+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/notation/relations/stareq_2.ma".
-include "static_2/syntax/term.ma".
-
-(* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
-
-inductive tdeq: relation term ≝
-| tdeq_sort: ∀s1,s2. tdeq (⋆s1) (⋆s2)
-| tdeq_lref: ∀i. tdeq (#i) (#i)
-| tdeq_gref: ∀l. tdeq (§l) (§l)
-| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq V1 V2 → tdeq T1 T2 → tdeq (②{I}V1.T1) (②{I}V2.T2)
-.
-
-interpretation
- "context-free sort-irrelevant equivalence (term)"
- 'StarEq T1 T2 = (tdeq T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tdeq_refl: reflexive … tdeq.
-#T elim T -T /2 width=1 by tdeq_pair/
-* /2 width=1 by tdeq_lref, tdeq_gref/
-qed.
-
-lemma tdeq_sym: symmetric … tdeq.
-#T1 #T2 #H elim H -T1 -T2
-/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tdeq_inv_sort1_aux: ∀X,Y. X ≛ Y → ∀s1. X = ⋆s1 →
- ∃s2. Y = ⋆s2.
-#X #Y * -X -Y
-[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
-| #i #s #H destruct
-| #l #s #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
-]
-qed-.
-
-lemma tdeq_inv_sort1: ∀Y,s1. ⋆s1 ≛ Y →
- ∃s2. Y = ⋆s2.
-/2 width=4 by tdeq_inv_sort1_aux/ qed-.
-
-fact tdeq_inv_lref1_aux: ∀X,Y. X ≛ Y → ∀i. X = #i → Y = #i.
-#X #Y * -X -Y //
-[ #s1 #s2 #j #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
-]
-qed-.
-
-lemma tdeq_inv_lref1: ∀Y,i. #i ≛ Y → Y = #i.
-/2 width=5 by tdeq_inv_lref1_aux/ qed-.
-
-fact tdeq_inv_gref1_aux: ∀X,Y. X ≛ Y → ∀l. X = §l → Y = §l.
-#X #Y * -X -Y //
-[ #s1 #s2 #k #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-]
-qed-.
-
-lemma tdeq_inv_gref1: ∀Y,l. §l ≛ Y → Y = §l.
-/2 width=5 by tdeq_inv_gref1_aux/ qed-.
-
-fact tdeq_inv_pair1_aux: ∀X,Y. X ≛ Y → ∀I,V1,T1. X = ②{I}V1.T1 →
- ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
-#X #Y * -X -Y
-[ #s1 #s2 #J #W1 #U1 #H destruct
-| #i #J #W1 #U1 #H destruct
-| #l #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma tdeq_inv_pair1: ∀I,V1,T1,Y. ②{I}V1.T1 ≛ Y →
- ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
-/2 width=3 by tdeq_inv_pair1_aux/ qed-.
-
-lemma tdeq_inv_sort2: ∀X1,s2. X1 ≛ ⋆s2 →
- ∃s1. X1 = ⋆s1.
-#X1 #s2 #H
-elim (tdeq_inv_sort1 X1 s2)
-/2 width=2 by tdeq_sym, ex_intro/
-qed-.
-
-lemma tdeq_inv_pair2: ∀I,X1,V2,T2. X1 ≛ ②{I}V2.T2 →
- ∃∃V1,T1. V1 ≛ V2 & T1 ≛ T2 & X1 = ②{I}V1.T1.
-#I #X1 #V2 #T2 #H
-elim (tdeq_inv_pair1 I V2 T2 X1)
-[ #V1 #T1 #HV #HT #H destruct ]
-/3 width=5 by tdeq_sym, ex3_2_intro/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tdeq_inv_pair: ∀I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛ ②{I2}V2.T2 →
- ∧∧ I1 = I2 & V1 ≛ V2 & T1 ≛ T2.
-#I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
-#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma tdeq_inv_pair_xy_x: ∀I,V,T. ②{I}V.T ≛ V → ⊥.
-#I #V elim V -V
-[ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
-]
-qed-.
-
-lemma tdeq_inv_pair_xy_y: ∀I,T,V. ②{I}V.T ≛ T → ⊥.
-#I #T elim T -T
-[ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tdeq_fwd_atom1: ∀I,Y. ⓪{I} ≛ Y → ∃J. Y = ⓪{J}.
-* #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
-/3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma tdeq_dec: ∀T1,T2. Decidable (T1 ≛ T2).
-#T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ /3 width=1 by tdeq_sort, or_introl/
-|2,3,13:
- @or_intror #H
- elim (tdeq_inv_sort1 … H) -H #x #H destruct
-|4,6,14:
- @or_intror #H
- lapply (tdeq_inv_lref1 … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (tdeq_inv_gref1 … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ elim (IHV V2) -IHV #HV12
- elim (IHT T2) -IHT #HT12
- [ /3 width=1 by tdeq_pair, or_introl/ ]
- ]
- @or_intror #H
- elim (tdeq_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Negated inversion lemmas *************************************************)
-
-lemma tdneq_inv_pair: ∀I1,I2,V1,V2,T1,T2.
- (②{I1}V1.T1 ≛ ②{I2}V2.T2 → ⊥) →
- ∨∨ I1 = I2 → ⊥
- | (V1 ≛ V2 → ⊥)
- | (T1 ≛ T2 → ⊥).
-#I1 #I2 #V1 #V2 #T1 #T2 #H12
-elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
-elim (tdeq_dec V1 V2) /3 width=1 by or3_intro1/
-elim (tdeq_dec T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
-qed-.
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