X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcpr.ma;h=ad4fcb3135663f1c02fd4ed5f5d0a825be5ab376;hb=52e675f555f559c047d5449db7fc89a51b977d35;hp=69b591178b2cfee77a672b356657694b7cb4ef3a;hpb=29973426e0227ee48368d1c24dc0c17bf2baef77;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma index 69b591178..ad4fcb313 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma @@ -12,254 +12,243 @@ (* *) (**************************************************************************) -include "basic_2/notation/relations/pred_3.ma". -include "basic_2/grammar/cl_shift.ma". -include "basic_2/relocation/ldrop_append.ma". -include "basic_2/substitution/lsubr.ma". +include "basic_2/notation/relations/pred_4.ma". +include "basic_2/grammar/genv.ma". +include "basic_2/static/lsubr.ma". (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************) +(* activate genv *) (* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *) (* Note: cpr_flat: does not hold in basic_1 *) -inductive cpr: lenv → relation term ≝ -| cpr_atom : ∀I,L. cpr L (⓪{I}) (⓪{I}) -| cpr_delta: ∀L,K,V,V2,W2,i. - ⇩[0, i] L ≡ K. ⓓV → cpr K V V2 → - ⇧[0, i + 1] V2 ≡ W2 → cpr L (#i) W2 -| cpr_bind : ∀a,I,L,V1,V2,T1,T2. - cpr L V1 V2 → cpr (L.ⓑ{I}V1) T1 T2 → - cpr L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) -| cpr_flat : ∀I,L,V1,V2,T1,T2. - cpr L V1 V2 → cpr L T1 T2 → - cpr L (ⓕ{I} V1. T1) (ⓕ{I}V2.T2) -| cpr_zeta : ∀L,V,T1,T,T2. cpr (L.ⓓV) T1 T → - ⇧[0, 1] T2 ≡ T → cpr L (+ⓓV.T1) T2 -| cpr_tau : ∀L,V,T1,T2. cpr L T1 T2 → cpr L (ⓝV.T1) T2 -| cpr_beta : ∀a,L,V1,V2,W1,W2,T1,T2. - cpr L V1 V2 → cpr L W1 W2 → cpr (L.ⓛW1) T1 T2 → - cpr L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2) -| cpr_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2. - cpr L V1 V → ⇧[0, 1] V ≡ V2 → cpr L W1 W2 → cpr (L.ⓓW1) T1 T2 → - cpr L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2) +inductive cpr: relation4 genv lenv term term ≝ +| cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I}) +| cpr_delta: ∀G,L,K,V,V2,W2,i. + ⇩[i] L ≡ K. ⓓV → cpr G K V V2 → + ⇧[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2 +| cpr_bind : ∀a,I,G,L,V1,V2,T1,T2. + cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 → + cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2) +| cpr_flat : ∀I,G,L,V1,V2,T1,T2. + cpr G L V1 V2 → cpr G L T1 T2 → + cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2) +| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T → + ⇧[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2 +| cpr_eps : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2 +| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2. + cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 → + cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2) +| cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2. + cpr G L V1 V → ⇧[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 → + cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2) . interpretation "context-sensitive parallel reduction (term)" - 'PRed L T1 T2 = (cpr L T1 T2). + 'PRed G L T1 T2 = (cpr G L T1 T2). (* Basic properties *********************************************************) -lemma lsubr_cpr_trans: lsub_trans … cpr lsubr. -#L1 #T1 #T2 #H elim H -L1 -T1 -T2 +lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr. +#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 [ // -| #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 - elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -L1 * /3 width=6/ -|3,7: /4 width=1/ -|4,6: /3 width=1/ -|5,8: /4 width=3/ +| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12 + elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 * + /3 width=6 by cpr_delta/ +|3,7: /4 width=1 by lsubr_bind, cpr_bind, cpr_beta/ +|4,6: /3 width=1 by cpr_flat, cpr_eps/ +|5,8: /4 width=3 by lsubr_bind, cpr_zeta, cpr_theta/ ] qed-. (* Basic_1: was by definition: pr2_free *) -lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2. -#T1 #T2 #HT12 #L +lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2. +#G #T1 #T2 #HT12 #L lapply (lsubr_cpr_trans … HT12 L ?) // qed. (* Basic_1: includes by definition: pr0_refl *) -lemma cpr_refl: ∀T,L. ⦃G, L⦄ ⊢ T ➡ T. -#T elim T -T // * /2 width=1/ +lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T. +#G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/ qed. (* Basic_1: was: pr2_head_1 *) -lemma cpr_pair_sn: ∀I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → +lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T. -* /2 width=1/ qed. +* /2 width=1 by cpr_bind, cpr_flat/ qed. -lemma cpr_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) → +lemma cpr_delift: ∀G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓓV) → ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2. -#K #V #T1 elim T1 -T1 -[ * #i #L #d #HLK /2 width=4/ - elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ] +#G #K #V #T1 elim T1 -T1 +[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/ + #i #L #d #HLK elim (lt_or_eq_or_gt i d) + #Hid [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ] destruct elim (lift_total V 0 (i+1)) #W #HVW - elim (lift_split … HVW i i) // /3 width=6/ + elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/ | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2 - [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/ - | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/ + [ elim (IHU1 (L. ⓑ{I}W1) (d+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/ + | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/ ] ] qed-. -lemma cpr_append: l_appendable_sn … cpr. -#K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/ /2 width=3/ -#K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L -lapply (ldrop_fwd_length_lt2 … HK0) #H -@(cpr_delta … (L@@K0) V1 … HVW2) // -@(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *) -qed. - (* Basic inversion lemmas ***************************************************) -fact cpr_inv_atom1_aux: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} → +fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I} ∨ - ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & - K ⊢ V ➡ V2 & - ⇧[O, i + 1] V2 ≡ T2 & - I = LRef i. -#L #T1 #T2 * -L -T1 -T2 -[ #I #L #J #H destruct /2 width=1/ -| #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8/ -| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct -| #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct -| #L #V #T1 #T #T2 #_ #_ #J #H destruct -| #L #V #T1 #T2 #_ #J #H destruct -| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct -| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct + ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 & + ⇧[O, i + 1] V2 ≡ T2 & I = LRef i. +#G #L #T1 #T2 * -G -L -T1 -T2 +[ #I #G #L #J #H destruct /2 width=1 by or_introl/ +| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/ +| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct +| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct +| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct +| #G #L #V #T1 #T2 #_ #J #H destruct +| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct +| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct ] qed-. -lemma cpr_inv_atom1: ∀I,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 → +lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 → T2 = ⓪{I} ∨ - ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV & - K ⊢ V ➡ V2 & - ⇧[O, i + 1] V2 ≡ T2 & - I = LRef i. + ∃∃K,V,V2,i. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 & + ⇧[O, i + 1] V2 ≡ T2 & I = LRef i. /2 width=3 by cpr_inv_atom1_aux/ qed-. (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *) -lemma cpr_inv_sort1: ∀L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k. -#L #T2 #k #H +lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k. +#G #L #T2 #k #H elim (cpr_inv_atom1 … H) -H // * #K #V #V2 #i #_ #_ #_ #H destruct qed-. (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *) -lemma cpr_inv_lref1: ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 → +lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 → T2 = #i ∨ - ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV & - K ⊢ V ➡ V2 & + ∃∃K,V,V2. ⇩[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 & ⇧[O, i + 1] V2 ≡ T2. -#L #T2 #i #H -elim (cpr_inv_atom1 … H) -H /2 width=1/ -* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/ +#G #L #T2 #i #H +elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/ +* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/ qed-. -lemma cpr_inv_gref1: ∀L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p. -#L #T2 #p #H +lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p. +#G #L #T2 #p #H elim (cpr_inv_atom1 … H) -H // * #K #V #V2 #i #_ #_ #_ #H destruct qed-. -fact cpr_inv_bind1_aux: ∀L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 → +fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → ( - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & - L. ⓑ{I}V1 ⊢ T1 ➡ T2 & + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 & U2 = ⓑ{a,I}V2.T2 ) ∨ - ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr. -#L #U1 #U2 * -L -U1 -U2 -[ #I #L #b #J #W1 #U1 #H destruct -| #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct -| #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5/ -| #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct -| #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3/ -| #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct -| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct -| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct + ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & + a = true & I = Abbr. +#G #L #U1 #U2 * -L -U1 -U2 +[ #I #G #L #b #J #W1 #U1 #H destruct +| #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct +| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/ +| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct +| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/ +| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct +| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct +| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct ] qed-. -lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → ( - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & - L. ⓑ{I}V1 ⊢ T1 ➡ T2 & +lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → ( + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 & U2 = ⓑ{a,I}V2.T2 ) ∨ - ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr. + ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & + a = true & I = Abbr. /2 width=3 by cpr_inv_bind1_aux/ qed-. (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *) -lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → ( - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & - L. ⓓV1 ⊢ T1 ➡ T2 & +lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → ( + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 & U2 = ⓓ{a}V2.T2 ) ∨ - ∃∃T. L.ⓓV1 ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true. -#a #L #V1 #T1 #U2 #H -elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/ + ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true. +#a #G #L #V1 #T1 #U2 #H +elim (cpr_inv_bind1 … H) -H * +/3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/ qed-. (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *) -lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 → - ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & L.