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 *)
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15 include "basic_2/computation/cprs_lift.ma".
16 include "basic_2/computation/cprs_cprs.ma".
17 include "basic_2/conversion/cpc_cpc.ma".
18 include "basic_2/equivalence/cpcs_cprs.ma".
20 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
22 (* Advanced inversion lemmas ************************************************)
24 lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
25 ∃∃T. L ⊢ T1 ➡* T & L ⊢ T2 ➡* T.
26 #L #T1 #T2 #H @(cpcs_ind … H) -T2
28 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
29 [ elim (cprs_strip … HT0 … HT2) -T #T #HT0 #HT2
30 lapply (cprs_strap1 … HT10 … HT0) -T0 /2 width=3/
31 | lapply (cprs_strap2 … HT2 … HT0) -T /2 width=3/
36 (* Basic_1: was: pc3_gen_sort *)
37 lemma cpcs_inv_sort: ∀L,k1,k2. L ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
39 elim (cpcs_inv_cprs … H) -H #T #H1
40 >(cprs_inv_sort1 … H1) -T #H2
41 lapply (cprs_inv_sort1 … H2) -L #H destruct //
44 (* Basic_1: was: pc3_gen_sort_abst *)
45 lemma cpcs_inv_sort_abst: ∀L,W,T,k. L ⊢ ⋆k ⬌* ⓛW.T → ⊥.
47 elim (cpcs_inv_cprs … H) -H #X #H1
48 >(cprs_inv_sort1 … H1) -X #H2
49 elim (cprs_inv_abst1 Abst W … H2) -H2 #W0 #T0 #_ #_ #H destruct
52 (* Basic_1: was: pc3_gen_abst *)
53 lemma cpcs_inv_abst: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → ∀I,V.
54 L ⊢ W1 ⬌* W2 ∧ L. ②{I}V ⊢ T1 ⬌* T2.
55 #L #W1 #W2 #T1 #T2 #H #I #V
56 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
57 elim (cprs_inv_abst1 I V … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
58 elim (cprs_inv_abst1 I V … H2) -H2 #W #T #HW2 #HT2 #H destruct /3 width=3/
61 (* Basic_1: was: pc3_gen_abst_shift *)
62 lemma cpcs_inv_abst_shift: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → ∀W.
63 L ⊢ W1 ⬌* W2 ∧ L. ⓛW ⊢ T1 ⬌* T2.
64 #L #W1 #W2 #T1 #T2 #H #W
65 lapply (cpcs_inv_abst … H Abst W) -H //
68 (* Basic_1: was: pc3_gen_lift *)
69 lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
70 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
71 L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
72 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
73 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
74 elim (cprs_inv_lift … HLK … HTU1 … HU1) -U1 #T #HTU #HT1
75 elim (cprs_inv_lift … HLK … HTU2 … HU2) -L -U2 #X #HXU
76 >(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/
79 (* Advanced properties ******************************************************)
81 (* Basic_1: was only: pc3_thin_dx *)
82 lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
83 ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
84 #L #V1 #V2 #HV12 #T1 #T2 #HT12 #I
85 elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2
86 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
89 lemma cpcs_abst: ∀L,V1,V2. L ⊢ V1 ⬌* V2 →
90 ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛV1. T1 ⬌* ⓛV2. T2.
91 #L #V1 #V2 #HV12 #V #T1 #T2 #HT12
92 elim (cpcs_inv_cprs … HV12) -HV12
93 elim (cpcs_inv_cprs … HT12) -HT12
94 /3 width=6 by cprs_div, cprs_abst/ (**) (* /3 width=6/ is a bit slow *)
97 lemma cpcs_abbr_dx: ∀L,V,T1,T2. L.ⓓV ⊢ T1 ⬌* T2 → L ⊢ ⓓV. T1 ⬌* ⓓV. T2.
99 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *)
102 lemma cpcs_bind_dx: ∀I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 →
103 L ⊢ ⓑ{I}V. T1 ⬌* ⓑ{I}V. T2.
104 * /2 width=1/ /2 width=2/ qed.
106 lemma cpcs_abbr_sn: ∀L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓓV1. T ⬌* ⓓV2. T.
108 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_abbr1/ (**) (* /3 width=5/ is a bit slow *)
111 (* Basic_1: was: pc3_lift *)
112 lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
113 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
114 K ⊢ T1 ⬌* T2 → L ⊢ U1 ⬌* U2.
115 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
116 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
117 elim (lift_total T d e) #U #HTU
118 lapply (cprs_lift … HLK … HTU1 … HT1 … HTU) -T1 #HU1
119 lapply (cprs_lift … HLK … HTU2 … HT2 … HTU) -K -T2 -T -d -e /2 width=3/
122 lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 →
123 ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0.
126 (* Main properties **********************************************************)
128 (* Basic_1: was pc3_t *)
129 theorem cpcs_trans: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
132 theorem cpcs_canc_sn: ∀L,T,T1,T2. L ⊢ T ⬌* T1 → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
133 /3 width=3 by cpcs_trans, cprs_comm/ qed. (**) (* /3 width=3/ is too slow *)
135 theorem cpcs_canc_dx: ∀L,T,T1,T2. L ⊢ T1 ⬌* T → L ⊢ T2 ⬌* T → L ⊢ T1 ⬌* T2.
136 /3 width=3 by cpcs_trans, cprs_comm/ qed. (**) (* /3 width=3/ is too slow *)
138 lemma cpcs_abbr1: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV1 ⊢ T1 ⬌* T2 →
139 L ⊢ ⓓV1. T1 ⬌* ⓓV2. T2.
140 #L #V1 #V2 #HV12 #T1 #T2 #HT12
141 @(cpcs_trans … (ⓓV1.T2)) /2 width=1/
144 lemma cpcs_abbr2: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓓV2 ⊢ T1 ⬌* T2 →
145 L ⊢ ⓓV1. T1 ⬌* ⓓV2. T2.
146 #L #V1 #V2 #HV12 #T1 #T2 #HT12
147 @(cpcs_trans … (ⓓV2.T1)) /2 width=1/