+++ /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 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( ⦃ L1, break T1 ⦄ > break ⦃ L2 , break T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTerm $L1 $T1 $L2 $T2 }.
-
-include "basic_2/substitution/ldrop.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-inductive csup: bi_relation lenv term ≝
-| csup_lref : ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → csup L (#i) K V
-| csup_bind_sn: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) L V
-| csup_bind_dx: ∀a,I,L,V,T. csup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
-| csup_flat_sn: ∀I,L,V,T. csup L (ⓕ{I}V.T) L V
-| csup_flat_dx: ∀I,L,V,T. csup L (ⓕ{I}V.T) L T
-.
-
-interpretation
- "structural predecessor (closure)"
- 'SupTerm L1 T1 L2 T2 = (csup L1 T1 L2 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact csup_inv_atom1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → ∀J. T1 = ⓪{J} →
- ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #K #V #i #HLK #J #H destruct /2 width=4/
-| #a #I #L #V #T #J #H destruct
-| #a #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-| #I #L #V #T #J #H destruct
-]
-qed-.
-
-lemma csup_inv_atom1: ∀J,L1,L2,T2. ⦃L1, ⓪{J}⦄ > ⦃L2, T2⦄ →
- ∃∃I,i. ⇩[0, i] L1 ≡ L2.ⓑ{I}T2 & J = LRef i.
-/2 width=3 by csup_inv_atom1_aux/ qed-.
-
-fact csup_inv_bind1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
- ∀b,J,W,U. T1 = ⓑ{b,J}W.U →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #K #V #i #_ #b #J #W #U #H destruct
-| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #a #I #L #V #T #b #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #b #J #W #U #H destruct
-| #I #L #V #T #b #J #W #U #H destruct
-]
-qed-.
-
-lemma csup_inv_bind1: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ > ⦃L2, T2⦄ →
- (L2 = L1 ∧ T2 = W) ∨
- (L2 = L1.ⓑ{J}W ∧ T2 = U).
-/2 width=4 by csup_inv_bind1_aux/ qed-.
-
-fact csup_inv_flat1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
- ∀J,W,U. T1 = ⓕ{J}W.U →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2
-[ #I #L #K #V #i #_ #J #W #U #H destruct
-| #a #I #L #V #T #J #W #U #H destruct
-| #a #I #L #V #T #J #W #U #H destruct
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-| #I #L #V #T #J #W #U #H destruct /3 width=1/
-]
-qed-.
-
-lemma csup_inv_flat1: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ > ⦃L2, T2⦄ →
- L2 = L1 ∧ (T2 = W ∨ T2 = U).
-/2 width=4 by csup_inv_flat1_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma csup_fwd_cw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ → #{L2, T2} < #{L1, T1}.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /width=1/ /2 width=4 by ldrop_pair2_fwd_cw/
-qed-.
-
-lemma csup_fwd_ldrop: ∀L1,L2,T1,T2. ⦃L1, T1⦄ > ⦃L2, T2⦄ →
- ∃i. ⇩[0, i] L1 ≡ L2 ∨ ⇩[0, i] L2 ≡ L1.
-#L1 #L2 #T1 #T2 * -L1 -L2 -T1 -T2 /3 width=2/ /4 width=2/
-#I #L1 #K1 #V1 #i #HLK1
-lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /3 width=2/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lift_csup_trans_eq: ∀T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∀L,U2. ⦃L, U1⦄ > ⦃L, U2⦄ →
- ∃T2. ⇧[d, e] T2 ≡ U2.
-#T1 #U1 #d #e * -T1 -U1 -d -e
-[5: #a #I #V1 #W1 #T1 #U1 #d #e #HVW1 #_ #L #X #H
- elim (csup_inv_bind1 … H) -H *
- [ #_ #H destruct /2 width=2/
- | #H elim (discr_lpair_x_xy … H)
- ]
-|6: #I #V1 #W1 #T1 #U1 #d #e #HVW1 #HUT1 #L #X #H
- elim (csup_inv_flat1 … H) -H #_ * #H destruct /2 width=2/
-]
-#i #d #e [2,3: #_ ] #L #X #H
-elim (csup_inv_atom1 … H) -H #I #j #HL #H destruct
-lapply (ldrop_pair2_fwd_cw … HL X) -HL #H
-elim (lt_refl_false … H)
-qed-.
-(*
-lemma lift_csup_trans_gt: ∀L1,L2,U1,U2. ⦃L1, U1⦄ > ⦃L2, U2⦄ →
- ⇩[0, 1] L2 ≡ L1 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
- ∃T2. ⇧[d + 1, e] T2 ≡ U2.
-#L1 #L2 #U1 #U2 * -L1 -L2 -U1 -U2
-[ #I #L1 #K1 #V #i #HLK1 #HKL1
- lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
- lapply (ldrop_fwd_lw … HKL1) -HKL1 #HKL1
- lapply (transitive_le … HLK1 HKL1) -L1 normalize #H
-
-
-| #a
-| #a
-]
-#I #L1 #W1 #U1 #HL1
-
-
-
- #X #d #e #H
- lapply (ldrop_inv_refl … HL1) -HL1
-| #a #I #L1 #W1 #U1 #j #HL1 #X #d #e #H
- lapply (ldrop_inv_ldrop1 … HL1)
-
- elim (lift_inv_bind2 … H) -H #W2 #U2 #HW21 #HU21 #H destruct
-
-
- /3 width=2/ /4 width=2/
-
-*)
-
-
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma csup_inv_lref2_be: ∀L,U,i. ⦃L, U⦄ > ⦃L, #i⦄ →
- ∀T,d,e. ⇧[d, e] T ≡ U → d ≤ i → i < d + e → ⊥.
-#L #U #i #H #T #d #e #HTU #Hdi #Hide
-elim (lift_csup_trans_eq … HTU … H) -H -T #T #H
-elim (lift_inv_lref2_be … H ? ?) //
-qed-.
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