ⓛV1 ⊢ T1 ➡ T2 & +lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 → + ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 & U2 = ⓛ{a}V2.T2. -#a #L #V1 #T1 #U2 #H +#a #G #L #V1 #T1 #U2 #H elim (cpr_inv_bind1 … H) -H * -[ /3 width=5/ +[ /3 width=5 by ex3_2_intro/ | #T #_ #_ #_ #H destruct ] qed-. -fact cpr_inv_flat1_aux: ∀L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 → +fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 → ∀I,V1,U1. U = ⓕ{I}V1.U1 → ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 & U2 = ⓕ{I} V2. T2 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast) | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 & - L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 & + ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 & - ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 & + ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 & U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl. -#L #U #U2 * -L -U -U2 -[ #I #L #J #W1 #U1 #H destruct -| #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct -| #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct -| #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5/ -| #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct -| #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1/ -| #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11/ -| #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13/ +#G #L #U #U2 * -L -U -U2 +[ #I #G #L #J #W1 #U1 #H destruct +| #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct +| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct +| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/ +| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct +| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/ +| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or4_intro2, ex6_6_intro/ +| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or4_intro3, ex7_7_intro/ ] qed-. -lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 → +lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 → ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 & U2 = ⓕ{I}V2.T2 | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast) | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 & - L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 & + ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 & - ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 & + ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 & U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl. /2 width=3 by cpr_inv_flat1_aux/ qed-. (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *) -lemma cpr_inv_appl1: ∀L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 → +lemma cpr_inv_appl1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 → ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 & U2 = ⓐV2.T2 | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 & - L.ⓛW1 ⊢ T1 ➡ T2 & + ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2 | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 & - ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 & + ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 & U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2. -#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H * -[ /3 width=5/ +#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H * +[ /3 width=5 by or3_intro0, ex3_2_intro/ | #_ #H destruct -| /3 width=11/ -| /3 width=13/ +| /3 width=11 by or3_intro1, ex5_6_intro/ +| /3 width=13 by or3_intro2, ex6_7_intro/ ] qed-. (* Note: the main property of simple terms *) -lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ → +lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ → ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 & U = ⓐV2. T2. -#L #V1 #T1 #U #H #HT1 +#G #L #V1 #T1 #U #H #HT1 elim (cpr_inv_appl1 … H) -H * [ /2 width=5/ | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct @@ -270,14 +259,13 @@ elim (cpr_inv_appl1 … H) -H * qed-. (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *) -lemma cpr_inv_cast1: ∀L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → ( +lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → ( ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 & U2 = ⓝ V2. T2 - ) ∨ - ⦃G, L⦄ ⊢ U1 ➡ U2. -#L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H * -[ /3 width=5/ -| /2 width=1/ + ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2. +#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H * +[ /3 width=5 by ex3_2_intro, or_introl/ +| /2 width=1 by or_intror/ | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct | #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct ] @@ -285,30 +273,13 @@ qed-. (* Basic forward lemmas *****************************************************) -lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b. +lemma cpr_fwd_bind1_minus: ∀I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b. ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 & T = -ⓑ{I}V2.T2. -#I #L #V1 #T1 #T #H #b +#I #G #L #V1 #T1 #T #H #b elim (cpr_inv_bind1 … H) -H * -[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/ -| #T2 #_ #_ #H destruct -] -qed-. - -lemma cpr_fwd_shift1: ∀L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T → - ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2. -#L1 @(lenv_ind_dx … L1) -L1 normalize -[ #L #T1 #T #HT1 - @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *) -| #I #L1 #V1 #IH #L #T1 #X - >shift_append_assoc normalize #H - elim (cpr_inv_bind1 … H) -H * - [ #V0 #T0 #_ #HT10 #H destruct - elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct - >append_length >HL12 -HL12 - @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *) - | #T #_ #_ #H destruct - ] +[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/ +| #T2 #_ #_ #H destruct ] qed-. @@ -319,6 +290,6 @@ qed-. pr2_gen_ctail pr2_ctail *) (* Basic_1: removed local theorems 4: - pr0_delta_tau pr0_cong_delta + pr0_delta_eps pr0_cong_delta pr2_free_free pr2_free_delta *)