From: Ferruccio Guidi Date: Mon, 10 Oct 2011 17:39:08 +0000 (+0000) Subject: refactoring ... X-Git-Tag: make_still_working~2224 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=55dc00c1c44cc21c7ae179cb9df03e7446002c46;p=helm.git refactoring ... --- diff --git a/matita/matita/contribs/lambda_delta/Basic_2/Basic-1.txt b/matita/matita/contribs/lambda_delta/Basic_2/Basic-1.txt new file mode 100644 index 000000000..b45870449 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/Basic-1.txt @@ -0,0 +1,468 @@ +# waiting #################################################################### + +aplus/props aplus_reg_r +aplus/props aplus_assoc +aplus/props aplus_asucc +aplus/props aplus_sort_O_S_simpl +aplus/props aplus_sort_S_S_simpl +aplus/props aplus_asort_O_simpl +aplus/props aplus_asort_le_simpl +aplus/props aplus_asort_simpl +aplus/props aplus_ahead_simpl +aplus/props aplus_asucc_false +aplus/props aplus_inj +aprem/fwd aprem_gen_sort +aprem/fwd aprem_gen_head_O +aprem/fwd aprem_gen_head_S +aprem/props aprem_repl +aprem/props aprem_asucc +arity/aprem arity_aprem +arity/cimp arity_cimp_conf +arity/fwd arity_gen_sort +arity/fwd arity_gen_lref +arity/fwd arity_gen_bind +arity/fwd arity_gen_abst +arity/fwd arity_gen_appl +arity/fwd arity_gen_cast +arity/fwd arity_gen_appls +arity/fwd arity_gen_lift +arity/lift1 arity_lift1 +arity/pr3 arity_sred_wcpr0_pr0 +arity/pr3 arity_sred_wcpr0_pr1 +arity/pr3 arity_sred_pr2 +arity/pr3 arity_sred_pr3 +arity/props node_inh +arity/props arity_lift +arity/props arity_mono +arity/props arity_repellent +arity/props arity_appls_cast +arity/props arity_appls_abbr +arity/props arity_appls_bind +arity/subst0 arity_gen_cvoid_subst0 +arity/subst0 arity_gen_cvoid +arity/subst0 arity_fsubst0 +arity/subst0 arity_subst0 +asucc/fwd asucc_gen_sort +asucc/fwd asucc_gen_head +cnt/props cnt_lift +C/props clt_wf__q_ind +C/props clt_wf_ind +C/props chead_ctail +C/props clt_thead (ctail) +C/props c_tail_ind +csuba/arity csuba_arity +csuba/arity csuba_arity_rev +csuba/arity arity_appls_appl +csuba/clear csuba_clear_conf +csuba/clear csuba_clear_trans +csuba/drop csuba_drop_abbr +csuba/drop csuba_drop_abst +csuba/drop csuba_drop_abst_rev +csuba/drop csuba_drop_abbr_rev +csuba/fwd csuba_gen_abbr +csuba/fwd csuba_gen_void +csuba/fwd csuba_gen_abst +csuba/fwd csuba_gen_flat +csuba/fwd csuba_gen_bind +csuba/fwd csuba_gen_abst_rev +csuba/fwd csuba_gen_void_rev +csuba/fwd csuba_gen_abbr_rev +csuba/fwd csuba_gen_flat_rev +csuba/fwd csuba_gen_bind_rev +csuba/getl csuba_getl_abbr +csuba/getl csuba_getl_abst +csuba/getl csuba_getl_abst_rev +csuba/getl csuba_getl_abbr_rev +csuba/props csuba_refl +csubc/arity csubc_arity_conf +csubc/arity csubc_arity_trans +csubc/clear csubc_clear_conf +csubc/csuba csubc_csuba +csubc/drop1 drop1_csubc_trans +csubc/drop1 csubc_drop1_conf_rev +csubc/drop csubc_drop_conf_O +csubc/drop drop_csubc_trans +csubc/drop csubc_drop_conf_rev +csubc/fwd csubc_gen_sort_l +csubc/fwd csubc_gen_head_l +csubc/fwd csubc_gen_sort_r +csubc/fwd csubc_gen_head_r +csubc/getl csubc_getl_conf +csubc/props csubc_refl +csubt/clear csubt_clear_conf +csubt/csuba csubt_csuba +csubt/drop csubt_drop_flat +csubt/drop csubt_drop_abbr +csubt/drop csubt_drop_abst +csubt/fwd csubt_gen_abbr +csubt/fwd csubt_gen_abst +csubt/fwd csubt_gen_flat +csubt/fwd csubt_gen_bind +csubt/getl csubt_getl_abbr +csubt/getl csubt_getl_abst +csubt/pc3 csubt_pr2 +csubt/pc3 csubt_pc3 +csubt/props csubt_refl +csubt/ty3 csubt_ty3 +csubt/ty3 csubt_ty3_ld +csubv/clear csubv_clear_conf +csubv/clear csubv_clear_conf_void +csubv/drop csubv_drop_conf +csubv/getl csubv_getl_conf +csubv/getl csubv_getl_conf_void +csubv/props csubv_bind_same +csubv/props csubv_refl +drop1/fwd drop1_gen_pnil +drop1/fwd drop1_gen_pcons +drop1/getl drop1_getl_trans +drop1/props drop1_skip_bind +drop1/props drop1_cons_tail +drop1/props drop1_trans +drop/props drop_ctail +ex0/props aplus_gz_le +ex0/props aplus_gz_ge +ex0/props next_plus_gz +ex0/props leqz_leq +ex0/props leq_leqz +ex1/props ex1__leq_sort_SS +ex1/props ex1_arity +ex1/props ex1_ty3 +ex2/props ex2_nf2 +ex2/props ex2_arity +fsubst0/fwd fsubst0_gen_base +leq/asucc asucc_repl +leq/asucc asucc_inj +leq/asucc leq_asucc +leq/asucc leq_ahead_asucc_false +leq/asucc leq_asucc_false +leq/fwd leq_gen_sort1 +leq/fwd leq_gen_head1 +leq/fwd leq_gen_sort2 +leq/fwd leq_gen_head2 +leq/props ahead_inj_snd +leq/props leq_refl +leq/props leq_eq +leq/props leq_sym +leq/props leq_trans +leq/props leq_ahead_false_1 +leq/props leq_ahead_false_2 +lift1/fwd lift1_sort +lift1/fwd lift1_lref +lift1/fwd lift1_bind +lift1/fwd lift1_flat +lift1/fwd lift1_cons_tail +lift1/fwd lifts1_flat +lift1/fwd lifts1_nil +lift1/fwd lifts1_cons +lift1/props lift1_lift1 +lift1/props lift1_xhg +lift1/props lifts1_xhg +lift1/props lift1_free +lift/props thead_x_lift_y_y +lift/props lifts_tapp +lift/props lifts_inj +llt/props lweight_repl +llt/props llt_repl +llt/props llt_trans +llt/props llt_head_sx +llt/props llt_head_dx +llt/props llt_wf__q_ind +llt/props llt_wf_ind +next_plus/props next_plus_assoc +next_plus/props next_plus_next +next_plus/props next_plus_lt +nf2/arity arity_nf2_inv_all +nf2/dec nf2_dec +nf2/fwd nf2_gen_lref +nf2/fwd nf2_gen_abst +nf2/fwd nf2_gen_cast +nf2/fwd nf2_gen_beta +nf2/fwd nf2_gen_flat +nf2/fwd nf2_gen__nf2_gen_aux +nf2/fwd nf2_gen_abbr +nf2/fwd nf2_gen_void +nf2/iso nf2_iso_appls_lref +nf2/lift1 nf2_lift1 +nf2/pr3 nf2_pr3_unfold +nf2/pr3 nf2_pr3_confluence +nf2/props nf2_sort +nf2/props nf2_csort_lref +nf2/props nf2_abst +nf2/props nf2_abst_shift +nf2/props nfs2_tapp +nf2/props nf2_appls_lref +nf2/props nf2_appl_lref +nf2/props nf2_lref_abst +nf2/props nf2_lift +pc1/props pc1_pr0_r +pc1/props pc1_pr0_x +pc1/props pc1_refl +pc1/props pc1_pr0_u +pc1/props pc1_s +pc1/props pc1_head_1 +pc1/props pc1_head_2 +pc1/props pc1_t +pc1/props pc1_pr0_u2 +pc1/props pc1_head +pc3/dec pc3_dec +pc3/dec pc3_abst_dec +pc3/fsubst0 pc3_pr2_fsubst0 +pc3/fsubst0 pc3_pr2_fsubst0_back +pc3/fsubst0 pc3_fsubst0 +pc3/fwd pc3_gen_sort +pc3/fwd pc3_gen_abst +pc3/fwd pc3_gen_abst_shift +pc3/fwd pc3_gen_lift +pc3/fwd pc3_gen_not_abst +pc3/fwd pc3_gen_lift_abst +pc3/fwd pc3_gen_sort_abst +pc3/left pc3_ind_left__pc3_left_pr3 +pc3/left pc3_ind_left__pc3_left_trans +pc3/left pc3_ind_left__pc3_left_sym +pc3/left pc3_ind_left__pc3_left_pc3 +pc3/left pc3_ind_left__pc3_pc3_left +pc3/left pc3_ind_left +pc3/nf2 pc3_nf2 +pc3/nf2 pc3_nf2_unfold +pc3/pc1 pc3_pc1 +pc3/props clear_pc3_trans +pc3/props pc3_pr2_r +pc3/props pc3_pr2_x +pc3/props pc3_pr3_r +pc3/props pc3_pr3_x +pc3/props pc3_pr3_t +pc3/props pc3_refl +pc3/props pc3_s +pc3/props pc3_thin_dx +pc3/props pc3_head_1 +pc3/props pc3_head_2 +pc3/props pc3_pr2_u +pc3/props pc3_t +pc3/props pc3_pr2_u2 +pc3/props pc3_pr3_conf +pc3/props pc3_head_12 +pc3/props pc3_head_21 +pc3/props pc3_pr0_pr2_t +pc3/props pc3_pr2_pr2_t +pc3/props pc3_pr2_pr3_t +pc3/props pc3_pr3_pc3_t +pc3/props pc3_lift +pc3/props pc3_eta +pc3/subst1 pc3_gen_cabbr +pc3/wcpr0 pc3_wcpr0__pc3_wcpr0_t_aux +pc3/wcpr0 pc3_wcpr0_t +pc3/wcpr0 pc3_wcpr0 +pr0/fwd pr0_gen_void +pr0/dec nf0_dec +pr0/subst1 pr0_subst1_back +pr0/subst1 pr0_subst1_fwd +pr1/pr1 pr1_strip +pr1/pr1 pr1_confluence +pr1/props pr1_pr0 +pr1/props pr1_t +pr1/props pr1_head_1 +pr1/props pr1_head_2 +pr1/props pr1_comp +pr1/props pr1_eta +pr2/clen pr2_gen_ctail + +# check ###################################################################### + +pr2/fwd pr2_gen_void +pr2/props pr2_ctail + + +# waiting #################################################################### + +pr3/fwd pr3_gen_sort +pr3/fwd pr3_gen_abst +pr3/fwd pr3_gen_cast +pr3/fwd pr3_gen_lift +pr3/fwd pr3_gen_lref +pr3/fwd pr3_gen_void +pr3/fwd pr3_gen_abbr +pr3/fwd pr3_gen_appl +pr3/fwd pr3_gen_bind +pr3/iso pr3_iso_appls_abbr +pr3/iso pr3_iso_appls_cast +pr3/iso pr3_iso_appl_bind +pr3/iso pr3_iso_appls_appl_bind +pr3/iso pr3_iso_appls_bind +pr3/iso pr3_iso_beta +pr3/iso pr3_iso_appls_beta +pr3/pr1 pr3_pr1 +pr3/pr3 pr3_strip +pr3/pr3 pr3_confluence +pr3/props clear_pr3_trans +pr3/props pr3_pr2 +pr3/props pr3_t +pr3/props pr3_thin_dx +pr3/props pr3_head_1 +pr3/props pr3_head_2 +pr3/props pr3_head_21 +pr3/props pr3_head_12 +pr3/props pr3_cflat +pr3/props pr3_flat +pr3/props pr3_pr0_pr2_t +pr3/props pr3_pr2_pr2_t +pr3/props pr3_pr2_pr3_t +pr3/props pr3_pr3_pr3_t +pr3/props pr3_lift +pr3/props pr3_eta +pr3/subst1 pr3_subst1 +pr3/subst1 pr3_gen_cabbr +pr3/wcpr0 pr3_wcpr0_t +sc3/arity sc3_arity_csubc +sc3/arity sc3_arity +sc3/props sc3_arity_gen +sc3/props sc3_repl +sc3/props sc3_lift +sc3/props sc3_lift1 +sc3/props sc3_abbr +sc3/props sc3_cast +sc3/props sc3_props__sc3_sn3_abst +sc3/props sc3_sn3 +sc3/props sc3_abst +sc3/props sc3_bind +sc3/props sc3_appl +sn3/fwd sn3_gen_bind +sn3/fwd sn3_gen_flat +sn3/fwd sn3_gen_head +sn3/fwd sn3_gen_cflat +sn3/fwd sn3_gen_lift +sn3/lift1 sns3_lifts1 +sn3/nf2 sn3_nf2 +sn3/nf2 nf2_sn3 +sn3/props sn3_pr3_trans +sn3/props sn3_pr2_intro +sn3/props sn3_cast +sn3/props sn3_cflat +sn3/props sn3_shift +sn3/props sn3_change +sn3/props sn3_gen_def +sn3/props sn3_cdelta +sn3/props sn3_cpr3_trans +sn3/props sn3_bind +sn3/props sn3_beta +sn3/props sn3_appl_lref +sn3/props sn3_appl_abbr +sn3/props sn3_appl_cast +sn3/props