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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 (* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
19 inductive ltpss_sn: nat → nat → relation lenv ≝
20 | ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
21 | ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
22 | ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
23 ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
24 ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
25 | ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
26 ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
27 ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
30 interpretation "parallel unfold (local environment, sn variant)"
31 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
33 (* Basic inversion lemmas ***************************************************)
35 fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
36 #d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
37 [ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
38 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
39 >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
43 lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
46 fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
47 L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
48 #d #e #L1 #L2 * -d -e -L1 -L2
50 | #L #I #V #H destruct
51 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
52 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
56 lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
59 fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
60 ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
61 ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
62 K1 ⊢ V1 ▶* [0, e - 1] V2 &
64 #d #e #L1 #L2 * -d -e -L1 -L2
65 [ #d #e #_ #_ #K1 #I #V1 #H destruct
66 | #L1 #I #V #_ #H elim (lt_refl_false … H)
67 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
68 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
72 lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
73 ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
74 K1 ⊢ V1 ▶* [0, e - 1] V2 &
78 fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
79 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
80 ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
81 K1 ⊢ V1 ▶* [d - 1, e] V2 &
83 #d #e #L1 #L2 * -d -e -L1 -L2
84 [ #d #e #_ #I #K1 #V1 #H destruct
85 | #L #I #V #H elim (lt_refl_false … H)
86 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
87 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
91 lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
92 ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
93 K1 ⊢ V1 ▶* [d - 1, e] V2 &
97 fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
98 L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
99 #d #e #L1 #L2 * -d -e -L1 -L2
101 | #L #I #V #H destruct
102 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
103 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
107 lemma ltpss_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
110 fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
111 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
112 ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
113 K1 ⊢ V1 ▶* [0, e - 1] V2 &
115 #d #e #L1 #L2 * -d -e -L1 -L2
116 [ #d #e #_ #_ #K1 #I #V1 #H destruct
117 | #L1 #I #V #_ #H elim (lt_refl_false … H)
118 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
119 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
123 lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
124 ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
125 K1 ⊢ V1 ▶* [0, e - 1] V2 &
129 fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
130 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
131 ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
132 K1 ⊢ V1 ▶* [d - 1, e] V2 &
134 #d #e #L1 #L2 * -d -e -L1 -L2
135 [ #d #e #_ #I #K2 #V2 #H destruct
136 | #L #I #V #H elim (lt_refl_false … H)
137 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
138 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
142 lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
143 ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
144 K1 ⊢ V1 ▶* [d - 1, e] V2 &
148 (* Basic properties *********************************************************)
150 lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
151 L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
152 L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
155 lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
156 L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
157 L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
160 lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
161 L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
162 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
163 #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
164 >(plus_minus_m_m e 1) /2 width=1/
167 lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
168 L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
169 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
170 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
171 >(plus_minus_m_m d 1) /2 width=1/
174 lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
175 L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
176 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
179 lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
180 L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
181 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
184 lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
186 #L #I #V #IHL * /2 width=1/ * /2 width=1/
189 lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
190 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
191 #L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
192 [ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
193 lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
194 lapply (lt_to_le_to_lt 0 … Hde2) // #He2
195 lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
196 | #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
197 >plus_plus_comm_23 in Hde12; #Hde12
198 elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
199 [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
200 lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
201 | -Hd21 normalize in Hde12;
202 lapply (lt_to_le_to_lt 0 … Hde12) // #He2
203 lapply (le_plus_to_minus_r … Hde12) -Hde12 /3 width=5/
208 lemma ltpss_sn_weak_all: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
209 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
210 // /3 width=2/ /3 width=3/
213 fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
214 ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
215 L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
216 #K1 #K2 #d #x #H elim H -K1 -K2 -d -x
217 [ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
218 lapply (le_n_O_to_eq … H2) -H2 #H destruct //
219 | #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
220 | #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
221 lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
222 /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
223 | #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
224 lapply (le_plus_to_le_r … H2) -H2 #Hd
225 /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
229 lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
230 ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
231 L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
232 /2 width=1 by ltpss_sn_append_le_aux/ qed.
234 lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
235 ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
236 L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
237 #K1 #K2 #d #e #H elim H -K1 -K2 -d -e
238 [ #d #e #L1 #L2 <minus_n_O //
239 | #K #I #V #L1 #L2 #_ #H
240 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
241 | #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
242 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
243 | #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
244 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
245 lapply (le_plus_to_le_r … H) -H /3 width=1/
249 (* Basic forward lemmas *****************************************************)
251 lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
252 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e