(**************************************************************************) (* ___ *) (* ||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/substitution/cofrees_lift.ma". include "basic_2/substitution/llpx_sn_alt1.ma". (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****) (* alternative definition of llpx_sn (not recursive) *) definition llpx_sn_alt2: relation4 bind2 lenv term term → relation4 ynat term lenv lenv ≝ λR,d,T,L1,L2. |L1| = |L2| ∧ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (L1 ⊢ i ~ϵ 𝐅*[d]⦃T⦄ → ⊥) → ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2 ). (* Main properties **********************************************************) lemma cpy_inv_nlift2_be: ∀G,L,U1,U2,d. ⦃G, L⦄ ⊢ U1 ▶[d, ∞] U2 → ∀i. d ≤ yinj i → ∀K,s. ⇩[s, i, 1] L ≡ K → (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) → (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥). #G #L #U1 #U2 #d #HU12 #i #Hdi #K #s #HLK #HnU2 #T1 #HTU1 elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) /2 width=2 by/ qed-. lemma cpy_inv_nlift2_ge: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 → ∀i. d ≤ yinj i → (* yinj i + yinj 1 ≤ d + e → *) (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) → ∃∃j. d ≤ yinj j & j ≤ i & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥). #G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e [ #I #G #L #d #e #i #Hdi (* #Hide *) #H @(ex3_intro … i) /2 width=2 by/ | #I #G #L #K #V #W #j #d #e #Hdj #Hjde #HLK #HVW #i #Hdi (* #Hide *) #HnW elim (le_or_ge i j) #Hij [2: @(ex3_intro … j) /2 width=7 by lift_inv_lref2_be/ ] elim (lift_split … HVW i j) -HVW // #X #_ #H elim HnW -HnW // | #a #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_bind … H) -H [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 // #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=9 by nlift_bind_sn/ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/ #j #Hdj #Hji >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ] #HnW1 @(ex3_intro … (j-1)) /3 width=9 by nlift_bind_dx, yle_pred, monotonic_pred/ ] | #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_flat … H) -H [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 // #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=8 by nlift_flat_sn/ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 // #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=8 by nlift_flat_dx/ ] ] qed-. axiom frees_fwd_nlift_ge: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → d ≤ yinj i → ∃∃j. d ≤ yinj j & j ≤ i & (∀T. ⇧[j, 1] T ≡ U → ⊥). (* lemma frees_ind_nlift: ∀L,d. ∀R:relation2 term nat. (∀U1,i. d ≤ yinj i → (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1 i) → (∀U1,U2,i,j. d ≤ yinj j → j ≤ i → ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) → R U2 i → R U1 j) → ∀U,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U i. #L #d #R #IH1 #IH2 #U1 #i #Hdi #H @(frees_ind … H) -U1 /3 width=4 by/ #U1 #U2 #HU12 #HnU2 #HU2 @(IH2 … HU12 … HU2) qed-. *)(* lemma frees_fwd_nlift: ∀L,d. ∀R:relation2 term nat. ( ∀U1,j. (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥) ∨ (∃∃U2,i. d ≤ yinj j → j < i & (L ⊢ j ~ϵ 𝐅*[d]⦃U1⦄ → ⊥) & ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 & R U1 i ) → d ≤ yinj j → R U1 j ) → ∀U,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U i. #L #d #R #IHR #U1 #j #Hdj #H elim (frees_inv_gen … H) -H #U2 #H generalize in match Hdj; -Hdj generalize in match j; -j @(cpys_ind … H) -U2 [ #j #Hdj #HnU1 @IHR -IHR /3 width=2 by or_introl/ | #U0 #U2 #HU10 #HU02 #IHU10 #j #Hdj #HnU2 elim (cpy_inv_nlift2_ge … HU02 … Hdj HnU2) -HU02 -HnU2 #i #Hdi #Hij #HnU0 lapply (IHU10 … HnU0) // -IHU10 #Hi @IHR -IHR // -Hdj @or_intror lemma frees_fwd_nlift: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → ∃∃j. d ≤ yinj j & j ≤ i & (∀T. ⇧[j, 1] T ≡ U → ⊥). #L #U1 #d #i #Hdi #H #U2 #H #HnU2 @(cpys_ind_dx … H) -U1 [ @(ex3_intro … i) /2 width=2 by/ ] -Hdi -HnU2 #U1 #U0 #HU10 #_ * #j #Hdj #Hji #HnU0 elim (cpy_inv_nlift2_ge … HU10 … Hdj HnU0) -U0 -Hdj /3 width=5 by transitive_le, ex3_intro/ qed-. *) theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2. #R #L1 #L2 #U1 #d #H elim (llpx_sn_inv_alt1 … H) -H #HL12 #IH @conj // -HL12 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_fwd_nlift_ge … H Hdi) -H -Hdi #j #Hdj #Hji #HnU1 lapply (ldrop_fwd_length_lt2 … HLK1) #HL1 lapply (ldrop_fwd_length_lt2 … HLK2) #HL2 lapply (le_to_lt_to_lt … Hji HL1) -HL1 #HL1 lapply (le_to_lt_to_lt … Hji HL2) -HL2 #HL2 elim (ldrop_O1_lt L1 j) // #Z1 #Y1 #X1 #HLY1 elim (ldrop_O1_lt L2 j) // #Z2 #Y2 #X2 #HLY2 generalize in match V2; -V2 generalize in match V1; -V1 generalize in match K2; -K2 generalize in match K1; -K1 generalize in match I2; -I2 generalize in match I1; -I1 generalize in match IH; -IH @(frees_ind_nlift … Hdi H) -U1 -i [ #U1 #i #Hdi #HnU1 #IH #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 elim (IH … HnU1 HLK1 HLK2) -IH -HnU1 -HLK1 -HLK2 /2 width=1 by conj/ | #U1 #U2 #i #j #Hdj #Hji #HU12 #HnU2 #IHU12 #IH #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 (* *) @(IHU12) … HLK1 HLK2) @(IHU12 … HLK1 HLK2) -IHU02 -I1 -I2 -K1 -K2 -V1 -V2 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH -HLK1 -HLK2 // elim (frees_fwd_nlift … HnU1) // -HnU1 -Hdi #j #Hdj #Hji #HnU1 lapply (ldrop_fwd_length_lt2 … HLK1) #HL1 lapply (ldrop_fwd_length_lt2 … HLK2) #HL2 lapply (le_to_lt_to_lt … Hji HL1) -HL1 #HL1 lapply (le_to_lt_to_lt … Hji HL2) -HL2 #HL2 elim (ldrop_O1_lt L1 j) // #Z1 #Y1 #X1 #HLY1 elim (ldrop_O1_lt L2 j) // #Z2 #Y2 #X2 #HLY2 elim (IH … HnU1 HLY1 HLY2) // #H #HX12 #HY12 destruct theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2. #R #L1 #L2 #U1 #d #H @(llpx_sn_ind_alt1 … H) -L1 -L2 -U1 -d #L1 #L2 #U1 #d #HL12 #IH @conj // -HL12 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1 #U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1 [ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/ | #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2 @(cpy_inv_nlift2_be … HU10) /2 width=3 by/ theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T2,d. llpx_sn R d T2 L1 L2 → ∀T1. ⦃⋆, L1⦄ ⊢ T1 ▶*[d, ∞] T2 → llpx_sn_alt2 R d T1 L1 L2. #R #L1 #L2 #U2 #d #H elim (llpx_sn_inv_alt1 … H) -H #HL12 #IH #U1 #HU12 @conj // -HL12 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1 #U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1 [ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/ | #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2 @(cpy_inv_nlift2_be … HU10) /2 width=3 by/ theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2. #R #L1 #L2 #U1 #d #H elim (llpx_sn_inv_alt1 … H) -H #HL12 #IH @conj // -HL12 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1 #U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1 [ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/ | #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2 @(cpy_inv_nlift2_be … HU10) /2 width=3 by/