From: Ferruccio Guidi Date: Fri, 15 Jun 2012 21:22:03 +0000 (+0000) Subject: - relation between native type and atomic arity proced X-Git-Tag: make_still_working~1637 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=69644bb333b2862a5ff2ff434df8830e854e3385;p=helm.git - relation between native type and atomic arity proced - higher order native typing defined - some minor results towards subject reduction for native typing --- diff --git a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt b/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt index 544232544..c5fe6def0 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt +++ b/matita/matita/contribs/lambda_delta/basic_2/basic_1.txt @@ -209,9 +209,7 @@ ty3/arity ty3_arity ty3/arity_props ty3_predicative ty3/arity_props ty3_repellent ty3/arity_props ty3_acyclic -ty3/arity_props ty3_sn3 ty3/dec ty3_inference -ty3/fwd ty3_gen_appl ty3/fwd tys3_gen_nil ty3/fwd tys3_gen_cons ty3/fwd_nf2 ty3_gen_appl_nf2 diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma index a71c773bc..5d4d74399 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma @@ -130,6 +130,16 @@ lapply (IHV1 T1 T1 ?) -IHV1 // #HV1 @(cprs_trans … HV1) -HV1 /2 width=1/ qed. +lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2. + L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 → L ⊢ ⓐV1.ⓛW.T1 ➡* ⓓV2.T2. +#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 @(cprs_ind … HT12) -T2 +[ /3 width=1/ +| -HV12 #T #T2 #_ #HT2 #IHT1 + lapply (cpr_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 + @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/ +] +qed. + (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *) lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_aaa.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_aaa.ma new file mode 100644 index 000000000..962856983 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_aaa.ma @@ -0,0 +1,49 @@ +(**************************************************************************) +(* ___ *) +(* ||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/computation/csn_aaa.ma". +include "basic_2/equivalence/lcpcs_aaa.ma". +include "basic_2/dynamic/nta.ma". + +(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) + +(* Forward lemmas on atomic arity assignment for terms **********************) + +lemma nta_fwd_aaa: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃∃A. L ⊢ T ⁝ A & L ⊢ U ⁝ A. +#h #L #T #U #H elim H -L -T -U +[ /2 width=3/ +| #L #K #V #W #U #i #HLK #_ #HWU * #B #HV #HW + lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ +| #L #K #W #V #U #i #HLK #_ #HWU * #B #HW #_ -V + lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/ +| * #L #V #W #T #U #_ #_ * #B #HV #HW * #A #HT #HU + [ /3 width=3/ | /3 width=5/ ] +| #L #V #W #T #U #_ #_ * #B #HV #HW * #X #H1 #H2 + elim (aaa_inv_abst … H1) -H1 #B1 #A1 #HW1 #HT #H destruct + elim (aaa_inv_abst … H2) -H2 #B2 #A #_ #HU #H destruct + lapply (aaa_mono … HW1 … HW) -HW1 #H destruct /4 width=5/ +| #L #V #W #T #U #_ #_ * #X #HT #HUX * #A #H #_ -W + elim (aaa_inv_appl … H) -H #B #HV #HUA + lapply (aaa_mono … HUA … HUX) -HUX #H destruct /3 width=5/ +| #L #T #U #_ * #A #HT #HU /3 width=3/ +| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #X #HT #HU1 * #A #HU2 #_ + lapply (aaa_cpcs_mono … HU12 … HU1 … HU2) -U1 #H destruct /2 width=3/ +] +qed-. + +(* Note: this is the stong normalization property *) +(* Basic_1: was only: ty3_sn3 *) +theorem nta_fwd_csn: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → L ⊢ ⬇* T ∧ L ⊢ ⬇* U. +#h #L #T #U #H elim (nta_fwd_aaa … H) -H /3 width=2/ +qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_lift.ma index 0631ca4f2..57e06a1e9 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_lift.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_lift.ma @@ -99,7 +99,7 @@ lemma nta_inv_cast1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓝY.X : U → ⦃h, L⦄ ⊢ (* Advanced forvard lemmas **************************************************) -fact nta_fwd_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. T = ⓐY.X → +fact nta_fwd_pure1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. T = ⓐY.X → ∃∃V,W. