--- /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/approxeq_2.ma".
+include "static_2/syntax/term_weight.ma".
+
+lemma tw_le_pair_dx (I): ∀V,T. ♯{T} < ♯{②{I}V.T}.
+#I #V #T /2 width=1 by le_S_S/
+qed.
+
+(* 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_sn_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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