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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 *)
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/unfold/tpss.ma".
17 (* PARALLEL UNFOLD ON LOCAL ENVIRONMENTS ************************************)
19 (* Basic_1: includes: csubst1_bind *)
20 inductive ltpss: nat → nat → relation lenv ≝
21 | ltpss_atom : ∀d,e. ltpss d e (⋆) (⋆)
22 | ltpss_pair : ∀L,I,V. ltpss 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
23 | ltpss_tpss2: ∀L1,L2,I,V1,V2,e.
24 ltpss 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶* V2 →
25 ltpss 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
26 | ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
27 ltpss d e L1 L2 → L2 ⊢ V1 [d, e] ▶* V2 →
28 ltpss (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
31 interpretation "parallel unfold (local environment)"
32 'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
34 (* Basic properties *********************************************************)
36 lemma ltpss_tps2: ∀L1,L2,I,V1,V2,e.
37 L1 [0, e] ▶* L2 → L2 ⊢ V1 [0, e] ▶ V2 →
38 L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
41 lemma ltpss_tps1: ∀L1,L2,I,V1,V2,d,e.
42 L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶ V2 →
43 L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
46 lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
47 L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶* V2 →
48 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
49 #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
50 >(plus_minus_m_m e 1) /2 width=1/
53 lemma ltpss_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
54 L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶* V2 →
55 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
56 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
57 >(plus_minus_m_m d 1) /2 width=1/
60 lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
61 L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
62 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
65 lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
66 L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 →
67 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
70 (* Basic_1: was by definition: csubst1_refl *)
71 lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
73 #L #I #V #IHL * /2 width=1/ * /2 width=1/
76 lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
77 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
78 // /3 width=2/ /3 width=3/
81 (* Basic forward lemmas *****************************************************)
83 lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
84 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
88 (* Basic inversion lemmas ***************************************************)
90 fact ltpss_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → e = 0 → L1 = L2.
91 #d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
92 [ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
93 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
94 >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
98 lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
101 fact ltpss_inv_atom1_aux: ∀d,e,L1,L2.
102 L1 [d, e] ▶* L2 → L1 = ⋆ → L2 = ⋆.
103 #d #e #L1 #L2 * -d -e -L1 -L2
105 | #L #I #V #H destruct
106 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
107 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
111 lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
114 fact ltpss_inv_tpss21_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → d = 0 → 0 < e →
115 ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
116 ∃∃K2,V2. K1 [0, e - 1] ▶* K2 &
117 K2 ⊢ V1 [0, e - 1] ▶* V2 &
119 #d #e #L1 #L2 * -d -e -L1 -L2
120 [ #d #e #_ #_ #K1 #I #V1 #H destruct
121 | #L1 #I #V #_ #H elim (lt_refl_false … H)
122 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
123 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
127 lemma ltpss_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶* L2 → 0 < e →
128 ∃∃K2,V2. K1 [0, e - 1] ▶* K2 & K2 ⊢ V1 [0, e - 1] ▶* V2 &
132 fact ltpss_inv_tpss11_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → 0 < d →
133 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
134 ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
135 K2 ⊢ V1 [d - 1, e] ▶* V2 &
137 #d #e #L1 #L2 * -d -e -L1 -L2
138 [ #d #e #_ #I #K1 #V1 #H destruct
139 | #L #I #V #H elim (lt_refl_false … H)
140 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
141 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
145 lemma ltpss_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶* L2 → 0 < d →
146 ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
147 K2 ⊢ V1 [d - 1, e] ▶* V2 &
151 fact ltpss_inv_atom2_aux: ∀d,e,L1,L2.
152 L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
153 #d #e #L1 #L2 * -d -e -L1 -L2
155 | #L #I #V #H destruct
156 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
157 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
161 lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
164 fact ltpss_inv_tpss22_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → d = 0 → 0 < e →
165 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
166 ∃∃K1,V1. K1 [0, e - 1] ▶* K2 &
167 K2 ⊢ V1 [0, e - 1] ▶* V2 &
169 #d #e #L1 #L2 * -d -e -L1 -L2
170 [ #d #e #_ #_ #K1 #I #V1 #H destruct
171 | #L1 #I #V #_ #H elim (lt_refl_false … H)
172 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
173 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
177 lemma ltpss_inv_tpss22: ∀e,L1,K2,I,V2. L1 [0, e] ▶* K2. ⓑ{I} V2 → 0 < e →
178 ∃∃K1,V1. K1 [0, e - 1] ▶* K2 & K2 ⊢ V1 [0, e - 1] ▶* V2 &
182 fact ltpss_inv_tpss12_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → 0 < d →
183 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
184 ∃∃K1,V1. K1 [d - 1, e] ▶* K2 &
185 K2 ⊢ V1 [d - 1, e] ▶* V2 &
187 #d #e #L1 #L2 * -d -e -L1 -L2
188 [ #d #e #_ #I #K2 #V2 #H destruct
189 | #L #I #V #H elim (lt_refl_false … H)
190 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
191 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
195 lemma ltpss_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶* K2. ⓑ{I} V2 → 0 < d →
196 ∃∃K1,V1. K1 [d - 1, e] ▶* K2 &
197 K2 ⊢ V1 [d - 1, e] ▶* V2 &
201 (* Basic_1: removed theorems 27:
202 csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
203 csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
204 csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
205 csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
206 csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
207 csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
208 csubst1_head csubst1_flat csubst1_gen_head
209 csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1