Basic_2A1: uses: scpds_cprs_trans *)
107 theorem cpms_cprs_trans (n) (h) (G) (L):
108 ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[n,h] T →
109 ∀T2. ❪G,L❫ ⊢ T ➡*[h] T2 → ❪G,L❫ ⊢ T1 ➡*[n,h] T2.
110 #n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
111 /2 width=3 by cpms_trans/ qed-.
113 (* Advanced inversion lemmas ************************************************)
115 lemma cpms_inv_appl_sn (n) (h) (G) (L):
116 ∀V1,T1,X2. ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
118 ❪G,L❫ ⊢ V1 ➡*[h] V2 & ❪G,L❫ ⊢ T1 ➡*[n,h] T2 &
121 ❪G,L❫ ⊢ T1 ➡*[n1,h] ⓛ[p]W.T & ❪G,L❫ ⊢ ⓓ[p]ⓝW.V1.T ➡*[n2,h] X2 &
123 | ∃∃n1,n2,p,V0,V2,V,T.
124 ❪G,L❫ ⊢ V1 ➡*[h] V0 & ⇧*[1] V0 ≘ V2 &
125 ❪G,L❫ ⊢ T1 ➡*[n1,h] ⓓ[p]V.T & ❪G,L❫ ⊢ ⓓ[p]V.ⓐV2.T ➡*[n2,h] X2 &
127 #n #h #G #L #V1 #T1 #U2 #H
128 @(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
129 #n1 #n2 #U #U2 #_ * *
130 [ #V0 #T0 #HV10 #HT10 #H #HU2 destruct
131 elim (cpm_inv_appl1 … HU2) -HU2 *
132 [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cpms_step_dx, or3_intro0, ex3_2_intro/
133 | #p #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
134 lapply (cprs_step_dx … HV10 … HV02) -V0 #HV12
135 lapply (lsubr_cpm_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
136 /5 width=8 by cprs_flat_dx, cpms_bind, cpm_cpms, lsubr_beta, ex3_5_intro, or3_intro1/
137 | #p #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
138 /6 width=12 by cprs_step_dx, cpm_cpms, cpm_appl, cpm_bind, ex5_7_intro, or3_intro2/
140 | #m1 #m2 #p #W #T #HT1 #HTU #H #HU2 destruct
141 lapply (cpms_step_dx … HTU … HU2) -U #H
142 @or3_intro1 @(ex3_5_intro … HT1 H) // (**) (* auto fails *)
143 | #m1 #m2 #p #V2 #W2 #V #T #HV12 #HVW2 #HT1 #HTU #H #HU2 destruct
144 lapply (cpms_step_dx … HTU … HU2) -U #H
145 @or3_intro2 @(ex5_7_intro … HV12 HVW2 HT1 H) // (**) (* auto fails *)
149 lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2 →
150 ∃∃T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T & ❪G,L❫ ⊢ T ➡*[n2,h] T2.
151 #h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/
152 #n1 #IH #n2 #T1 #T2 <plus_S1 #H
153 elim (cpms_inv_succ_sn … H) -H #T0 #HT10 #HT02
154 elim (IH … HT02) -IH -HT02 #T #HT0 #HT2
155 lapply (cpms_trans … HT10 … HT0) -T0 #HT1
156 /2 width=3 by ex2_intro/
159 (* Advanced main properties *************************************************)
161 theorem cpms_cast (n) (h) (G) (L):
162 ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
163 ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[n,h] U2 →
164 ❪G,L❫ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
165 #n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
166 [ /3 width=3 by cpms_cast_sn/
167 | #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H
168 elim (cpms_inv_plus … H) -H #U #HU1 #HU2
169 /3 width=3 by cpms_trans, cpms_cast_sn/
173 theorem cpms_trans_swap (h) (G) (L) (T1):
174 ∀n1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T → ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 →
175 ∃∃T0. ❪G,L❫ ⊢ T1 ➡*[n2,h] T0 & ❪G,L❫ ⊢ T0 ➡*[n1,h] T2.
176 #h #G #L #T1 #n1 #T #HT1 #n2 #T2 #HT2
177 lapply (cpms_trans … HT1 … HT2) -T <commutative_plus #HT12
178 /2 width=1 by cpms_inv_plus/