sn3_appl_bind +sn3/props sn3_appl_appl +sn3/props sn3_appl_beta +sn3/props sn3_appl_appls +sn3/props sn3_appls_lref +sn3/props sn3_appls_cast +sn3/props sn3_appls_bind +sn3/props sn3_appls_beta +sn3/props sn3_lift +sn3/props sn3_abbr +sn3/props sn3_appls_abbr +sn3/props sns3_lifts +sty0/fwd sty0_gen_sort +sty0/fwd sty0_gen_lref +sty0/fwd sty0_gen_bind +sty0/fwd sty0_gen_appl +sty0/fwd sty0_gen_cast +sty0/props sty0_lift +sty0/props sty0_correct +sty1/cnt sty1_cnt +sty1/props sty1_trans +sty1/props sty1_bind +sty1/props sty1_appl +sty1/props sty1_lift +sty1/props sty1_correct +sty1/props sty1_abbr +sty1/props sty1_cast2 +subst0/dec dnf_dec2 +subst0/dec dnf_dec +subst1/props subst1_ex +subst/fwd subst_sort +subst/fwd subst_lref_lt +subst/fwd subst_lref_eq +subst/fwd subst_lref_gt +subst/fwd subst_head +subst/props subst_lift_SO +subst/props subst_subst0 +T/dec terms_props__bind_dec +T/dec bind_dec_not +T/dec terms_props__flat_dec +T/dec terms_props__kind_dec +T/dec term_dec +T/dec binder_dec +T/dec abst_dec +tlist/props tslt_wf__q_ind +tlist/props tslt_wf_ind +tlist/props theads_tapp +tlist/props tcons_tapp_ex +tlist/props tlist_ind_rev +T/props not_abbr_abst +T/props not_void_abst +T/props not_abbr_void +T/props not_abst_void +T/props thead_x_y_y +T/props tweight_lt +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/fsubst0 ty3_fsubst0 +ty3/fsubst0 ty3_csubst0 +ty3/fsubst0 ty3_subst0 +ty3/fwd ty3_gen_sort +ty3/fwd ty3_gen_lref +ty3/fwd ty3_gen_bind +ty3/fwd ty3_gen_appl +ty3/fwd ty3_gen_cast +ty3/fwd tys3_gen_nil +ty3/fwd tys3_gen_cons +ty3/fwd_nf2 ty3_gen_appl_nf2 +ty3/fwd_nf2 ty3_inv_lref_nf2_pc3 +ty3/fwd_nf2 ty3_inv_lref_nf2 +ty3/fwd_nf2 ty3_inv_appls_lref_nf2 +ty3/fwd_nf2 ty3_inv_lref_lref_nf2 +ty3/nf2 ty3_nf2_inv_abst_premise_csort +ty3/nf2 ty3_nf2_inv_all +ty3/nf2 ty3_nf2_inv_sort +ty3/nf2 ty3_nf2_gen__ty3_nf2_inv_abst_aux +ty3/nf2 ty3_nf2_inv_abst +ty3/pr3 ty3_sred_wcpr0_pr0 +ty3/pr3 ty3_sred_pr0 +ty3/pr3 ty3_sred_pr1 +ty3/pr3 ty3_sred_pr2 +ty3/pr3 ty3_sred_pr3 +ty3/pr3_props ty3_cred_pr2 +ty3/pr3_props ty3_cred_pr3 +ty3/pr3_props ty3_gen_lift +ty3/pr3_props ty3_tred +ty3/pr3_props ty3_sconv_pc3 +ty3/pr3_props ty3_sred_back +ty3/pr3_props ty3_sconv +ty3/props ty3_lift +ty3/props ty3_correct +ty3/props ty3_unique +ty3/props ty3_gen_abst_abst +ty3/props ty3_typecheck +ty3/props ty3_getl_subst0 +ty3/sty0 ty3_sty0 +ty3/subst1 ty3_gen_cabbr +ty3/subst1 ty3_gen_cvoid +wf3/clear wf3_clear_conf +wf3/clear clear_wf3_trans +wf3/fwd wf3_gen_sort1 +wf3/fwd wf3_gen_bind1 +wf3/fwd wf3_gen_flat1 +wf3/fwd wf3_gen_head2 +wf3/getl wf3_getl_conf +wf3/getl getl_wf3_trans +wf3/props wf3_mono +wf3/props wf3_total +wf3/props ty3_shift1 +wf3/props wf3_idem +wf3/props wf3_ty3 +wf3/ty3 wf3_pr2_conf +wf3/ty3 wf3_pr3_conf +wf3/ty3 wf3_pc3_conf +wf3/ty3 wf3_ty3_conf diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_shift.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_shift.ma new file mode 100644 index 000000000..d72b6c4cb --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_shift.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/lenv.ma". + +(* SHIFT OF A CLOSURE *******************************************************) + +let rec shift L T on L ≝ match L with +[ LAtom ⇒ T +| LPair L I V ⇒ shift L (𝕓{I} V. T) +]. + +interpretation "shift (closure)" 'Append L T = (shift L T). diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_weight.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_weight.ma new file mode 100644 index 000000000..35bf32a9d --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/cl_weight.ma @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/lenv_weight.ma". +include "Basic-2/grammar/cl_shift.ma". + +(* WEIGHT OF A CLOSURE ******************************************************) + +definition cw: lenv → term → ? ≝ λL,T. #[L] + #[T]. + +interpretation "weight (closure)" 'Weight L T = (cw L T). + +(* Basic properties *********************************************************) + +(* Basic-1: was: flt_wf__q_ind *) + +(* Basic-1: was: flt_wf_ind *) +axiom cw_wf_ind: ∀R:lenv→term→Prop. + (∀L2,T2. (∀L1,T1. #[L1,T1] < #[L2,T2] → R L1 T1) → R L2 T2) → + ∀L,T. R L T. + +(* Basic-1: was: flt_shift *) +lemma cw_shift: ∀K,I,V,T. #[K. 𝕓{I} V, T] < #[K, 𝕔{I} V. T]. +normalize // +qed. + +lemma tw_shift: ∀L,T. #[L, T] ≤ #[L @ T]. +#L elim L // +#K #I #V #IHL #T +@transitive_le [3: @IHL |2: /2/ | skip ] +qed. + +(* Basic-1: removed theorems 6: + flt_thead_sx flt_thead_dx flt_arith0 flt_arith1 flt_arith2 flt_trans +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/item.