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 notation "hvbox( T1 break ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
16 non associative with precedence 45
17 for @{ 'PSubstStar $T1 $d $e $T2 }.
19 include "basic_2/unfold/tpss.ma".
21 (* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
23 (* Basic_1: includes: csubst1_bind *)
24 inductive ltpss_dx: nat → nat → relation lenv ≝
25 | ltpss_dx_atom : ∀d,e. ltpss_dx d e (⋆) (⋆)
26 | ltpss_dx_pair : ∀L,I,V. ltpss_dx 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
27 | ltpss_dx_tpss2: ∀L1,L2,I,V1,V2,e.
28 ltpss_dx 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
29 ltpss_dx 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
30 | ltpss_dx_tpss1: ∀L1,L2,I,V1,V2,d,e.
31 ltpss_dx d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
32 ltpss_dx (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
35 interpretation "parallel unfold (local environment, dx variant)"
36 'PSubstStar L1 d e L2 = (ltpss_dx d e L1 L2).
38 (* Basic inversion lemmas ***************************************************)
40 fact ltpss_dx_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
41 #d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
42 [ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
43 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
44 >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
48 lemma ltpss_dx_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
51 fact ltpss_dx_inv_atom1_aux: ∀d,e,L1,L2.
52 L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
53 #d #e #L1 #L2 * -d -e -L1 -L2
55 | #L #I #V #H destruct
56 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
57 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
61 lemma ltpss_dx_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
64 fact ltpss_dx_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
65 ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
66 ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
67 K2 ⊢ V1 ▶* [0, e - 1] V2 &
69 #d #e #L1 #L2 * -d -e -L1 -L2
70 [ #d #e #_ #_ #K1 #I #V1 #H destruct
71 | #L1 #I #V #_ #H elim (lt_refl_false … H)
72 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
73 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
77 lemma ltpss_dx_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
78 ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
79 K2 ⊢ V1 ▶* [0, e - 1] V2 &
83 fact ltpss_dx_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
84 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
85 ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
86 K2 ⊢ V1 ▶* [d - 1, e] V2 &
88 #d #e #L1 #L2 * -d -e -L1 -L2
89 [ #d #e #_ #I #K1 #V1 #H destruct
90 | #L #I #V #H elim (lt_refl_false … H)
91 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
92 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
96 lemma ltpss_dx_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
97 ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
98 K2 ⊢ V1 ▶* [d - 1, e] V2 &
102 fact ltpss_dx_inv_atom2_aux: ∀d,e,L1,L2.
103 L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
104 #d #e #L1 #L2 * -d -e -L1 -L2
106 | #L #I #V #H destruct
107 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
108 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
112 lemma ltpss_dx_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
115 fact ltpss_dx_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
116 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
117 ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
118 K2 ⊢ V1 ▶* [0, e - 1] V2 &
120 #d #e #L1 #L2 * -d -e -L1 -L2
121 [ #d #e #_ #_ #K1 #I #V1 #H destruct
122 | #L1 #I #V #_ #H elim (lt_refl_false … H)
123 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
124 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
128 lemma ltpss_dx_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
129 ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
130 K2 ⊢ V1 ▶* [0, e - 1] V2 &
134 fact ltpss_dx_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
135 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
136 ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
137 K2 ⊢ V1 ▶* [d - 1, e] V2 &
139 #d #e #L1 #L2 * -d -e -L1 -L2
140 [ #d #e #_ #I #K2 #V2 #H destruct
141 | #L #I #V #H elim (lt_refl_false … H)
142 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
143 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
147 lemma ltpss_dx_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
148 ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
149 K2 ⊢ V1 ▶* [d - 1, e] V2 &
153 (* Basic properties *********************************************************)
155 lemma ltpss_dx_tps2: ∀L1,L2,I,V1,V2,e.
156 L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
157 L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
160 lemma ltpss_dx_tps1: ∀L1,L2,I,V1,V2,d,e.
