(* *)
(**************************************************************************)
-include "ground/xoa/ex_3_2.ma".
-include "static_2/notation/relations/stareq_2.ma".
-include "static_2/syntax/term.ma".
+include "static_2/notation/relations/approxeq_2.ma".
+include "static_2/syntax/teqg.ma".
(* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
-inductive teqx: relation term ≝
-| teqx_sort: ∀s1,s2. teqx (⋆s1) (⋆s2)
-| teqx_lref: ∀i. teqx (#i) (#i)
-| teqx_gref: ∀l. teqx (§l) (§l)
-| teqx_pair: ∀I,V1,V2,T1,T2. teqx V1 V2 → teqx T1 T2 → teqx (②[I]V1.T1) (②[I]V2.T2)
-.
+definition sfull: relation2 nat nat ≝
+ λs1,s2. ⊤.
+
+definition teqx: relation term ≝
+ teqg sfull.
interpretation
- "context-free sort-irrelevant equivalence (term)"
- 'StarEq T1 T2 = (teqx T1 T2).
+ "context-free sort-irrelevant equivalence (term)"
+ 'ApproxEq T1 T2 = (teqx T1 T2).
(* Basic properties *********************************************************)
-lemma teqx_refl: reflexive … teqx.
-#T elim T -T /2 width=1 by teqx_pair/
-* /2 width=1 by teqx_lref, teqx_gref/
-qed.
-
-lemma teqx_sym: symmetric … teqx.
-#T1 #T2 #H elim H -T1 -T2
-/2 width=3 by teqx_sort, teqx_lref, teqx_gref, teqx_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
+lemma sfull_dec:
+ ∀s1,s2. Decidable (sfull s1 s2).
+/2 width=1 by or_introl/ qed.
-fact teqx_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 teqx_pair:
+ ∀V1,V2. V1 ≅ V2 → ∀T1,T2. T1 ≅ T2 →
+ ∀I. ②[I]V1.T1 ≅ ②[I]V2.T2.
+/2 width=1 by teqg_pair/ qed.
-lemma teqx_inv_sort1: ∀Y,s1. ⋆s1 ≛ Y →
- ∃s2. Y = ⋆s2.
-/2 width=4 by teqx_inv_sort1_aux/ qed-.
+lemma teqx_refl:
+ reflexive … teqx.
+/2 width=1 by teqg_refl/ qed.
-fact teqx_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 teqx_sym:
+ symmetric … teqx.
+/2 width=1 by teqg_sym/ qed-.
-lemma teqx_inv_lref1: ∀Y,i. #i ≛ Y → Y = #i.
-/2 width=5 by teqx_inv_lref1_aux/ qed-.
+lemma teqg_teqx (S):
+ ∀T1,T2. T1 ≛[S] T2 → T1 ≅ T2.
+/2 width=3 by teqg_co/ qed.
-fact teqx_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-.
+(* Basic inversion lemmas ***************************************************)
-lemma teqx_inv_gref1: ∀Y,l. §l ≛ Y → Y = §l.
-/2 width=5 by teqx_inv_gref1_aux/ qed-.
-
-fact teqx_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/
-]
+lemma teqx_inv_sort1:
+ ∀X2,s1. ⋆s1 ≅ X2 →
+ ∃s2. X2 = ⋆s2.
+#X1 #s1 #H elim (teqg_inv_sort1 … H) -H /2 width=2 by ex_intro/
qed-.
-
-lemma teqx_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 teqx_inv_pair1_aux/ qed-.
-
-lemma teqx_inv_sort2: ∀X1,s2. X1 ≛ ⋆s2 →
- ∃s1. X1 = ⋆s1.
-#X1 #s2 #H
-elim (teqx_inv_sort1 X1 s2)
-/2 width=2 by teqx_sym, ex_intro/
+(*
+lemma teqx_inv_lref1:
+ ∀X,i. #i ≅ X → X = #i.
+/2 width=5 by teqg_inv_lref1/ qed-.
+
+lemma teqx_inv_gref1:
+ ∀X,l. §l ≅ X → X = §l.
+/2 width=5 by teqg_inv_gref1/ qed-.
+*)
+lemma teqx_inv_pair1:
+ ∀I,V1,T1,X2. ②[I]V1.T1 ≅ X2 →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & X2 = ②[I]V2.T2.
