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/relocation/ldrop.ma".
16 include "basic_2/relocation/lpx_sn.ma".
18 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
20 (* alternative definition of lpx_sn *)
21 inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
22 | lpx_sn_alt_intro: ∀L1,L2.
23 (∀I1,I2,K1,K2,V1,V2,i.
24 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
26 (∀I1,I2,K1,K2,V1,V2,i.
27 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → lpx_sn_alt R K1 K2
28 ) → |L1| = |L2| → lpx_sn_alt R L1 L2
31 (* compact definition of lpx_sn_alt *****************************************)
33 lemma lpx_sn_alt_ind_alt: ∀R. ∀S:relation lenv.
34 (∀L1,L2. |L1| = |L2| → (
36 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
37 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2 & S K1 K2
39 ∀L1,L2. lpx_sn_alt R L1 L2 → S L1 L2.
40 #R #S #IH #L1 #L2 #H elim H -L1 -L2
41 #L1 #L2 #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
42 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (H1 … HLK1 HLK2) -H1
43 /3 width=7 by and4_intro/
46 lemma lpx_sn_alt_inv_alt: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
49 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
50 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2.
51 #R #L1 #L2 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
52 #L1 #L2 #HL12 #IH @conj // -HL12
53 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
54 /2 width=1 by and3_intro/
57 lemma lpx_sn_alt_intro_alt: ∀R,L1,L2. |L1| = |L2| →
58 (∀I1,I2,K1,K2,V1,V2,i.
59 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
60 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2
61 ) → lpx_sn_alt R L1 L2.
62 #R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
63 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
64 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
67 (* Basic forward lemmas ******************************************************)
69 lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
70 #R #L1 #L2 #H elim (lpx_sn_alt_inv_alt … H) //
73 (* Basic inversion lemmas ***************************************************)
75 lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
76 #R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
77 normalize /2 width=1 by length_inv_zero_sn/
80 lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
81 ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
82 #R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_alt … H) -H
83 #H #IH elim (length_inv_pos_sn … H) -H
84 #I2 #K2 #V2 #HK12 #H destruct
85 elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
88 lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
89 #R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
90 normalize /2 width=1 by length_inv_zero_dx/
93 lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
94 ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
95 #R #I2 #L1 #K2 #V2 #H elim (lpx_sn_alt_inv_alt … H) -H
96 #H #IH elim (length_inv_pos_dx … H) -H
97 #I1 #K1 #V1 #HK12 #H destruct
98 elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
101 (* Basic properties *********************************************************)
103 lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
104 #R @lpx_sn_alt_intro_alt //
105 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
109 lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
110 lpx_sn_alt R L1 L2 → R L1 V1 V2 →
111 lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
112 #R #I #L1 #L2 #V1 #V2 #H #HV12 elim (lpx_sn_alt_inv_alt … H) -H
113 #HL12 #IH @lpx_sn_alt_intro_alt normalize //
114 #I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
116 lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
117 lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
118 /4 width=3 by lpx_sn_alt_intro_alt, and3_intro/
119 | -HL12 -HV12 /3 width=5 by ldrop_inv_drop1/
123 (* Main properties **********************************************************)
125 theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
126 #R #L1 #L2 #H elim H -L1 -L2
127 /2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
130 (* Main inversion lemmas ****************************************************)
132 theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
134 [ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
135 | #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
136 #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
140 (* alternative definition of lpx_sn *****************************************)
142 lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
143 (∀I1,I2,K1,K2,V1,V2,i.
144 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
145 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2
147 #R #L1 #L2 #HL12 #IH @lpx_sn_alt_inv_lpx_sn
148 @lpx_sn_alt_intro_alt // -HL12
149 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
150 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_lpx_sn_alt, and3_intro/
153 lemma lpx_sn_ind_alt: ∀R. ∀S:relation lenv.
154 (∀L1,L2. |L1| = |L2| → (
155 ∀I1,I2,K1,K2,V1,V2,i.
156 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
157 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2 & S K1 K2
159 ∀L1,L2. lpx_sn R L1 L2 → S L1 L2.
160 #R #S #IH1 #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
161 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
162 #L1 #L2 #HL12 #IH2 @IH1 -IH1 // -HL12
163 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH2 … HLK1 HLK2) -IH2 -HLK1 -HLK2
164 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and4_intro/
167 lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
169 ∀I1,I2,K1,K2,V1,V2,i.
170 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
171 ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2.
172 #R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
173 #H elim (lpx_sn_alt_inv_alt … H) -H
175 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
176 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and3_intro/