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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 "ground/ynat/ynat_lt.ma".
16 include "basic_2A/notation/relations/midiso_4.ma".
17 include "basic_2A/grammar/lenv_length.ma".
19 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
21 inductive lreq: relation4 ynat ynat lenv lenv ≝
22 | lreq_atom: ∀l,m. lreq l m (⋆) (⋆)
23 | lreq_zero: ∀I1,I2,L1,L2,V1,V2.
24 lreq 0 0 L1 L2 → lreq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
25 | lreq_pair: ∀I,L1,L2,V,m. lreq 0 m L1 L2 →
26 lreq 0 (↑m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
27 | lreq_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
28 lreq l m L1 L2 → lreq (↑l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
32 "equivalence (local environment)"
33 'MidIso l m L1 L2 = (lreq l m L1 L2).
35 (* Basic properties *********************************************************)
37 lemma lreq_pair_lt: ∀I,L1,L2,V,m. L1 ⩬[0, ↓m] L2 → 0 < m →
38 L1.ⓑ{I}V ⩬[0, m] L2.ⓑ{I}V.
39 #I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lreq_pair/
42 lemma lreq_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[↓l, m] L2 → 0 < l →
43 L1.ⓑ{I1}V1 ⩬[l, m] L2. ⓑ{I2}V2.
44 #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lreq_succ/
47 lemma lreq_pair_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 →
48 ∀I,V. L1.ⓑ{I}V ⩬[0, ∞] L2.ⓑ{I}V.
49 #L1 #L2 #HL12 #I #V lapply (lreq_pair I … V … HL12) -HL12 //
52 lemma lreq_refl: ∀L,l,m. L ⩬[l, m] L.
54 #L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
55 #Hl destruct /2 width=1 by lreq_succ/
56 #m elim (ynat_cases … m) [| * #x ]
57 #Hm destruct /2 width=1 by lreq_zero, lreq_pair/
60 lemma lreq_O2: ∀L1,L2,l. |L1| = |L2| → L1 ⩬[l, yinj 0] L2.
61 #L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
62 * // [1,3: #L2 #I2 #V2 ] #l normalize
63 [1,3: <plus_n_Sm #H destruct ]
64 #H lapply (injective_plus_l … H) -H #HL12
65 elim (ynat_cases l) /3 width=1 by lreq_zero/
66 * /3 width=1 by lreq_succ/
69 lemma lreq_sym: ∀l,m. symmetric … (lreq l m).
70 #l #m #L1 #L2 #H elim H -L1 -L2 -l -m
71 /2 width=1 by lreq_zero, lreq_pair, lreq_succ/
74 (* Basic inversion lemmas ***************************************************)
76 fact lreq_inv_atom1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → L1 = ⋆ → L2 = ⋆.
77 #L1 #L2 #l #m * -L1 -L2 -l -m //
78 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
79 | #I #L1 #L2 #V #m #_ #H destruct
80 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
84 lemma lreq_inv_atom1: ∀L2,l,m. ⋆ ⩬[l, m] L2 → L2 = ⋆.
85 /2 width=5 by lreq_inv_atom1_aux/ qed-.
87 fact lreq_inv_zero1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
88 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
89 ∃∃J2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
90 #L1 #L2 #l #m * -L1 -L2 -l -m
91 [ #l #m #J1 #K1 #W1 #H destruct
92 | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
93 /2 width=5 by ex2_3_intro/
94 | #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
95 elim (ysucc_inv_O_dx … H)
96 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
97 elim (ysucc_inv_O_dx … H)
101 lemma lreq_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⩬[0, 0] L2 →
102 ∃∃I2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
103 /2 width=9 by lreq_inv_zero1_aux/ qed-.
105 fact lreq_inv_pair1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
106 ∀J,K1,W. L1 = K1.ⓑ{J}W → l = 0 → 0 < m →
107 ∃∃K2. K1 ⩬[0, ↓m] K2 & L2 = K2.ⓑ{J}W.
