+++ /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 "ground/xoa/ex_1_2.ma".
-include "ground/xoa/ex_3_2.ma".
-include "static_2/notation/relations/approxeq_2.ma".
-include "static_2/syntax/term_weight.ma".
-
-(* SORT-IRRELEVANT WHD EQUIVALENCE ON TERMS *********************************)
-
-inductive tweq: relation term ≝
-| tweq_sort: ∀s1,s2. tweq (⋆s1) (⋆s2)
-| tweq_lref: ∀i. tweq (#i) (#i)
-| tweq_gref: ∀l. tweq (§l) (§l)
-| tweq_abbr: ∀p,V1,V2,T1,T2. (p=Ⓣ→tweq T1 T2) → tweq (ⓓ[p]V1.T1) (ⓓ[p]V2.T2)
-| tweq_abst: ∀p,V1,V2,T1,T2. tweq (ⓛ[p]V1.T1) (ⓛ[p]V2.T2)
-| tweq_appl: ∀V1,V2,T1,T2. tweq T1 T2 → tweq (ⓐV1.T1) (ⓐV2.T2)
-| tweq_cast: ∀V1,V2,T1,T2. tweq V1 V2 → tweq T1 T2 → tweq (ⓝV1.T1) (ⓝV2.T2)
-.
-
-interpretation
- "context-free tail sort-irrelevant equivalence (term)"
- 'ApproxEq T1 T2 = (tweq T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tweq_abbr_pos: ∀V1,V2,T1,T2. T1 ≅ T2 → +ⓓV1.T1 ≅ +ⓓV2.T2.
-/3 width=1 by tweq_abbr/ qed.
-
-lemma tweq_abbr_neg: ∀V1,V2,T1,T2. -ⓓV1.T1 ≅ -ⓓV2.T2.
-#V1 #V2 #T1 #T2
-@tweq_abbr #H destruct
-qed.
-
-lemma tweq_refl: reflexive … tweq.
-#T elim T -T * [||| #p * | * ]
-/2 width=1 by tweq_sort, tweq_lref, tweq_gref, tweq_abbr, tweq_abst, tweq_appl, tweq_cast/
-qed.
-
-lemma tweq_sym: symmetric … tweq.
-#T1 #T2 #H elim H -T1 -T2
-/3 width=3 by tweq_sort, tweq_lref, tweq_gref, tweq_abbr, tweq_abst, tweq_appl, tweq_cast/
-qed-.
-
-(* Left basic inversion lemmas **********************************************)
-
-fact tweq_inv_sort_sn_aux:
- ∀X,Y. X ≅ Y → ∀s1. X = ⋆s1 → ∃s2. Y = ⋆s2.
-#X #Y * -X -Y
-[1 : #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
-|2,3: #i #s #H destruct
-|4 : #p #V1 #V2 #T1 #T2 #_ #s #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #s #H destruct
-|6 : #V1 #V2 #T1 #T2 #_ #s #H destruct
-|7 : #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
-]
-qed-.
-
-lemma tweq_inv_sort_sn:
- ∀Y,s1. ⋆s1 ≅ Y → ∃s2. Y = ⋆s2.
-/2 width=4 by tweq_inv_sort_sn_aux/ qed-.
-
-fact tweq_inv_lref_sn_aux:
- ∀X,Y. X ≅ Y → ∀i. X = #i → Y = #i.
-#X #Y * -X -Y
-[1 : #s1 #s2 #j #H destruct
-|2,3: //
-|4 : #p #V1 #V2 #T1 #T2 #_ #j #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #j #H destruct
-|6 : #V1 #V2 #T1 #T2 #_ #j #H destruct
-|7 : #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
-]
-qed-.
-
-lemma tweq_inv_lref_sn: ∀Y,i. #i ≅ Y → Y = #i.
-/2 width=5 by tweq_inv_lref_sn_aux/ qed-.
-
-fact tweq_inv_gref_sn_aux:
- ∀X,Y. X ≅ Y → ∀l. X = §l → Y = §l.
-#X #Y * -X -Y
-[1 : #s1 #s2 #k #H destruct
-|2,3: //
-|4 : #p #V1 #V2 #T1 #T2 #_ #k #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #k #H destruct
-|6 : #V1 #V2 #T1 #T2 #_ #k #H destruct
-|7 : #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
-]
-qed-.
-
-lemma tweq_inv_gref_sn:
- ∀Y,l. §l ≅ Y → Y = §l.