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/item.ma new file mode 100644 index 000000000..ead47b546 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/item.ma @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* THE FORMAL SYSTEM λδ - MATITA SOURCE FILES + * Specification started: 2011 April 17 + * Confluence of context-sensitive parallel reduction closed: 2011 September 21 + * Confluence of context-free parallel reduction closed: 2011 September 6 + * - Patience on me so that I gain peace and perfection! - + * [ suggested invocation to start formal specifications with ] + *) + +include "Ground-2/list.ma". +include "Ground-2/star.ma". +include "Basic-2/notation.ma". + +(* ITEMS ********************************************************************) + +(* atomic items *) +inductive item0: Type[0] ≝ + | Sort: nat → item0 (* sort: starting at 0 *) + | LRef: nat → item0 (* reference by index: starting at 0 *) +. + +(* binary binding items *) +inductive bind2: Type[0] ≝ + | Abbr: bind2 (* abbreviation *) + | Abst: bind2 (* abstraction *) +. + +(* binary non-binding items *) +inductive flat2: Type[0] ≝ + | Appl: flat2 (* application *) + | Cast: flat2 (* explicit type annotation *) +. + +(* binary items *) +inductive item2: Type[0] ≝ + | Bind: bind2 → item2 (* binding item *) + | Flat: flat2 → item2 (* non-binding item *) +. + +coercion item2_of_bind2: ∀I:bind2.item2 ≝ Bind on _I:bind2 to item2. + +coercion item2_of_flat2: ∀I:flat2.item2 ≝ Flat on _I:flat2 to item2. + +(* Basic-1: removed theorems 19: + s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc + s_arith0 s_arith1 + r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1 +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv.ma new file mode 100644 index 000000000..dfec09808 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv.ma @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term.ma". + +(* LOCAL ENVIRONMENTS *******************************************************) + +(* local environments *) +inductive lenv: Type[0] ≝ +| LAtom: lenv (* empty *) +| LPair: lenv → bind2 → term → lenv (* binary binding construction *) +. + +interpretation "sort (local environment)" 'Star = LAtom. + +interpretation "environment binding construction (binary)" 'DBind L I T = (LPair L I T). diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_length.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_length.ma new file mode 100644 index 000000000..23e445907 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_length.ma @@ -0,0 +1,24 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/lenv.ma". + +(* LENGTH OF A LOCAL ENVIRONMENT ********************************************) + +let rec length L ≝ match L with +[ LAtom ⇒ 0 +| LPair L _ _ ⇒ length L + 1 +]. + +interpretation "length (local environment)" 'card L = (length L). diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_weight.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_weight.ma new file mode 100644 index 000000000..9f45673ba --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/lenv_weight.ma @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term_weight.ma". +include "Basic-2/grammar/lenv.ma". + +(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************) + +let rec lw L ≝ match L with +[ LAtom ⇒ 0 +| LPair L _ V ⇒ lw L + #[V] +]. + +interpretation "weight (local environment)" 'Weight L = (lw L). + +(* Basic-1: removed theorems 2: clt_cong clt_head *) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/leq.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/leq.ma new file mode 100644 index 000000000..ac5503d28 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/leq.ma @@ -0,0 +1,61 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/lenv_length.ma". + +(* LOCAL ENVIRONMENT EQUALITY ***********************************************) + +inductive leq: nat → nat → relation lenv ≝ +| leq_sort: ∀d,e. leq d e (⋆) (⋆) +| leq_OO: ∀L1,L2. leq 0 0 L1 L2 +| leq_eq: ∀L1,L2,I,V,e. leq 0 e L1 L2 → + leq 0 (e + 1) (L1. 𝕓{I} V) (L2.𝕓{I} V) +| leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e. + leq d e L1 L2 → leq (d + 1) e (L1. 𝕓{I1} V1) (L2. 