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 @{ 'PSubstStarSn $T1 $d $e $T2 }.
19 include "basic_2/unfold/tpss.ma".
21 (* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
23 inductive ltpss_sn: nat → nat → relation lenv ≝
24 | ltpss_sn_atom : ∀d,e. ltpss_sn d e (⋆) (⋆)
25 | ltpss_sn_pair : ∀L,I,V. ltpss_sn 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
26 | ltpss_sn_tpss2: ∀L1,L2,I,V1,V2,e.
27 ltpss_sn 0 e L1 L2 → L1 ⊢ V1 ▶* [0, e] V2 →
28 ltpss_sn 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
29 | ltpss_sn_tpss1: ∀L1,L2,I,V1,V2,d,e.
30 ltpss_sn d e L1 L2 → L1 ⊢ V1 ▶* [d, e] V2 →
31 ltpss_sn (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
34 interpretation "parallel unfold (local environment, sn variant)"
35 'PSubstStarSn L1 d e L2 = (ltpss_sn d e L1 L2).
37 (* Basic inversion lemmas ***************************************************)
39 fact ltpss_sn_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → e = 0 → L1 = L2.
40 #d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
41 [ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
42 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
43 >(IHL12 ?) -IHL12 // >(tpss_inv_refl_O2 … HV12) //
47 lemma ltpss_sn_inv_refl_O2: ∀d,L1,L2. L1 ⊢ ▶* [d, 0] L2 → L1 = L2.
50 fact ltpss_sn_inv_atom1_aux: ∀d,e,L1,L2.
51 L1 ⊢ ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
52 #d #e #L1 #L2 * -d -e -L1 -L2
54 | #L #I #V #H destruct
55 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct
56 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
60 lemma ltpss_sn_inv_atom1: ∀d,e,L2. ⋆ ⊢ ▶* [d, e] L2 → L2 = ⋆.
63 fact ltpss_sn_inv_tpss21_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
64 ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
65 ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
66 K1 ⊢ V1 ▶* [0, e - 1] V2 &
68 #d #e #L1 #L2 * -d -e -L1 -L2
69 [ #d #e #_ #_ #K1 #I #V1 #H destruct
70 | #L1 #I #V #_ #H elim (lt_refl_false … H)
71 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct /2 width=5/
72 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
76 lemma ltpss_sn_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [0, e] L2 → 0 < e →
77 ∃∃K2,V2. K1 ⊢ ▶* [0, e - 1] K2 &
78 K1 ⊢ V1 ▶* [0, e - 1] V2 &
82 fact ltpss_sn_inv_tpss11_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
83 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
84 ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
85 K1 ⊢ V1 ▶* [d - 1, e] V2 &
87 #d #e #L1 #L2 * -d -e -L1 -L2
88 [ #d #e #_ #I #K1 #V1 #H destruct
89 | #L #I #V #H elim (lt_refl_false … H)
90 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
91 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct /2 width=5/
95 lemma ltpss_sn_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ▶* [d, e] L2 → 0 < d →
96 ∃∃K2,V2. K1 ⊢ ▶* [d - 1, e] K2 &
97 K1 ⊢ V1 ▶* [d - 1, e] V2 &
101 fact ltpss_sn_inv_atom2_aux: ∀d,e,L1,L2.
102 L1 ⊢ ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
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_sn_inv_atom2: ∀d,e,L1. L1 ⊢ ▶* [d, e] ⋆ → L1 = ⋆.
114 fact ltpss_sn_inv_tpss22_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → d = 0 → 0 < e →
115 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
116 ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
117 K1 ⊢ 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 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
123 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
127 lemma ltpss_sn_inv_tpss22: ∀e,L1,K2,I,V2. L1 ⊢ ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
128 ∃∃K1,V1. K1 ⊢ ▶* [0, e - 1] K2 &
129 K1 ⊢ V1 ▶* [0, e - 1] V2 &
133 fact ltpss_sn_inv_tpss12_aux: ∀d,e,L1,L2. L1 ⊢ ▶* [d, e] L2 → 0 < d →
134 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
135 ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
136 K1 ⊢ V1 ▶* [d - 1, e] V2 &
138 #d #e #L1 #L2 * -d -e -L1 -L2
139 [ #d #e #_ #I #K2 #V2 #H destruct
140 | #L #I #V #H elim (lt_refl_false … H)
141 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
142 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
146 lemma ltpss_sn_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ⊢ ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
147 ∃∃K1,V1. K1 ⊢ ▶* [d - 1, e] K2 &
148 K1 ⊢ V1 ▶* [d - 1, e] V2 &
152 (* Basic properties *********************************************************)
154 lemma ltpss_sn_tps2: ∀L1,L2,I,V1,V2,e.
155 L1 ⊢ ▶* [0, e] L2 → L1 ⊢ V1 ▶ [0, e] V2 →
156 L1. ⓑ{I} V1 ⊢ ▶* [0, e + 1] L2. ⓑ{I} V2.
159 lemma ltpss_sn_tps1: ∀L1,L2,I,V1,V2,d,e.