161 L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
162 L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
165 lemma ltpss_dx_tpss2_lt: ∀L1,L2,I,V1,V2,e.
166 L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
167 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
168 #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
169 >(plus_minus_m_m e 1) /2 width=1/
172 lemma ltpss_dx_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
173 L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
174 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
175 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
176 >(plus_minus_m_m d 1) /2 width=1/
179 lemma ltpss_dx_tps2_lt: ∀L1,L2,I,V1,V2,e.
180 L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
181 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
184 lemma ltpss_dx_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
185 L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
186 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
189 (* Basic_1: was by definition: csubst1_refl *)
190 lemma ltpss_dx_refl: ∀L,d,e. L ▶* [d, e] L.
192 #L #I #V #IHL * /2 width=1/ * /2 width=1/
195 lemma ltpss_dx_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
196 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
197 #L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
198 [ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
199 lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
200 lapply (lt_to_le_to_lt 0 … Hde2) // #He2
201 lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
202 | #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
203 >plus_plus_comm_23 in Hde12; #Hde12
204 elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
205 [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
206 lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
207 | -Hd21 normalize in Hde12;
208 lapply (lt_to_le_to_lt 0 … Hde12) // #He2
209 lapply (le_plus_to_minus_r … Hde12) -Hde12
210 /3 width=5 by ltpss_dx_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
215 lemma ltpss_dx_weak_full: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
216 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
217 // /3 width=2/ /3 width=3/
220 fact ltpss_dx_append_le_aux: ∀K1,K2,d,x. K1 ▶* [d, x] K2 → x = |K1| - d →
221 ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
222 L1 @@ K1 ▶* [d, x + e] L2 @@ K2.
223 #K1 #K2 #d #x #H elim H -K1 -K2 -d -x
224 [ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
225 lapply (le_n_O_to_eq … H2) -H2 #H destruct //
226 | #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
227 | #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
228 lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
229 /4 width=5 by ltpss_dx_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
230 | #K1 #K2 #I #V1 #V2 #d #x #_ #HV12 #IHK12 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #H1 #L1 #L2 #e #HL12 #H2 destruct
231 lapply (le_plus_to_le_r … H2) -H2 #Hd
232 /4 width=5 by ltpss_dx_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
236 lemma ltpss_dx_append_le: ∀K1,K2,d. K1 ▶* [d, |K1| - d] K2 →
237 ∀L1,L2,e. L1 ▶* [0, e] L2 → d ≤ |K1| →
238 L1 @@ K1 ▶* [d, |K1| - d + e] L2 @@ K2.
239 /2 width=1 by ltpss_dx_append_le_aux/ qed.
241 lemma ltpss_dx_append_zero: ∀K1,K2. K1 ▶* [0, |K1|] K2 →
242 ∀L1,L2,e. L1 ▶* [0, e] L2 →
243 L1 @@ K1 ▶* [0, |K1| + e] L2 @@ K2.
246 lemma ltpss_dx_append_ge: ∀K1,K2,d,e. K1 ▶* [d, e] K2 →
247 ∀L1,L2. L1 ▶* [d - |K1|, e] L2 → |K1| ≤ d →
248 L1 @@ K1 ▶* [d, e] L2 @@ K2.
249 #K1 #K2 #d #e #H elim H -K1 -K2 -d -e
250 [ #d #e #L1 #L2 <minus_n_O //
251 | #K #I #V #L1 #L2 #_ #H
252 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
253 | #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
254 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
255 | #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
256 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
257 lapply (le_plus_to_le_r … H) -H /3 width=1/
261 (* Basic forward lemmas *****************************************************)
263 lemma ltpss_dx_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
264 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
268 (* Basic_1: removed theorems 28:
269 csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
270 csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
271 csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
272 csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
273 csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
274 csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
275 csubst1_head csubst1_flat csubst1_gen_head
276 csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1