-/2 width=5 by tweq_inv_gref_sn_aux/ qed-.
-
-fact tweq_inv_abbr_sn_aux:
- ∀X,Y. X ≅ Y → ∀p,V1,T1. X = ⓓ[p]V1.T1 →
- ∃∃V2,T2. p = Ⓣ → T1 ≅ T2 & Y = ⓓ[p]V2.T2.
-#X #Y * -X -Y
-[1 : #s1 #s2 #q #W1 #U1 #H destruct
-|2,3: #i #q #W1 #U1 #H destruct
-|4 : #p #V1 #V2 #T1 #T2 #HT #q #W1 #U1 #H destruct /3 width=4 by ex2_2_intro/
-|5 : #p #V1 #V2 #T1 #T2 #q #W1 #U1 #H destruct
-|6 : #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
-|7 : #V1 #V2 #T1 #T2 #_ #_ #q #W1 #U1 #H destruct
-]
-qed-.
-
-lemma tweq_inv_abbr_sn:
- ∀p,V1,T1,Y. ⓓ[p]V1.T1 ≅ Y →
- ∃∃V2,T2. p = Ⓣ → T1 ≅ T2 & Y = ⓓ[p]V2.T2.
-/2 width=4 by tweq_inv_abbr_sn_aux/ qed-.
-
-fact tweq_inv_abst_sn_aux:
- ∀X,Y. X ≅ Y → ∀p,V1,T1. X = ⓛ[p]V1.T1 →
- ∃∃V2,T2. Y = ⓛ[p]V2.T2.
-#X #Y * -X -Y
-[1 : #s1 #s2 #q #W1 #U1 #H destruct
-|2,3: #i #q #W1 #U1 #H destruct
-|4 : #p #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #q #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
-|6 : #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
-|7 : #V1 #V2 #T1 #T2 #_ #_ #q #W1 #U1 #H destruct
-]
-qed-.
-
-lemma tweq_inv_abst_sn:
- ∀p,V1,T1,Y. ⓛ[p]V1.T1 ≅ Y →
- ∃∃V2,T2. Y = ⓛ[p]V2.T2.
-/2 width=5 by tweq_inv_abst_sn_aux/ qed-.
-
-fact tweq_inv_appl_sn_aux:
- ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓐV1.T1 →
- ∃∃V2,T2. T1 ≅ T2 & Y = ⓐV2.T2.
-#X #Y * -X -Y
-[1 : #s1 #s2 #W1 #U1 #H destruct
-|2,3: #i #W1 #U1 #H destruct
-|4 : #p #V1 #V2 #T1 #T2 #HT #W1 #U1 #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #W1 #U1 #H destruct
-|6 : #V1 #V2 #T1 #T2 #HT #W1 #U1 #H destruct /2 width=4 by ex2_2_intro/
-|7 : #V1 #V2 #T1 #T2 #_ #_ #W1 #U1 #H destruct
-]
-qed-.
-
-lemma tweq_inv_appl_sn:
- ∀V1,T1,Y. ⓐV1.T1 ≅ Y →
- ∃∃V2,T2. T1 ≅ T2 & Y = ⓐV2.T2.
-/2 width=4 by tweq_inv_appl_sn_aux/ qed-.
-
-fact tweq_inv_cast_sn_aux:
- ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓝV1.T1 →
- ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
-#X #Y * -X -Y
-[1 : #s1 #s2 #W1 #U1 #H destruct
-|2,3: #i #W1 #U1 #H destruct
-|4 : #p #V1 #V2 #T1 #T2 #_ #W1 #U1 #H destruct
-|5 : #p #V1 #V2 #T1 #T2 #W1 #U1 #H destruct
-|6 : #V1 #V2 #T1 #T2 #_ #W1 #U1 #H destruct
-|7 : #V1 #V2 #T1 #T2 #HV #HT #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma tweq_inv_cast_sn:
- ∀V1,T1,Y. ⓝV1.T1 ≅ Y →
- ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
-/2 width=3 by tweq_inv_cast_sn_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tweq_inv_abbr_pos_sn:
- ∀V1,T1,Y. +ⓓV1.T1 ≅ Y → ∃∃V2,T2. T1 ≅ T2 & Y = +ⓓV2.T2.
-#V1 #V2 #Y #H
-elim (tweq_inv_abbr_sn … H) -H #V2 #T2
-/3 width=4 by ex2_2_intro/
-qed-.