160 L1 ⊢ ▶* [d, e] L2 → L1 ⊢ V1 ▶ [d, e] V2 →
161 L1. ⓑ{I} V1 ⊢ ▶* [d + 1, e] L2. ⓑ{I} V2.
164 lemma ltpss_sn_tpss2_lt: ∀L1,L2,I,V1,V2,e.
165 L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶* [0, e - 1] V2 →
166 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
167 #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
168 >(plus_minus_m_m e 1) /2 width=1/
171 lemma ltpss_sn_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
172 L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶* [d - 1, e] V2 →
173 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
174 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
175 >(plus_minus_m_m d 1) /2 width=1/
178 lemma ltpss_sn_tps2_lt: ∀L1,L2,I,V1,V2,e.
179 L1 ⊢ ▶* [0, e - 1] L2 → L1 ⊢ V1 ▶ [0, e - 1] V2 →
180 0 < e → L1. ⓑ{I} V1 ⊢ ▶* [0, e] L2. ⓑ{I} V2.
183 lemma ltpss_sn_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
184 L1 ⊢ ▶* [d - 1, e] L2 → L1 ⊢ V1 ▶ [d - 1, e] V2 →
185 0 < d → L1. ⓑ{I} V1 ⊢ ▶* [d, e] L2. ⓑ{I} V2.
188 lemma ltpss_sn_refl: ∀L,d,e. L ⊢ ▶* [d, e] L.
190 #L #I #V #IHL * /2 width=1/ * /2 width=1/
193 lemma ltpss_sn_weak: ∀L1,L2,d1,e1. L1 ⊢ ▶* [d1, e1] L2 →
194 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ⊢ ▶* [d2, e2] L2.
195 #L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
196 [ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
197 lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
198 lapply (lt_to_le_to_lt 0 … Hde2) // #He2
199 lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
200 | #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
201 >plus_plus_comm_23 in Hde12; #Hde12
202 elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
203 [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
204 lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
205 | -Hd21 normalize in Hde12;
206 lapply (lt_to_le_to_lt 0 … Hde12) // #He2
207 lapply (le_plus_to_minus_r … Hde12) -Hde12
208 /3 width=5 by ltpss_sn_tpss2_lt, tpss_weak/ (**) (* /3 width=5/ used to work *)
213 lemma ltpss_sn_weak_full: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [0, |L1|] L2.
214 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
215 // /3 width=2/ /3 width=3/
218 fact ltpss_sn_append_le_aux: ∀K1,K2,d,x. K1 ⊢ ▶* [d, x] K2 → x = |K1| - d →
219 ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
220 L1 @@ K1 ⊢ ▶* [d, x + e] L2 @@ K2.
221 #K1 #K2 #d #x #H elim H -K1 -K2 -d -x
222 [ #d #x #H1 #L1 #L2 #e #HL12 #H2 destruct
223 lapply (le_n_O_to_eq … H2) -H2 #H destruct //
224 | #K #I #V <minus_n_O normalize <plus_n_Sm #H destruct
225 | #K1 #K2 #I #V1 #V2 #x #_ #HV12 <minus_n_O #IHK12 <minus_n_O #H #L1 #L2 #e #HL12 #_
226 lapply (injective_plus_l … H) -H #H destruct >plus_plus_comm_23
227 /4 width=5 by ltpss_sn_tpss2, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
228 | #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
229 lapply (le_plus_to_le_r … H2) -H2 #Hd
230 /4 width=5 by ltpss_sn_tpss1, tpss_append, tpss_weak, monotonic_le_plus_r/ (**) (* too slow without trace *)
234 lemma ltpss_sn_append_le: ∀K1,K2,d. K1 ⊢ ▶* [d, |K1| - d] K2 →
235 ∀L1,L2,e. L1 ⊢ ▶* [0, e] L2 → d ≤ |K1| →
236 L1 @@ K1 ⊢ ▶* [d, |K1| - d + e] L2 @@ K2.
237 /2 width=1 by ltpss_sn_append_le_aux/ qed.
239 lemma ltpss_sn_append_ge: ∀K1,K2,d,e. K1 ⊢ ▶* [d, e] K2 →
240 ∀L1,L2. L1 ⊢ ▶* [d - |K1|, e] L2 → |K1| ≤ d →
241 L1 @@ K1 ⊢ ▶* [d, e] L2 @@ K2.
242 #K1 #K2 #d #e #H elim H -K1 -K2 -d -e
243 [ #d #e #L1 #L2 <minus_n_O //
244 | #K #I #V #L1 #L2 #_ #H
245 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
246 | #K1 #K2 #I #V1 #V2 #e #_ #_ #_ #L1 #L2 #_ #H
247 lapply (le_n_O_to_eq … H) -H normalize <plus_n_Sm #H destruct
248 | #K1 #K2 #I #V1 #V2 #d #e #_ #HV12 #IHK12 #L1 #L2
249 normalize <minus_le_minus_minus_comm // <minus_plus_m_m #HL12 #H
250 lapply (le_plus_to_le_r … H) -H /3 width=1/
254 (* Basic forward lemmas *****************************************************)
256 lemma ltpss_sn_fwd_length: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → |L1| = |L2|.
257 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e