(**************************************************************************) (* ___ *) (* ||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 "basic_2/notation/relations/lrsubeq_4.ma". include "basic_2/substitution/drop.ma". (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************) inductive lsuby: relation4 ynat ynat lenv lenv ≝ | lsuby_atom: ∀L,l,m. lsuby l m L (⋆) | lsuby_zero: ∀I1,I2,L1,L2,V1,V2. lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) | lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 → lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V) | lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m. lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2) . interpretation "local environment refinement (extended substitution)" 'LRSubEq L1 l m L2 = (lsuby l m L1 L2). (* Basic properties *********************************************************) lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m → L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V. #I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/ qed. lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l → L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2. #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/ qed. lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 → ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V. #L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 // qed. lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L. #L elim L -L // #L #I #V #IHL #l elim (ynat_cases … l) [| * #x ] #Hl destruct /2 width=1 by lsuby_succ/ #m elim (ynat_cases … m) [| * #x ] #Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/ qed. lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, 0] L2. #L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * [ #l #H elim (ylt_yle_false … H) -H // | #L1 #I1 #V1 #l #H lapply (yle_inv_succ … H) -H #HL12 elim (ynat_cases l) /3 width=1 by lsuby_zero/ * /3 width=1 by lsuby_succ/ ] qed. lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1. #l #m #L1 #L2 #H elim H -l -m -L1 -L2 [ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 // | /2 width=1 by lsuby_O2/ | #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H /3 width=1 by lsuby_pair/ | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H /3 width=1 by lsuby_succ/ ] qed-. (* Basic inversion lemmas ***************************************************) fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆. #L1 #L2 #l #m * -L1 -L2 -l -m // [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct | #I1 #I2 #L1 #L2 #V #m #_ #H destruct | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct ] qed-. lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆. /2 width=5 by lsuby_inv_atom1_aux/ qed-. fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 → L2 = ⋆ ∨ ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2. #L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ [ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct /3 width=5 by ex2_3_intro, or_intror/ | #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H elim (ysucc_inv_O_dx … H) | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H elim (ysucc_inv_O_dx … H) ] qed-. lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 → L2 = ⋆ ∨ ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2. /2 width=9 by lsuby_inv_zero1_aux/ qed-. fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m → L2 = ⋆ ∨ ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W. #L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct /3 width=4 by ex2_2_intro, or_intror/ | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H elim (ysucc_inv_O_dx … H) ] qed-. lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m → L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V. /2 width=6 by lsuby_inv_pair1_aux/ qed-. fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l → L2 = ⋆ ∨ ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2. #L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/ [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct /3 width=5 by ex2_3_intro, or_intror/ ] qed-. lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l → L2 = ⋆ ∨ ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2. /2 width=5 by lsuby_inv_succ1_aux/ qed-. fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 → ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1. #L1 #L2 #l #m * -L1 -L2 -l -m [ #L1 #l #m #J2 #K2 #W1 #H destruct | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct /2 width=5 by ex2_3_intro/ | #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H elim (ysucc_inv_O_dx … H) | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H elim (ysucc_inv_O_dx … H) ] qed-. lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 → ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1. /2 width=9 by lsuby_inv_zero2_aux/ qed-. fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m → ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W. #L1 #L2 #l #m * -L1 -L2 -l -m [ #L1 #l #m #J2 #K2 #W #H destruct | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct /2 width=4 by ex2_2_intro/ | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H elim (ysucc_inv_O_dx … H) ] qed-. lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m → ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V. /2 width=6 by lsuby_inv_pair2_aux/ qed-. fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l → ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1. #L1 #L2 #l #m * -L1 -L2 -l -m [ #L1 #l #m #J2 #K2 #W2 #H destruct | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H elim (ylt_yle_false … H) // | #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct /2 width=5 by ex2_3_intro/ ] qed-. lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l → ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1. /2 width=5 by lsuby_inv_succ2_aux/ qed-. (* Basic forward lemmas *****************************************************) lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|. #L1 #L2 #l #m #H elim H -L1 -L2 -l -m /2 width=1 by yle_succ/ qed-. (* Properties on basic slicing **********************************************) lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W → l ≤ i → ∀m0. i + ⫯m0 = l + m → ∃∃I1,K1. K1 ⊆[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W. #L1 #L2 #l #m #H elim H -L1 -L2 -l -m [ #L1 #l #m #J2 #K2 #W #s #i #H elim (drop_inv_atom1 … H) -H #H destruct | #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #m0 >yplus_O2 >yplus_succ2 #H elim (ysucc_inv_O_dx … H) | #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ #m0 >yplus_succ2 >yplus_succ2 #H0 lapply (ysucc_inv_inj … H0) -H0 elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ] [ destruct -I2 /2 width=4 by drop_pair, ex2_2_intro/ | lapply (ylt_inv_O1 … Hi) #H yplus_succ1 >yplus_succ2 #H lapply (ysucc_inv_inj … H) -H