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15 include "basic_2/notation/relations/lazyeq_4.ma".
16 include "basic_2/syntax/item_sd.ma".
17 include "basic_2/syntax/lenv.ma".
19 (* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
21 inductive tdeq (h) (o): relation term ≝
22 | tdeq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tdeq h o (⋆s1) (⋆s2)
23 | tdeq_lref: ∀i. tdeq h o (#i) (#i)
24 | tdeq_gref: ∀l. tdeq h o (§l) (§l)
25 | tdeq_pair: ∀I,V1,V2,T1,T2. tdeq h o V1 V2 → tdeq h o T1 T2 → tdeq h o (②{I}V1.T1) (②{I}V2.T2)
29 "degree-based equivalence (terms)"
30 'LazyEq h o T1 T2 = (tdeq h o T1 T2).
32 definition cdeq: ∀h. sd h → relation3 lenv term term ≝
35 (* Basic properties *********************************************************)
37 lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
38 #h #o #T elim T -T /2 width=1 by tdeq_pair/
39 * /2 width=1 by tdeq_lref, tdeq_gref/
40 #s elim (deg_total h o s) /2 width=3 by tdeq_sort/
43 lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
44 #h #o #T1 #T2 #H elim H -T1 -T2
45 /2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
48 (* Basic inversion lemmas ***************************************************)
50 fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀s1. X = ⋆s1 →
51 ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
53 [ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
56 | #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
60 lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y →
61 ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
62 /2 width=3 by tdeq_inv_sort1_aux/ qed-.
64 fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀i. X = #i → Y = #i.
65 #h #o #X #Y * -X -Y //
66 [ #s1 #s2 #d #_ #_ #j #H destruct
67 | #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
71 lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≡[h, o] Y → Y = #i.
72 /2 width=5 by tdeq_inv_lref1_aux/ qed-.
74 fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀l. X = §l → Y = §l.
75 #h #o #X #Y * -X -Y //
76 [ #s1 #s2 #d #_ #_ #k #H destruct
77 | #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
81 lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≡[h, o] Y → Y = §l.
82 /2 width=5 by tdeq_inv_gref1_aux/ qed-.
84 fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≡[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
85 ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2.
87 [ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
88 | #i #J #W1 #U1 #H destruct
89 | #l #J #W1 #U1 #H destruct
90 | #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
94 lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≡[h, o] Y →
95 ∃∃V2,T2. V1 ≡[h, o] V2 & T1 ≡[h, o] T2 & Y = ②{I}V2.T2.
96 /2 width=3 by tdeq_inv_pair1_aux/ qed-.
98 (* Advanced inversion lemmas ************************************************)
100 lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≡[h, o] Y → ∀d. deg h o s1 d →
101 ∃∃s2. deg h o s2 d & Y = ⋆s2.
102 #h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H
103 #s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
106 lemma tdeq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≡[h, o] ②{I2}V2.T2 →
107 ∧∧ I1 = I2 & V1 ≡[h, o] V2 & T1 ≡[h, o] T2.
108 #h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
109 #V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
112 lemma tdeq_inv_pair_xy_y: ∀h,o,I,T,V. ②{I}V.T ≡[h, o] T → ⊥.
113 #h #o #I #T elim T -T
114 [ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
115 | #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
119 (* Basic forward lemmas *****************************************************)
121 lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≡[h, o] Y → ∃J. Y = ⓪{J}.
122 #h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
123 /3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/