𝕓{I2} V2) +. + +interpretation "local environment equality" 'Eq L1 d e L2 = (leq d e L1 L2). + +definition leq_repl_dx: ∀S. (lenv → relation S) → Prop ≝ λS,R. + ∀L1,s1,s2. R L1 s1 s2 → + ∀L2,d,e. L1 [d, e]≈ L2 → R L2 s1 s2. + +(* Basic properties *********************************************************) + +lemma TC_leq_repl_dx: ∀S,R. leq_repl_dx S R → leq_repl_dx S (λL. (TC … (R L))). +#S #R #HR #L1 #s1 #s2 #H elim H -H s2 +[ /3 width=5/ +| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12 + lapply (HR … Hs2 … HL12) -HR Hs2 HL12 /3/ +] +qed. + +lemma leq_refl: ∀d,e,L. L [d, e] ≈ L. +#d elim d -d +[ #e elim e -e // #e #IHe #L elim L -L /2/ +| #d #IHd #e #L elim L -L /2/ +] +qed. + +lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1. +#L1 #L2 #d #e #H elim H -H L1 L2 d e /2/ +qed. + +lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d → + ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2. + +#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/ +qed. + +(* Basic inversion lemmas ***************************************************) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/sh.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/sh.ma new file mode 100644 index 000000000..bec437a75 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/sh.ma @@ -0,0 +1,23 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "Ground-2/list.ma". + +(* SORT HIERARCHY ***********************************************************) + +(* sort hierarchy specifications *) +record sh: Type[0] ≝ { + next: nat → nat; (* next sort in the hierarchy *) + next_lt: ∀k. k < next k (* strict monotonicity condition *) +}. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/term.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term.ma new file mode 100644 index 000000000..3b6614361 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term.ma @@ -0,0 +1,35 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/item.ma". + +(* TERMS ********************************************************************) + +(* terms *) +inductive term: Type[0] ≝ + | TAtom: item0 → term (* atomic item construction *) + | TPair: item2 → term → term → term (* binary item construction *) +. + +interpretation "sort (term)" 'Star k = (TAtom (Sort k)). + +interpretation "local reference (term)" 'LRef i = (TAtom (LRef i)). + +interpretation "term construction (atomic)" 'SItem I = (TAtom I). + +interpretation "term construction (binary)" 'SItem I T1 T2 = (TPair I T1 T2). + +interpretation "term binding construction (binary)" 'SBind I T1 T2 = (TPair (Bind I) T1 T2). + +interpretation "term flat construction (binary)" 'SFlat I T1 T2 = (TPair (Flat I) T1 T2). diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_simple.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_simple.ma new file mode 100644 index 000000000..c182ba360 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_simple.ma @@ -0,0 +1,36 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term.ma". + +(* SIMPLE (NEUTRAL) TERMS ***************************************************) + +inductive simple: term → Prop ≝ + | simple_atom: ∀I. simple (𝕒{I}) + | simple_flat: ∀I,V,T. simple (𝕗{I} V. T) +. + +interpretation "simple (term)" 'Simple T = (simple T). + +(* Basic inversion lemmas ***************************************************) + +fact simple_inv_bind_aux: ∀T. 𝕊[T] → ∀J,W,U. T = 𝕓{J} W. U → False. +#T * -T +[ #I #J #W #U #H destruct +| #I #V #T #J #W #U #H destruct +] +qed. + +lemma simple_inv_bind: ∀I,V,T. 𝕊[𝕓{I} V. T] → False. +/2 width=6/ qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_weight.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_weight.ma new file mode 100644 index 000000000..d383ded4d --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/term_weight.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term.ma". + +(* WEIGHT OF A TERM *********************************************************) + +let rec tw T ≝ match T with +[ TAtom _ ⇒ 1 +| TPair _ V T ⇒ tw V + tw T + 1 +]. + +interpretation "weight (term)" 'Weight T = (tw T). + +(* Basic properties *********************************************************) + +lemma tw_pos: ∀T. 1 ≤ #[T]. +#T elim T -T /2/ +qed. + +(* Basic eliminators ********************************************************) + +axiom tw_wf_ind: ∀R:term→Prop. + (∀T2. (∀T1. #[T1] < #[T2] → R T1) → R T2) → + ∀T. R T. + +(* Basic-1: removed theorems 11: + wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O + weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind + removed local theorems 1: q_ind +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/grammar/thom.ma b/matita/matita/contribs/lambda_delta/Basic_2/grammar/thom.ma new file mode 100644 index 000000000..23aa62e03 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/grammar/thom.ma @@ -0,0 +1,56 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term_simple.ma". + +(* HOMOMORPHIC TERMS ********************************************************) + +inductive thom: relation term ≝ + | thom_atom: ∀I. thom (𝕒{I}) (𝕒{I}) + | thom_abst: ∀V1,V2,T1,T2. thom (𝕔{Abst} V1. T1) (𝕔{Abst} V2. T2) + | thom_appl: ∀V1,V2,T1,T2. thom T1 T2 → 𝕊[T1] → 𝕊[T2] → + thom (𝕔{Appl} V1. T1) (𝕔{Appl} V2. T2) +. + +interpretation "homomorphic (term)" 'napart T1 T2 = (thom T1 T2). + +(* Basic properties *********************************************************) + +lemma thom_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1. +#T1 #T2 #H elim H -H T1 T2 /2/ +qed. + +lemma thom_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2. +#T1 #T2 #H elim H -H T1 T2 /2/ +qed. + +lemma thom_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1. +/3/ qed. + +lemma simple_thom_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝕊[T1] → 𝕊[T2]. +#T1 #T2 #H elim H -H T1 T2 // +#V1 #V2 #T1 #T2 #H +elim (simple_inv_bind … H) +qed. + +lemma simple_thom_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝕊[T2] → 𝕊[T1]. +/3/ qed. + +(* Basic inversion lemmas ***************************************************) + + +(* Basic-1: removed theorems 7: + iso_gen_sort iso_gen_lref iso_gen_head iso_refl iso_trans + iso_flats_lref_bind_false iso_flats_flat_bind_false +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/names.txt b/matita/matita/contribs/lambda_delta/Basic_2/names.txt new file mode 100644 index 000000000..c33d5a8ea --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/names.txt @@ -0,0 +1,22 @@ +NAMING CONVENTIONS FOR METAVARIABLES + +A,B : arity +C,D : candidate of reducibility +E,F,G : reserved: future use +H : reserved: transient premise +IH : reserved: inductive premise +I,J : item +K,L : local environment +M,N : reserved: future use +O : reserved: standard library +P,Q : reserved: future use +R : generic predicate (relation) +S : reserved: standard library +T,U,V,W: term +X,Y,Z : reserved: transient objet denoted by a capital letter + +d : relocation depth +e : relocation height +h : sort hierarchy parameter +i,j : local reference position index (de Bruijn's) +k : sort index diff --git a/matita/matita/contribs/lambda_delta/Basic_2/notation.ma b/matita/matita/contribs/lambda_delta/Basic_2/notation.ma new file mode 100644 index 000000000..a84fa8ff8 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/notation.ma @@ -0,0 +1,125 @@ +(**************************************************************************) +(* ___ *) +(* ||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 FOR THE FORMAL SYSTEM λδ ****************************************) + +(* Grammar ******************************************************************) + +notation "hvbox( ⋆ )" + non associative with precedence 90 + for @{ 'Star }. + +notation "hvbox( ⋆ term 90 k )" + non associative with precedence 90 + for @{ 'Star $k }. + +notation "hvbox( # term 90 k )" + non associative with precedence 90 + for @{ 'LRef $k }. + +notation "hvbox( 𝕒 { I } )" + non associative with precedence 90 + for @{ 'SItem $I }. + +notation "hvbox( 𝕔 { I } break term 90 T1 . break term 90 T )" + non associative with precedence 90 + for @{ 'SItem $I $T1 $T }. + +notation "hvbox( 𝕓 { I } break term 90 T1 . break term 90 T )" + non associative with precedence 90 + for @{ 'SBind $I $T1 $T }. + +notation "hvbox( 𝕗 { I } break term 90 T1 . break term 90 T )" + non associative with precedence 90 + for @{ 'SFlat $I $T1 $T }. + +notation "hvbox( T . break 𝕓 { I } break term 90 T1 )" + non associative with precedence 89 + for @{ 'DBind $T $I $T1 }. +(* +notation > "hvbox( T . break 𝕔 { I } break term 90 T1 )" + non associative with precedence 89 + for @{ 'DBind $T $I $T1 }. +*) (**) (* this breaks all parsing *) +notation "hvbox( # [ x ] )" + non associative with precedence 90 + for @{ 'Weight $x }. + +notation "hvbox( # [ x , break y ] )" + non associative with precedence 90 + for @{ 'Weight $x $y }. + +notation "hvbox( 𝕊 [ T ] )" + non associative with precedence 45 + for @{ 'Simple $T }. + +notation "hvbox( T1 break [ d , break e ] ≈ break T2 )" + non associative with precedence 45 + for @{ 'Eq $T1 $d $e $T2 }. + +(* Substitution *************************************************************) + +notation "hvbox( ↑ [ d , break e ] break T1 ≡ break T2 )" + non associative with precedence 45 + for @{ 'RLift $d $e $T1 $T2 }. + +notation "hvbox( ↓ [ d , break e ] break L1 ≡ break L2 )" + non associative with precedence 45 + for @{ 'RDrop $d $e $L1 $L2 }. + +notation "hvbox( T1 break [ d , break e ] ≫ break T2 )" + non associative with precedence 45 + for @{ 'PSubst $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≫ break T2 )" + non associative with precedence 45 + for @{ 'PSubst $L $T1 $d $e $T2 }. + +(* Unfold *******************************************************************) + +notation "hvbox( T1 break [ d , break e ] ≫* break T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $T1 $d $e $T2 }. + +notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≫* break T2 )" + non associative with precedence 45 + for @{ 'PSubstStar $L $T1 $d $e $T2 }. + +(* Reduction ****************************************************************) + +notation "hvbox( T1 ⇒ break T2 )" + non associative with precedence 45 + for @{ 'PRed $T1 $T2 }. + +notation "hvbox( L ⊢ break term 90 T1 ⇒ break T2 )" + non associative with precedence 45 + for @{ 'PRed $L $T1 $T2 }. + +notation "hvbox( L1 ⊢ ⇒ break L2 )" + non associative with precedence 45 + for @{ 'CPRed $L1 $L2 }. + +(* Computation **************************************************************) + +notation "hvbox( T1 ⇒* break T2 )" + non associative with precedence 45 + for @{ 'PRedStar $T1 $T2 }. + +notation "hvbox( L ⊢ break term 90 T1 ⇒* break T2 )" + non associative with precedence 45 + for @{ 'PRedStar $L $T1 $T2 }. + +notation "hvbox( L1 ⊢ ⇒* break L2 )" + non associative with precedence 45 + for @{ 'CPRedStar $L1 $L2 }. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr.ma new file mode 100644 index 000000000..9187765da --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr.ma @@ -0,0 +1,91 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/cl_shift.ma". +include "Basic-2/unfold/tpss.ma". +include "Basic-2/reduction/tpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) + +(* Basic-1: includes: pr2_delta1 *) +definition cpr: lenv → relation term ≝ + λL,T1,T2. ∃∃T. T1 ⇒ T & L ⊢ T [0, |L|] ≫* T2. + +interpretation + "context-sensitive parallel reduction (term)" + 'PRed L T1 T2 = (cpr L T1 T2). + +(* Basic properties *********************************************************) + +(* Basic-1: was by definition: pr2_free *) +lemma cpr_pr: ∀T1,T2. T1 ⇒ T2 → ∀L. L ⊢ T1 ⇒ T2. +/2/ qed. + +lemma cpr_tpss: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 ⇒ T2. +/3 width=5/ qed. + +lemma cpr_refl: ∀L,T. L ⊢ T ⇒ T. +/2/ qed. + +(* Note: new property *) +(* Basic-1: was only: pr2_thin_dx *) +lemma cpr_flat: ∀I,L,V1,V2,T1,T2. + L ⊢ V1 ⇒ V2 → L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{I} V1. T1 ⇒ 𝕗{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/ +qed. + +lemma cpr_cast: ∀L,V,T1,T2. + L ⊢ T1 ⇒ T2 → L ⊢ 𝕔{Cast} V. T1 ⇒ T2. +#L #V #T1 #T2 * /3/ +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Basic-1: was: pr2_gen_csort *) +lemma cpr_inv_atom: ∀T1,T2. ⋆ ⊢ T1 ⇒ T2 → T1 ⇒ T2. +#T1 #T2 * #T #HT normalize #HT2 +<(tpss_inv_refl_O2 … HT2) -HT2 // +qed. + +(* Basic-1: was: pr2_gen_sort *) +lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ⇒ T2 → T2 = ⋆k. +#L #T2 #k * #X #H +>(tpr_inv_atom1 … H) -H #H +>(tpss_inv_sort1 … H) -H // +qed. + +(* Basic-1: was: pr2_gen_cast *) +lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ 𝕔{Cast} V1. T1 ⇒ U2 → ( + ∃∃V2,T2. L ⊢ V1 ⇒ V2 & L ⊢ T1 ⇒ T2 & + U2 = 𝕔{Cast} V2. T2 + ) ∨ L ⊢ T1 ⇒ U2. +#L #V1 #T1 #U2 * #X #H #HU2 +elim (tpr_inv_cast1 … H) -H /3/ +* #V #T #HV1 #HT1 #H destruct -X; +elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct -U2 /4 width=5/ +qed. + +(* Basic-1: removed theorems 5: + pr2_head_1 pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans + Basic-1: removed local theorems 3: + pr2_free_free pr2_free_delta pr2_delta_delta +*) + +(* +pr2/fwd pr2_gen_appl +pr2/fwd pr2_gen_abbr +pr2/props pr2_change +pr2/subst1 pr2_subst1 +pr2/subst1 pr2_gen_cabbr +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr_cpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr_cpr.ma new file mode 100644 index 000000000..086209007 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/cpr_cpr.ma @@ -0,0 +1,58 @@ +(**************************************************************************) +(* ___ *) +(* ||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/reduction/tpr_tpr.ma". +include "Basic-2/reduction/cpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************) + +(* Advanced properties ******************************************************) + +lemma cpr_bind_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ⇒ V2 → T1 ⇒ T2 → + L ⊢ 𝕓{I} V1. T1 ⇒ 𝕓{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12 +@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2/ (* /3 width=5/ is too slow *) +qed. + +(* Basic-1: was only: pr2_gen_cbind *) +lemma cpr_bind_dx: ∀I,L,V1,V2,T1,T2. V1 ⇒ V2 → L. 