-
-lemma tweq_inv_abbr_neg_sn:
- ∀V1,T1,Y. -ⓓV1.T1 ≅ Y → ∃∃V2,T2. Y = -ⓓV2.T2.
-#V1 #V2 #Y #H
-elim (tweq_inv_abbr_sn … H) -H #V2 #T2 #_
-/2 width=3 by ex1_2_intro/
-qed-.
-
-lemma tweq_inv_abbr_pos_bi:
- ∀V1,V2,T1,T2. +ⓓV1.T1 ≅ +ⓓV2.T2 → T1 ≅ T2.
-#V1 #V2 #T1 #T2 #H
-elim (tweq_inv_abbr_pos_sn … H) -H #W2 #U2 #HTU #H destruct //
-qed-.
-
-lemma tweq_inv_appl_bi:
- ∀V1,V2,T1,T2. ⓐV1.T1 ≅ ⓐV2.T2 → T1 ≅ T2.
-#V1 #V2 #T1 #T2 #H
-elim (tweq_inv_appl_sn … H) -H #W2 #U2 #HTU #H destruct //
-qed-.
-
-lemma tweq_inv_cast_bi:
- ∀V1,V2,T1,T2. ⓝV1.T1 ≅ ⓝV2.T2 → ∧∧ V1 ≅ V2 & T1 ≅ T2.
-#V1 #V2 #T1 #T2 #H
-elim (tweq_inv_cast_sn … H) -H #W2 #U2 #HVW #HTU #H destruct
-/2 width=1 by conj/
-qed-.
-
-lemma tweq_inv_cast_xy_y: ∀T,V. ⓝV.T ≅ T → ⊥.
-@(f_ind … tw) #n #IH #T #Hn #V #H destruct
-elim (tweq_inv_cast_sn … H) -H #X1 #X2 #_ #HX2 #H destruct -V
-/2 width=4 by/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tweq_fwd_pair_sn (I):
- ∀V1,T1,X2. ②[I]V1.T1 ≅ X2 → ∃∃V2,T2. X2 = ②[I]V2.T2.
-* [ #p ] * [ cases p -p ] #V1 #T1 #X2 #H
-[ elim (tweq_inv_abbr_pos_sn … H) -H #V2 #T2 #_ #H
-| elim (tweq_inv_abbr_neg_sn … H) -H #V2 #T2 #H
-| elim (tweq_inv_abst_sn … H) -H #V2 #T2 #H
-| elim (tweq_inv_appl_sn … H) -H #V2 #T2 #_ #H
-| elim (tweq_inv_cast_sn … H) -H #V2 #T2 #_ #_ #H
-] /2 width=3 by ex1_2_intro/
-qed-.
-
-lemma tweq_fwd_pair_bi (I1) (I2):
- ∀V1,V2,T1,T2. ②[I1]V1.T1 ≅ ②[I2]V2.T2 → I1 = I2.
-#I1 #I2 #V1 #V2 #T1 #T2 #H
-elim (tweq_fwd_pair_sn … H) -H #W2 #U2 #H destruct //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma tweq_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 tweq_sort, or_introl/
-|2,3,13:
- @or_intror #H
- elim (tweq_inv_sort_sn … H) -H #x #H destruct
-|4,6,14:
- @or_intror #H
- lapply (tweq_inv_lref_sn … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tweq_inv_lref_sn … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (tweq_inv_gref_sn … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tweq_inv_gref_sn … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (tweq_fwd_pair_sn … H) -H #X1 #X2 #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ cases I2 -I2 [ #p ] * [ cases p -p ]
- [ elim (IHT T2) -IHT #HT12
- [ /3 width=1 by tweq_abbr_pos, or_introl/
- | /4 width=3 by tweq_inv_abbr_pos_bi, or_intror/
- ]
- | /3 width=1 by tweq_abbr_neg, or_introl/
- | /3 width=1 by tweq_abst, or_introl/
- | elim (IHT T2) -IHT #HT12
- [ /3 width=1 by tweq_appl, or_introl/
- | /4 width=3 by tweq_inv_appl_bi, or_intror/
- ]
- | elim (IHV V2) -IHV #HV12
- elim (IHT T2) -IHT #HT12
- [1: /3 width=1 by tweq_cast, or_introl/
- |*: @or_intror #H
- elim (tweq_inv_cast_bi … H) -H #HV12 #HT12
- /2 width=1 by/
- ]
- ]
- | /4 width=5 by tweq_fwd_pair_bi, or_intror/
- ]
-]
-qed-.
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