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & L ⊢ ⓐY.V ⬌* U. #h #L #T #U #H elim H -L -T -U [ #L #k #X #Y #H destruct @@ -116,7 +116,7 @@ fact nta_fwd_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. T = ⓐY.X ] qed. -lemma nta_fwd_appl1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓐY.X : U → +lemma nta_fwd_pure1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓐY.X : U → ∃∃V,W. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & L ⊢ ⓐY.V ⬌* U. /2 width=3/ qed-. diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_ltpr.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_ltpr.ma index 6ecc0bb81..3799dcbb8 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_ltpr.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_ltpr.ma @@ -12,31 +12,157 @@ (* *) (**************************************************************************) +(* FRAGMENT 0 +include "basic_2/unfold/delift_lift.ma". + + +include "basic_2/equivalence/cpcs_cpcs.ma". +include "basic_2/unfold/delift_delift.ma". + + + +axiom pippo1: ∀T1,V. ∃∃T2. ⓓV.T1 ➡ T2 & ⋆.ⓓV ⊢ T1 ▼*[0, 1] ≡ T2. +(* +#T1 #V elim (pippo0 (⋆.ⓓV) 0 1 ? T1) +[ #T2 #HT12 @(ex2_1_intro … HT12) + @tpr +*) + +axiom pippo: ∀L,V,T1. ∃∃T2. L ⊢ ⓓV.T1 ➡ T2 & L.ⓓV ⊢ T1 ▼*[0, 1] ≡ T2. + +axiom cprs_inv_appl1_cpcs: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → ( + ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 & + L ⊢ U2 ➡* ⓐV2. T2 + ) ∨ + ∃∃V2,W,T. L ⊢ V1 ➡* V2 & + L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ⬌* U2. +#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/ +#U #U2 #_ #HU2 * * +[ #V0 #T0 #HV10 #HT10 #HUT0 + elim (cprs_strip … HUT0 … HU2) -U #U #H #HU2 + elim (cpr_inv_appl1 … H) -H * + [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/ + | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct + lapply (cprs_strap1 … HV10 HV02) -V0 #HV12 + lapply (cprs_div ? (ⓓV2.T) ? ? ? HU2) -HU2 /2 width=1/ /3 width=6/ + | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct + lapply (cprs_strap1 … HV10 HV0) -V0 #HV1 + lapply (cprs_trans … HT10 (ⓓW2.T2) ?) -HT10 /2 width=1/ -W0 -T #HT1 + elim (pippo L W2 T2) #T3 #H1T3 #H2T3 + elim (pippo L W2 (ⓐV2.T2)) #X #H1X #H + elim (delift_inv_flat1 … H) -H #V3 #Y #HV23 #HY #H destruct + + + @or_introl @(ex3_2_intro … HV1 HT1) -HV1 -HT1 + + ] +| /4 width=8/ +] +qed-. + + +include "basic_2/computation/cprs_lcprs.ma". +include "basic_2/dynamic/nta.ma". + +axiom cprs_inv_appl_abst: ∀L,V,T,W,U. L ⊢ ⓐV.T ➡* ⓛW.U → + ∃∃V0,W0,T0,W1,U1. ⇧[O, 1] W ≡ W1 & + ⇧[O + 1, 1] U ≡ U1 & + L ⊢ V ➡* V0 & L ⊢ T ➡* ⓛW0.T0 & + L.ⓓV0 ⊢ T0 ➡* ⓛW1.U1. +#L #V #T #W #U #H +elim (cprs_inv_appl1 … H) -H * +[ #V0 #T0 #_ #_ #H destruct +| #V0 #W0 #T0 #HV0 #HT0 #H + elim (cprs_inv_abbr1 … H) -H * + [ #V1 #T1 #_ #_ #H destruct + | #X #H #HT01 + elim (lift_inv_bind1 … H) -H #W1 #U1 #HW1 #HU1 #H destruct /2 width=10/ + ] +| #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #H + elim (cprs_inv_abbr1 … H) -H * + [ #V3 #T3 #_ #_ #H destruct + | #X #H #HT3 + elim (lift_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H destruct /2 width=10/ + +axiom pippo: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. L ⊢ T ➡* ⓛX.Y → + ∃Z. L ⊢ U ➡* ⓛX.Z. +#h #L #T #U #H elim H -L -T -U +[ +| +| +| +| +| #L #V #W #T #U #HTU #HUW #IHTU #IHUW #X #Y #HTY + elim (cprs_inv_appl1 … HTY) -HTY * + [ #V0 #T0 #_ #_ #H destruct + | #V0 #W0 #T0 #HV0 #HT0 #HTY + elim (IHTU … HT0) -IHTU -HT0 #Z #HUZ + elim (cprs_inv_abbr1 … HTY) -HTY * + [ #V1 #T1 #_ #_ #H destruct #X0 + + include "basic_2/dynamic/nta_ltpss.ma". include "basic_2/dynamic/nta_thin.ma". include "basic_2/dynamic/lsubn_nta.ma". -lemma cpr_beta: ∀L,V1,V2,W,T1,T2. - V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2. -#L #V1 #V2 #W #T1 #T2 #HV12 * #T #HT1 #HT2 -lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2 -lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 -@(ex2_1_intro … (ⓓV2.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too skow *) -qed. - -lemma cpr_beta2: ∀L,V1,V2,W,T1,T2. - L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2. -#L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2 -lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2 -lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 -@(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too skow *) -qed. +include "basic_2/hod/ntas_lift.ma". + + elim (nta_inv_pure1 … HUW) -HUW #V0 #U0 #U1 #HV0 #HU0 #HU0W #HU01 + @(ex2_2_intro … HYW) + [2: + + +axiom pippo_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀Y,X. Z = ⓐY.X → + ∀W,T. L ⊢ X ➡* ⓛW.T → ⦃h, L⦄ ⊢ Y : W. +#h #L #Z #U #H elim H -L -Z -U +[ +| +| +| +| +| #L #V #W #T #U #HTU #_ #_ #IHUW #Y #X #H #W0 #T0 #HX destruct + lapply (IHUW Y U ? ?) -IHUW -W // #T + @(ex2_2_intro … HYW) + [2: + +axiom pippo: ∀h,L,V,X,U. ⦃h, L⦄ ⊢ ⓐV.X : U → + ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ V : W. +#h #L #V #X #Y #H + +*) (* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************) (* Properties on context-free parallel reduction for local environments ******) - -lemma nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → +(* +axiom nta_ltpr_cprs_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → + ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 : U. +#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U +[ #L1 #k #L2 #_ #T2 #H + >(cprs_inv_sort1 … H) -H // +| +| +| +| +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #T2 #H + elim (cprs_inv_appl1 … H) -H * + [ #V2 #T0 #HV12 #HT10 #H destruct + elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U + @(nta_conv … (ⓐV2.U1)) (* /2 width=1/*) [ /4 width=2/] (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #HV12 #HT10 #HT02 + lapply (IHTU1 … HL12 (ⓛW2.T0) ?) -IHTU1 /2 width=1/ -HT10 #H + elim (nta_inv_bind1 … H) -H #W #U0 #HW2 #HTU0 #HU01 + elim (cpcs_inv_abst1 … HU01) -HU01 #W #U #HU1 #HU0 + lapply (IHUW1 … HL12 (ⓐV2.ⓛW.U) ?) -IHUW1 -HL12 /2 width=1/ -HV12 #H + + + + elim (nta_fwd_pure1 … H) -H #W0 #U2 #HVU2 #H #HW01 + elim (nta_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H + elim (cpcs_inv_abst1 … H) -H #W4 #U4 +*) +(* +axiom nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. #h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U [ #L1 #k #L2 #_ #T2 #H @@ -92,25 +218,36 @@ lemma nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ elim (nta_fwd_correct … H1) #T #H2 elim (nta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21 - elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 + elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0 lapply (lsubn_nta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2 - @(nta_conv … (ⓓV2.U2)) - [2: /2 width=2/ - |3: -h -L1 -T0 -T2 -W -U0 - |4: /3 width=2/ - ] + @(nta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *) | #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct ] - - - - - -| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H - elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct - lapply (cpr_tpss … HV12) #HV - elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=4/ ] #U #HU - @(nta_conv … (ⓐV2.U1)) // /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *) +| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #X #H + elim (tpr_inv_appl1 … H) -H * + [ #V2 #T2 #HV12 #HT12 #H destruct + elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U + @(nta_conv … (ⓐV2.U1)) /2 width=1/ /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *) + | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct + lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 /2 width=1/ -T0 #H + elim (nta_inv_bind1 … H) -H #W #U2 #HW2 #HTU2 #HU21 + lapply (IHUW1 … HL12 (ⓐV2.U1) ?) -IHUW1 -HL12 /2 width=1/ #H + elim (nta_inv_pure1 … H) -H #V0 #U0 #U #HV20 #HU10 #HU0W1 #HU0 + @(nta_conv … (ⓓV2.U2)) + [2: @nta_bind // + @(lsubn_nta_trans … HTU2) @lsubn_abbr // +(* + lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB + lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 + lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ +*) +*) +(* +axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y → + ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W. + +*) +(* SEGMENT 2 | #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct lapply (cpr_tpss … HU12) /4 width=4/ @@ -118,39 +255,13 @@ lemma nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ @(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *) ] qed. +*) - - - - - - - - - - - - - - - - - - +(* SEGMENT 3 fact nta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 → ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U. -| #L1 #V1 #T1 #B #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct - elim (tpr_inv_appl1 … H) -H * - [ #V2 #T2 #HV12 #HT12 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ /2 width=3/ - | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct - elim (nta_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct - lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB - lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0 - lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/ | #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct elim (nta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0 lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2 @@ -172,8 +283,9 @@ qed. /2 width=9/ qed. -lemma nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. +axiom nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A. /2 width=5/ qed. -lemma nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. +axiom nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A. /2 width=5/ qed. +*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_nta.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_nta.ma index b36725d92..05eb6c55d 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_nta.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_nta.ma @@ -37,10 +37,10 @@ theorem nta_mono: ∀h,L,T,U1. ⦃h, L⦄ ⊢ T : U1 → ∀U2. ⦃h, L⦄ ⊢ T elim (nta_inv_bind1 … H) -H #W2 #U2 #_ #HTU2 #H @(cpcs_trans … H) -X /3 width=1/ | #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H - elim (nta_fwd_appl1 … H) -H #U2 #W2 #_ #HTU2 #H + elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H @(cpcs_trans … H) -X /3 width=1/ | #L #V #W1 #T #U1 #_ #_ #IHTU1 #_ #X #H - elim (nta_fwd_appl1 … H) -H #U2 #W2 #_ #HTU2 #H + elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H @(cpcs_trans … H) -X /3 width=1/ | #L #T #U1 #_ #_ #X #H elim (nta_inv_cast1 … H) -H // diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_thin.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_thin.ma index 96e84dd51..e2e568d3a 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_thin.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/nta_thin.ma @@ -12,7 +12,6 @@ (* *) (**************************************************************************) -include "basic_2/substitution/ldrop_sfr.ma". include "basic_2/unfold/delift_lift.ma". include "basic_2/unfold/thin_ldrop.ma". include "basic_2/equivalence/cpcs_delift.ma". diff --git a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma index bbf70c782..d67e66630 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/equivalence/cpcs_cpcs.ma @@ -65,6 +65,14 @@ lemma cpcs_inv_abst_shift: ∀L,W1,W2,T1,T2. L ⊢ ⓛW1.T1 ⬌* ⓛW2.T2 → lapply (cpcs_inv_abst … H Abst W) -H // qed. +lemma cpcs_inv_abst1: ∀L,W1,T1,T. L ⊢ ⓛW1.T1 ⬌* T → + ∃∃W2,T2. L ⊢ T ➡* ⓛW2.T2 & L ⊢ ⓛW1.T1 ➡* ⓛW2.T2. +#L #W1 #T1 #T #H +elim (cpcs_inv_cprs … H) -H #X #H1 #H2 +elim (cprs_inv_abst1 Abst W1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct +@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *) +qed-. + (* Basic_1: was: pc3_gen_lift *) lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 → @@ -86,6 +94,10 @@ elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *) qed. +lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 → + ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2. +/3 width=1/ qed. + lemma cpcs_abst: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛV1. T1 ⬌* ⓛV2. T2. #L #V1 #V2 #HV12 #V #T1 #T2 #HT12 @@ -111,6 +123,20 @@ qed. lemma cpcs_bind_sn: ∀I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{I}V1. T ⬌* ⓑ{I}V2. T. * /2 width=1/ /2 width=2/ qed. +lemma cpcs_beta_dx: ∀L,V1,V2,W,T1,T2. + L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛW.T1 ⬌* ⓓV2.T2. +#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 +elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2 +lapply (cprs_beta_dx … HV12 HT1) -HV12 -HT1 #HT1 +lapply (cprs_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 +@(cprs_div … HT1) /2 width=1/ +qed. + +lemma cpcs_beta_dx_tpr_rev: ∀L,V1,V2,W,T1,T2. + V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 → + L ⊢ ⓓV2.T2 ⬌* ⓐV1.