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15 include "static_2/notation/relations/stareq_2.ma".
16 include "static_2/syntax/term.ma".
18 (* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
20 inductive tdeq: relation term ≝
21 | tdeq_sort: ∀s1,s2. tdeq (⋆s1) (⋆s2)
22 | tdeq_lref: ∀i. tdeq (#i) (#i)
23 | tdeq_gref: ∀l. tdeq (§l) (§l)
24 | tdeq_pair: ∀I,V1,V2,T1,T2. tdeq V1 V2 → tdeq T1 T2 → tdeq (②{I}V1.T1) (②{I}V2.T2)
28 "context-free sort-irrelevant equivalence (term)"
29 'StarEq T1 T2 = (tdeq T1 T2).
31 (* Basic properties *********************************************************)
33 lemma tdeq_refl: reflexive … tdeq.
34 #T elim T -T /2 width=1 by tdeq_pair/
35 * /2 width=1 by tdeq_lref, tdeq_gref/
38 lemma tdeq_sym: symmetric … tdeq.
39 #T1 #T2 #H elim H -T1 -T2
40 /2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
43 (* Basic inversion lemmas ***************************************************)
45 fact tdeq_inv_sort1_aux: ∀X,Y. X ≛ Y → ∀s1. X = ⋆s1 →
48 [ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
51 | #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
55 lemma tdeq_inv_sort1: ∀Y,s1. ⋆s1 ≛ Y →
57 /2 width=4 by tdeq_inv_sort1_aux/ qed-.
59 fact tdeq_inv_lref1_aux: ∀X,Y. X ≛ Y → ∀i. X = #i → Y = #i.
61 [ #s1 #s2 #j #H destruct
62 | #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
66 lemma tdeq_inv_lref1: ∀Y,i. #i ≛ Y → Y = #i.
67 /2 width=5 by tdeq_inv_lref1_aux/ qed-.
69 fact tdeq_inv_gref1_aux: ∀X,Y. X ≛ Y → ∀l. X = §l → Y = §l.
71 [ #s1 #s2 #k #H destruct
72 | #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
76 lemma tdeq_inv_gref1: ∀Y,l. §l ≛ Y → Y = §l.
77 /2 width=5 by tdeq_inv_gref1_aux/ qed-.
79 fact tdeq_inv_pair1_aux: ∀X,Y. X ≛ Y → ∀I,V1,T1. X = ②{I}V1.T1 →
80 ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
82 [ #s1 #s2 #J #W1 #U1 #H destruct
83 | #i #J #W1 #U1 #H destruct
84 | #l #J #W1 #U1 #H destruct
85 | #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
89 lemma tdeq_inv_pair1: ∀I,V1,T1,Y. ②{I}V1.T1 ≛ Y →
90 ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
91 /2 width=3 by tdeq_inv_pair1_aux/ qed-.
93 lemma tdeq_inv_sort2: ∀X1,s2. X1 ≛ ⋆s2 →
96 elim (tdeq_inv_sort1 X1 s2)
97 /2 width=2 by tdeq_sym, ex_intro/
100 lemma tdeq_inv_pair2: ∀I,X1,V2,T2. X1 ≛ ②{I}V2.T2 →
101 ∃∃V1,T1. V1 ≛ V2 & T1 ≛ T2 & X1 = ②{I}V1.T1.
103 elim (tdeq_inv_pair1 I V2 T2 X1)
104 [ #V1 #T1 #HV #HT #H destruct ]
105 /3 width=5 by tdeq_sym, ex3_2_intro/
108 (* Advanced inversion lemmas ************************************************)
110 lemma tdeq_inv_pair: ∀I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛ ②{I2}V2.T2 →
111 ∧∧ I1 = I2 & V1 ≛ V2 & T1 ≛ T2.
112 #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
113 #V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
116 lemma tdeq_inv_pair_xy_x: ∀I,V,T. ②{I}V.T ≛ V → ⊥.
118 [ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
119 | #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
123 lemma tdeq_inv_pair_xy_y: ∀I,T,V. ②{I}V.T ≛ T → ⊥.
125 [ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
126 | #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
130 (* Basic forward lemmas *****************************************************)
132 lemma tdeq_fwd_atom1: ∀I,Y. ⓪{I} ≛ Y → ∃J. Y = ⓪{J}.
133 * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
134 /3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
137 (* Advanced properties ******************************************************)
139 lemma tdeq_dec: ∀T1,T2. Decidable (T1 ≛ T2).
140 #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
141 [ /3 width=1 by tdeq_sort, or_introl/
144 elim (tdeq_inv_sort1 … H) -H #x #H destruct
147 lapply (tdeq_inv_lref1 … H) -H #H destruct
149 elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
151 lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
154 lapply (tdeq_inv_gref1 … H) -H #H destruct
156 elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
158 lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
161 elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
163 elim (eq_item2_dec I1 I2) #HI12 destruct
164 [ elim (IHV V2) -IHV #HV12
165 elim (IHT T2) -IHT #HT12
166 [ /3 width=1 by tdeq_pair, or_introl/ ]
169 elim (tdeq_inv_pair … H) -H /2 width=1 by/
173 (* Negated inversion lemmas *************************************************)
175 lemma tdneq_inv_pair: ∀I1,I2,V1,V2,T1,T2.
176 (②{I1}V1.T1 ≛ ②{I2}V2.T2 → ⊥) →
180 #I1 #I2 #V1 #V2 #T1 #T2 #H12
181 elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
182 elim (tdeq_dec V1 V2) /3 width=1 by or3_intro1/
183 elim (tdeq_dec T1 T2) /4 width=1 by tdeq_pair, or3_intro2/