108 #L1 #L2 #l #m * -L1 -L2 -l -m
109 [ #l #m #J #K1 #W #H destruct
110 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
111 elim (ylt_yle_false … H) //
112 | #I #L1 #L2 #V #m #HL12 #J #K1 #W #H #_ #_ destruct
113 /2 width=3 by ex2_intro/
114 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J #K1 #W #_ #H
115 elim (ysucc_inv_O_dx … H)
119 lemma lreq_inv_pair1: ∀I,K1,L2,V,m. K1.ⓑ{I}V ⩬[0, m] L2 → 0 < m →
120 ∃∃K2. K1 ⩬[0, ↓m] K2 & L2 = K2.ⓑ{I}V.
121 /2 width=6 by lreq_inv_pair1_aux/ qed-.
123 lemma lreq_inv_pair: ∀I1,I2,L1,L2,V1,V2,m. L1.ⓑ{I1}V1 ⩬[0, m] L2.ⓑ{I2}V2 → 0 < m →
124 ∧∧ L1 ⩬[0, ↓m] L2 & I1 = I2 & V1 = V2.
125 #I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 … H) -H //
126 #Y #HL12 #H destruct /2 width=1 by and3_intro/
129 fact lreq_inv_succ1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
130 ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
131 ∃∃J2,K2,W2. K1 ⩬[↓l, m] K2 & L2 = K2.ⓑ{J2}W2.
132 #L1 #L2 #l #m * -L1 -L2 -l -m
133 [ #l #m #J1 #K1 #W1 #H destruct
134 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
135 elim (ylt_yle_false … H) //
136 | #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
137 elim (ylt_yle_false … H) //
138 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
139 /2 width=5 by ex2_3_intro/
143 lemma lreq_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⩬[l, m] L2 → 0 < l →
144 ∃∃I2,K2,V2. K1 ⩬[↓l, m] K2 & L2 = K2.ⓑ{I2}V2.
145 /2 width=5 by lreq_inv_succ1_aux/ qed-.
147 lemma lreq_inv_atom2: ∀L1,l,m. L1 ⩬[l, m] ⋆ → L1 = ⋆.
148 /3 width=3 by lreq_inv_atom1, lreq_sym/
151 lemma lreq_inv_succ: ∀I1,I2,L1,L2,V1,V2,l,m. L1.ⓑ{I1}V1 ⩬[l, m] L2.ⓑ{I2}V2 → 0 < l →
153 #I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … H) -H //
154 #Z #Y #X #HL12 #H destruct //
157 lemma lreq_inv_zero2: ∀I2,K2,L1,V2. L1 ⩬[0, 0] K2.ⓑ{I2}V2 →
158 ∃∃I1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
159 #I2 #K2 #L1 #V2 #H elim (lreq_inv_zero1 … (lreq_sym … H)) -H
160 /3 width=5 by lreq_sym, ex2_3_intro/
163 lemma lreq_inv_pair2: ∀I,K2,L1,V,m. L1 ⩬[0, m] K2.ⓑ{I}V → 0 < m →
164 ∃∃K1. K1 ⩬[0, ↓m] K2 & L1 = K1.ⓑ{I}V.
165 #I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 … (lreq_sym … H)) -H
166 /3 width=3 by lreq_sym, ex2_intro/
169 lemma lreq_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.ⓑ{I2}V2 → 0 < l →
170 ∃∃I1,K1,V1. K1 ⩬[↓l, m] K2 & L1 = K1.ⓑ{I1}V1.
171 #I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … (lreq_sym … H)) -H
172 /3 width=5 by lreq_sym, ex2_3_intro/
175 (* Basic forward lemmas *****************************************************)
177 lemma lreq_fwd_length: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → |L1| = |L2|.
178 #L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize //
181 (* Advanced inversion lemmas ************************************************)
183 fact lreq_inv_O_Y_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → l = 0 → m = ∞ → L1 = L2.
184 #L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
185 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
186 | /4 width=1 by eq_f3, ysucc_inv_Y_dx/
187 | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #_ #H elim (ysucc_inv_O_dx … H)
191 lemma lreq_inv_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 → L1 = L2.
192 /2 width=5 by lreq_inv_O_Y_aux/ qed-.