ⓛW.T1. +/4 width=1/ qed. + (* Note: it does not hold replacing |L1| with |L2| *) lemma cpcs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 → ∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ⬌* T2. diff --git a/matita/matita/contribs/lambda_delta/basic_2/hod/ntas.ma b/matita/matita/contribs/lambda_delta/basic_2/hod/ntas.ma new file mode 100644 index 000000000..c090997e6 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/hod/ntas.ma @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||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/dynamic/nta.ma". + +(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) + +definition ntas: sh → lenv → relation term ≝ + λh,L. star … (nta h L). + +interpretation "higher order native type assignment (term)" + 'NativeTypeStar h L T U = (ntas h L T U). + +(* Basic eliminators ********************************************************) +(* +lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 → + (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) → + ∀T2. L ⊢ T1 ➡* T2 → R T2. +#L #T1 #R #HT1 #IHT1 #T2 #HT12 +@(TC_star_ind … HT1 IHT1 … HT12) // +qed-. +*) +axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 → + (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) → + ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1. +(* +#h #L #T2 #R #HT2 #IHT2 #T1 #HT12 +@(star_ind_dx … HT2 IHT2 … HT12) // +qed-. +*) +(* Basic properties *********************************************************) + +lemma ntas_refl: ∀h,L,T. ⦃h, L⦄ ⊢ T :* T. +// qed. + +lemma ntas_strap1: ∀h,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2. +/2 width=3/ qed. + +lemma ntas_strap2: ∀h,L,T1,T,T2. + ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2. +/2 width=3/ qed. diff --git a/matita/matita/contribs/lambda_delta/basic_2/hod/ntas_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/hod/ntas_lift.ma new file mode 100644 index 000000000..1adced79d --- /dev/null +++ b/matita/matita/contribs/lambda_delta/basic_2/hod/ntas_lift.ma @@ -0,0 +1,71 @@ +(**************************************************************************) +(* ___ *) +(* ||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/dynamic/nta_lift.ma". +include "basic_2/hod/ntas.ma". + +(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************) + +(* Advanced properties on native type assignment for terms ******************) + +lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U → + ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U. +#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/ +qed. + +axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U → + ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z. +(* REQUIRES SUBJECT CONVERSION +#h #L #T #W #Y #H @(ntas_ind_dx … H) -T +[ #U #HYU + elim (nta_fwd_correct … HYU) #U0 #HU0 + elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U +*) + +(* Advanced inversion lemmas on native type assignment for terms ************) + +fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X → + ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +#h #L #Z #U #H elim H -L -Z -U +[ #L #k #X #Y #H destruct +| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct +| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct +| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct +| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/ +| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct + elim (IHUW U Y ?) -IHUW // /3 width=9/ +| #L #Z #U #_ #_ #X #Y #H destruct +| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct + elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT + lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/ +] +qed. + +(* Basic_1: was only: ty3_gen_appl *) +lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U → + ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & + L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T. +/2 width=3/ qed-. + +axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U → + ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W & + L ⊢ ⓐZ.ⓛY.W ⬌* U. +(* REQUIRES SUBJECT REDUCTION +#h #L #Z #Y #X #U #H +elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT +elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V +lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U +@(ex3_1_intro … HX0U) /2 width=2/ +*) diff --git a/matita/matita/contribs/lambda_delta/basic_2/notation.ma b/matita/matita/contribs/lambda_delta/basic_2/notation.ma index 8a947e683..31126679f 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/notation.