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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 *)
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15 include "basic_2/notation/relations/lazyeq_3.ma".
16 include "basic_2/grammar/ceq.ma".
17 include "basic_2/relocation/lexs.ma".
19 (* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
21 (* Basic_2A1: includes: lreq_atom lreq_zero lreq_pair lreq_succ *)
22 definition lreq: relation3 rtmap lenv lenv ≝ lexs ceq cfull.
25 "ranged equivalence (local environment)"
26 'LazyEq f L1 L2 = (lreq f L1 L2).
28 (* Basic properties *********************************************************)
30 lemma lreq_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ≡[f] L2).
31 /2 width=3 by lexs_eq_repl_back/ qed-.
33 lemma lreq_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ≡[f] L2).
34 /2 width=3 by lexs_eq_repl_fwd/ qed-.
36 lemma sle_lreq_trans: ∀f2,L1,L2. L1 ≡[f2] L2 →
37 ∀f1. f1 ⊆ f2 → L1 ≡[f1] L2.
38 /2 width=3 by sle_lexs_trans/ qed-.
40 (* Basic_2A1: includes: lreq_refl *)
41 lemma lreq_refl: ∀f. reflexive … (lreq f).
42 /2 width=1 by lexs_refl/ qed.
44 (* Basic_2A1: includes: lreq_sym *)
45 lemma lreq_sym: ∀f. symmetric … (lreq f).
46 #f #L1 #L2 #H elim H -L1 -L2 -f
47 /2 width=1 by lexs_next, lexs_push/
50 (* Basic inversion lemmas ***************************************************)
52 (* Basic_2A1: includes: lreq_inv_atom1 *)
53 lemma lreq_inv_atom1: ∀f,Y. ⋆ ≡[f] Y → Y = ⋆.
54 /2 width=4 by lexs_inv_atom1/ qed-.
56 (* Basic_2A1: includes: lreq_inv_pair1 *)
57 lemma lreq_inv_next1: ∀g,J,K1,Y,W1. K1.ⓑ{J}W1 ≡[⫯g] Y →
58 ∃∃K2. K1 ≡[g] K2 & Y = K2.ⓑ{J}W1.
59 #g #J #K1 #Y #W1 #H elim (lexs_inv_next1 … H) -H /2 width=3 by ex2_intro/
62 (* Basic_2A1: includes: lreq_inv_zero1 lreq_inv_succ1 *)
63 lemma lreq_inv_push1: ∀g,J,K1,Y,W1. K1.ⓑ{J}W1 ≡[↑g] Y →
64 ∃∃K2,W2. K1 ≡[g] K2 & Y = K2.ⓑ{J}W2.
65 #g #J #K1 #Y #W1 #H elim (lexs_inv_push1 … H) -H /2 width=4 by ex2_2_intro/ qed-.
67 (* Basic_2A1: includes: lreq_inv_atom2 *)
68 lemma lreq_inv_atom2: ∀f,X. X ≡[f] ⋆ → X = ⋆.
69 /2 width=4 by lexs_inv_atom2/ qed-.
71 (* Basic_2A1: includes: lreq_inv_pair2 *)
72 lemma lreq_inv_next2: ∀g,J,X,K2,W2. X ≡[⫯g] K2.ⓑ{J}W2 →
73 ∃∃K1. K1 ≡[g] K2 & X = K1.ⓑ{J}W2.
74 #g #J #X #K2 #W2 #H elim (lexs_inv_next2 … H) -H /2 width=3 by ex2_intro/ qed-.
76 (* Basic_2A1: includes: lreq_inv_zero2 lreq_inv_succ2 *)
77 lemma lreq_inv_push2: ∀g,J,X,K2,W2. X ≡[↑g] K2.ⓑ{J}W2 →
78 ∃∃K1,W1. K1 ≡[g] K2 & X = K1.ⓑ{J}W1.
79 #g #J #X #K2 #W2 #H elim (lexs_inv_push2 … H) -H /2 width=4 by ex2_2_intro/ qed-.
81 (* Basic_2A1: includes: lreq_inv_pair *)
82 lemma lreq_inv_next: ∀f,I1,I2,L1,L2,V1,V2.
83 L1.ⓑ{I1}V1 ≡[⫯f] (L2.ⓑ{I2}V2) →
84 ∧∧ L1 ≡[f] L2 & V1 = V2 & I1 = I2.
85 /2 width=1 by lexs_inv_next/ qed-.
87 (* Basic_2A1: includes: lreq_inv_succ *)
88 lemma lreq_inv_push: ∀f,I1,I2,L1,L2,V1,V2.
89 L1.ⓑ{I1}V1 ≡[↑f] (L2.ⓑ{I2}V2) →
91 #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push … H) -H /2 width=1 by conj/
94 lemma lreq_inv_tl: ∀f,I,L1,L2,V. L1 ≡[⫱f] L2 → L1.ⓑ{I}V ≡[f] L2.ⓑ{I}V.
95 /2 width=1 by lexs_inv_tl/ qed-.
97 (* Basic_2A1: removed theorems 5:
98 lreq_pair_lt lreq_succ_lt lreq_pair_O_Y lreq_O2 lreq_inv_O_Y