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/substitution/tps.ma".
17 (* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
19 (* Basic_1: includes: csubst1_bind *)
20 inductive ltps: nat → nat → relation lenv ≝
21 | ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
22 | ltps_pair: ∀L,I,V. ltps 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
23 | ltps_tps2: ∀L1,L2,I,V1,V2,e.
24 ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 →
25 ltps 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2
26 | ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
27 ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 →
28 ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
31 interpretation "parallel substritution (local environment)"
32 'PSubst L1 d e L2 = (ltps d e L1 L2).
34 (* Basic properties *********************************************************)
36 lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
37 L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
38 0 < e → L1. ⓑ{I} V1 [0, e] ▶ L2. ⓑ{I} V2.
39 #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
40 >(plus_minus_m_m e 1) /2 width=1/
43 lemma ltps_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
44 L1 [d - 1, e] ▶ L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 →
45 0 < d → L1. ⓑ{I} V1 [d, e] ▶ L2. ⓑ{I} V2.
46 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
47 >(plus_minus_m_m d 1) /2 width=1/
50 (* Basic_1: was by definition: csubst1_refl *)
51 lemma ltps_refl: ∀L,d,e. L [d, e] ▶ L.
53 #L #I #V #IHL * /2 width=1/ * /2 width=1/
56 lemma ltps_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → L1 [0, |L2|] ▶ L2.
57 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
58 // /3 width=2/ /3 width=3/
61 (* Basic forward lemmas *****************************************************)
63 lemma ltps_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → |L1| = |L2|.
64 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
68 (* Basic inversion lemmas ***************************************************)
70 fact ltps_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → e = 0 → L1 = L2.
71 #d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
72 [ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
73 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
74 >(IHL12 ?) -IHL12 // >(tps_inv_refl_O2 … HV12) //
78 lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶ L2 → L1 = L2.
81 fact ltps_inv_atom1_aux: ∀d,e,L1,L2.
82 L1 [d, e] ▶ L2 → L1 = ⋆ → L2 = ⋆.
83 #d #e #L1 #L2 * -d -e -L1 -L2
85 | #L #I #V #H destruct
86 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
87 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
91 lemma ltps_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶ L2 → L2 = ⋆.
94 fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
95 ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
96 ∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
97 K2 ⊢ V1 [0, e - 1] ▶ V2 &
99 #d #e #L1 #L2 * -d -e -L1 -L2
100 [ #d #e #_ #_ #K1 #I #V1 #H destruct
101 | #L1 #I #V #_ #H elim (lt_refl_false … H)
102 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
103 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
107 lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶ L2 → 0 < e →
108 ∃∃K2,V2. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
112 fact ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
113 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
114 ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
115 K2 ⊢ V1 [d - 1, e] ▶ V2 &
117 #d #e #L1 #L2 * -d -e -L1 -L2
118 [ #d #e #_ #I #K1 #V1 #H destruct
119 | #L #I #V #H elim (lt_refl_false … H)
120 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
121 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
125 lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶ L2 → 0 < d →
126 ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
127 K2 ⊢ V1 [d - 1, e] ▶ V2 &
131 fact ltps_inv_atom2_aux: ∀d,e,L1,L2.
132 L1 [d, e] ▶ L2 → L2 = ⋆ → L1 = ⋆.
133 #d #e #L1 #L2 * -d -e -L1 -L2
135 | #L #I #V #H destruct
136 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
137 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
141 lemma ltps_inv_atom2: ∀d,e,L1. L1 [d, e] ▶ ⋆ → L1 = ⋆.
144 fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
145 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
146 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
147 K2 ⊢ V1 [0, e - 1] ▶ V2 &
149 #d #e #L1 #L2 * -d -e -L1 -L2
150 [ #d #e #_ #_ #K1 #I #V1 #H destruct
151 | #L1 #I #V #_ #H elim (lt_refl_false … H)
152 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
153 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
157 lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
158 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
162 fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
163 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
164 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
165 K2 ⊢ V1 [d - 1, e] ▶ V2 &
167 #d #e #L1 #L2 * -d -e -L1 -L2
168 [ #d #e #_ #I #K2 #V2 #H destruct
169 | #L #I #V #H elim (lt_refl_false … H)
170 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
171 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
175 lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
176 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
177 K2 ⊢ V1 [d - 1, e] ▶ V2 &
181 (* Basic_1: removed theorems 27:
182 csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
183 csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
184 csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
185 csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
186 csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
187 csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
188 csubst1_head csubst1_flat csubst1_gen_head
189 csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1