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/notation.ma @@ -365,3 +365,9 @@ notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 :: break term 46 T2 )" notation "hvbox( h ⊢ break term 46 L1 : ⊑ break term 46 L2 )" non associative with precedence 45 for @{ 'CrSubEqN $h $L1 $L2 }. + +(* Higher order dynamic typing **********************************************) + +notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 :* break term 46 T2 )" + non associative with precedence 45 + for @{ 'NativeTypeStar $h $L $T1 $T2 }. diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma index 9994fdd6b..6db1a47aa 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma @@ -32,11 +32,22 @@ lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2. L.ⓛV ⊢ T1 ➡ T2 → L ⊢ ⓛV1. T1 ➡ ⓛV2. T2. #L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02 lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02 -@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ -V1 -T1 (**) (* explicit constructors *) -@tpss_bind // -V0 -@(tpss_lsubs_trans (L.ⓛV)) // -T0 -T2 /2 width=1/ +lapply (tpss_lsubs_trans … HT02 (L.ⓛV2) ?) -HT02 /2 width=1/ #HT02 +@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ (* explicit constructors *) qed. +lemma cpr_beta: ∀L,V1,V2,W,T1,T2. + L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2. +#L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2 +lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2 +lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2 +@(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *) +qed. + +lemma cpr_beta_dx: ∀L,V1,V2,W,T1,T2. + V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2. +/3 width=1/ qed. + (* Advanced inversion lemmas ************************************************) (* Basic_1: was: pr2_gen_lref *) diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma index a6846d3b8..1fc544b1b 100644 --- a/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma +++ b/matita/matita/contribs/lambda_delta/basic_2/unfold/delift_lift.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "basic_2/substitution/ldrop_sfr.ma". include "basic_2/unfold/tpss_lift.ma". include "basic_2/unfold/delift.ma". @@ -28,7 +29,38 @@ lapply (lift_trans_be … HV2 … HVU ? ?) -HV2 // >minus_plus commutative_plus in ⊢ (??%??→?); H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *) + elim (lift_total V2 0 d) /3 width=7/ +| #I #V1 #T1 #d #e #Hde #HL #H destruct + elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 + elim (IH (L.ⓑ{I}V1) T1 ? ? (d+1) e ? ? ?) -IH [3,6: // |2: skip |4,5: /2 width=1/ ] -Hde -HL #T2 #HT12 + lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/ +| #I #V1 #T1 #d #e #Hde #HL #H destruct + elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12 + elim (IH … T1 … Hde HL ?) -IH -Hde -HL [3,4: // |2: skip ] /3 width=2/ +] +qed. + +lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L → + ∃T2. L ⊢ T1 ▼*[d, e] ≡ T2. +/2 width=2/ qed-. + (* Advanced inversion lemmas ************************************************) lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i ▼*[d, e] ≡ U2 → i < d → U2 = #i. diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml b/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml index fff92a290..4aaba50b3 100644 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml +++ b/matita/matita/contribs/lambda_delta/ground_2/xoa.conf.xml @@ -28,6 +28,7 @@ 5 2 5 3 5 4 + 5 5 6 4 6 6 7 6 diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa.ma b/matita/matita/contribs/lambda_delta/ground_2/xoa.ma index 6fe0ca037..6214557bb 100644 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa.ma +++ b/matita/matita/contribs/lambda_delta/ground_2/xoa.ma @@ -128,6 +128,14 @@ inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4). +(* multiple existental quantifier (5, 5) *) + +inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝ + | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4). + (* multiple existental quantifier (6, 4) *) inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝ diff --git a/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma b/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma index 86f68b608..e1e2240ce 100644 --- a/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma +++ b/matita/matita/contribs/lambda_delta/ground_2/xoa_notation.ma @@ -154,6 +154,16 @@ notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term non associative with precedence 20 for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }. +(* multiple existental quantifier (5, 5) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }. + (* multiple existental quantifier (6, 4) *) notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"