+/2 width=3 by teqg_inv_pair1/ qed-.
+
+lemma teqx_inv_sort2:
+ ∀X1,s2. X1 ≅ ⋆s2 →
+ ∃s1. X1 = ⋆s1.
+#X1 #s2 #H elim (teqg_inv_sort2 … H) -H /2 width=2 by ex_intro/
qed-.
-lemma teqx_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 (teqx_inv_pair1 I V2 T2 X1)
-[ #V1 #T1 #HV #HT #H destruct ]
-/3 width=5 by teqx_sym, ex3_2_intro/
-qed-.
+lemma teqx_inv_pair2:
+ ∀I,X1,V2,T2. X1 ≅ ②[I]V2.T2 →
+ ∃∃V1,T1. V1 ≅ V2 & T1 ≅ T2 & X1 = ②[I]V1.T1.
+/2 width=1 by teqg_inv_pair2/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma teqx_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 (teqx_inv_pair1 … H) -H
-#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma teqx_inv_pair_xy_x: ∀I,V,T. ②[I]V.T ≛ V → ⊥.
-#I #V elim V -V
-[ #J #T #H elim (teqx_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #IHX #_ #T #H elim (teqx_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
-]
-qed-.
-
-lemma teqx_inv_pair_xy_y: ∀I,T,V. ②[I]V.T ≛ T → ⊥.
-#I #T elim T -T
-[ #J #V #H elim (teqx_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #_ #IHY #V #H elim (teqx_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
-]
-qed-.
-
+lemma teqx_inv_pair:
+ ∀I1,I2,V1,V2,T1,T2. ②[I1]V1.T1 ≅ ②[I2]V2.T2 →
+ ∧∧ I1 = I2 & V1 ≅ V2 & T1 ≅ T2.
+/2 width=1 by teqg_inv_pair/ qed-.
+(*
+lemma teqx_inv_pair_xy_x:
+ ∀I,V,T. ②[I]V.T ≅ V → ⊥.
+/2 width=5 by teqg_inv_pair_xy_x/ qed-.
+
+lemma teqx_inv_pair_xy_y:
+ ∀I,T,V. ②[I]V.T ≅ T → ⊥.
+/2 width=5 by teqg_inv_pair_xy_y/ qed-.
+*)
(* Basic forward lemmas *****************************************************)
-
-lemma teqx_fwd_atom1: ∀I,Y. ⓪[I] ≛ Y → ∃J. Y = ⓪[J].
-* #x #Y #H [ elim (teqx_inv_sort1 … H) -H ]
-/3 width=4 by teqx_inv_gref1, teqx_inv_lref1, ex_intro/
-qed-.
-
+(*
+lemma teqx_fwd_atom1:
+ ∀I,Y. ⓪[I] ≅ Y → ∃J. Y = ⓪[J].
+/2 width=3 by teqg_fwd_atom1/ qed-.
+*)
(* Advanced properties ******************************************************)
-lemma teqx_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 teqx_sort, or_introl/
-|2,3,13:
- @or_intror #H
- elim (teqx_inv_sort1 … H) -H #x #H destruct
-|4,6,14:
- @or_intror #H
- lapply (teqx_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 (teqx_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (teqx_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 (teqx_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (teqx_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 teqx_pair, or_introl/ ]
- ]
- @or_intror #H
- elim (teqx_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
+lemma teqx_dec:
+ ∀T1,T2. Decidable (T1 ≅ T2).
+/3 width=1 by teqg_dec, or_introl/ qed-.
(* Negated inversion lemmas *************************************************)
-lemma tneqx_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 (teqx_dec V1 V2) /3 width=1 by or3_intro1/
-elim (teqx_dec T1 T2) /4 width=1 by teqx_pair, or3_intro2/
-qed-.
+lemma tneqx_inv_pair:
+ ∀I1,I2,V1,V2,T1,T2.
+ (②[I1]V1.T1 ≅ ②[I2]V2.T2 → ⊥) →
+ ∨∨ I1 = I2 → ⊥
+ | (V1 ≅ V2 → ⊥)
+ | (T1 ≅ T2 → ⊥).
+/3 width=1 by tneqg_inv_pair, or_introl/ qed-.
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