𝕓{I} V2 ⊢ T1 ⇒ T2 → + L ⊢ 𝕓{I} V1. T1 ⇒ 𝕓{I} V2. T2. +#I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2 +elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 (tpr_inv_atom1 … H) -H #H +elim (tpss_inv_lref1 … H) -H /2/ +* /3 width=6/ +qed. + +(* Basic-1: was: pr2_gen_abst *) +lemma cpr_inv_abst1: ∀V1,T1,U2. 𝕔{Abst} V1. T1 ⇒ U2 → + ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2. +/2/ qed. + +(* Relocation properties ****************************************************) + +(* Basic-1: was: pr2_lift *) + +(* Basic-1: was: pr2_gen_lift *) + diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/lcpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/lcpr.ma new file mode 100644 index 000000000..f01daabce --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/lcpr.ma @@ -0,0 +1,41 @@ +(**************************************************************************) +(* ___ *) +(* ||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/reduction/cpr.ma". + +(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************) + +inductive lcpr: relation lenv ≝ +| lcpr_sort: lcpr (⋆) (⋆) +| lcpr_item: ∀K1,K2,I,V1,V2. + lcpr K1 K2 → K2 ⊢ V1 ⇒ V2 → lcpr (K1. 𝕓{I} V1) (K2. 𝕓{I} V2) (*𝕓*) +. + +interpretation + "context-sensitive parallel reduction (environment)" + 'CPRed L1 L2 = (lcpr L1 L2). + +(* Basic inversion lemmas ***************************************************) + +fact lcpr_inv_item1_aux: ∀L1,L2. L1 ⊢ ⇒ L2 → ∀K1,I,V1. L1 = K1. 𝕓{I} V1 → + ∃∃K2,V2. K1 ⊢ ⇒ K2 & K2 ⊢ V1 ⇒ V2 & L2 = K2. 𝕓{I} V2. +#L1 #L2 * -L1 L2 +[ #K1 #I #V1 #H destruct +| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct - K1 I V1 /2 width=5/ +] +qed. + +lemma lcpr_inv_item1: ∀K1,I,V1,L2. K1. 𝕓{I} V1 ⊢ ⇒ L2 → + ∃∃K2,V2. K1 ⊢ ⇒ K2 & K2 ⊢ V1 ⇒ V2 & L2 = K2. 𝕓{I} V2. +/2/ qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr.ma new file mode 100644 index 000000000..a9093a042 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr.ma @@ -0,0 +1,83 @@ +(**************************************************************************) +(* ___ *) +(* ||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/reduction/tpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +inductive ltpr: relation lenv ≝ +| ltpr_sort: ltpr (⋆) (⋆) +| ltpr_item: ∀K1,K2,I,V1,V2. + ltpr K1 K2 → V1 ⇒ V2 → ltpr (K1. 𝕓{I} V1) (K2. 𝕓{I} V2) (*𝕓*) +. + +interpretation + "context-free parallel reduction (environment)" + 'PRed L1 L2 = (ltpr L1 L2). + +(* Basic properties *********************************************************) + +lemma ltpr_refl: ∀L:lenv. L ⇒ L. +#L elim L -L /2/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact ltpr_inv_atom1_aux: ∀L1,L2. L1 ⇒ L2 → L1 = ⋆ → L2 = ⋆. +#L1 #L2 * -L1 L2 +[ // +| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct +] +qed. + +(* Basic-1: was: wcpr0_gen_sort *) +lemma ltpr_inv_atom1: ∀L2. ⋆ ⇒ L2 → L2 = ⋆. +/2/ qed. + +fact ltpr_inv_pair1_aux: ∀L1,L2. L1 ⇒ L2 → ∀K1,I,V1. L1 = K1. 𝕓{I} V1 → + ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2. +#L1 #L2 * -L1 L2 +[ #K1 #I #V1 #H destruct +| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct - K1 I V1 /2 width=5/ +] +qed. + +(* Basic-1: was: wcpr0_gen_head *) +lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. 𝕓{I} V1 ⇒ L2 → + ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2. +/2/ qed. + +fact ltpr_inv_atom2_aux: ∀L1,L2. L1 ⇒ L2 → L2 = ⋆ → L1 = ⋆. +#L1 #L2 * -L1 L2 +[ // +| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct +] +qed. + +lemma ltpr_inv_atom2: ∀L1. L1 ⇒ ⋆ → L1 = ⋆. +/2/ qed. + +fact ltpr_inv_pair2_aux: ∀L1,L2. L1 ⇒ L2 → ∀K2,I,V2. L2 = K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 ⇒ K2 & V1 ⇒ V2 & L1 = K1. 𝕓{I} V1. +#L1 #L2 * -L1 L2 +[ #K2 #I #V2 #H destruct +| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct -K2 I V2 /2 width=5/ +] +qed. + +lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ⇒ K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 ⇒ K2 & V1 ⇒ V2 & L1 = K1. 𝕓{I} V1. +/2/ qed. + +(* Basic-1: removed theorems 2: wcpr0_getl wcpr0_getl_back *) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr_drop.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr_drop.ma new file mode 100644 index 000000000..3c70e6a0b --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/ltpr_drop.ma @@ -0,0 +1,52 @@ +(**************************************************************************) +(* ___ *) +(* ||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/reduction/tpr_lift.ma". +include "Basic-2/reduction/ltpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************) + +(* Basic-1: was: wcpr0_drop *) +lemma ltpr_drop_conf: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀L2. L1 ⇒ L2 → + ∃∃K2. ↓[d, e] L2 ≡ K2 & K1 ⇒ K2. +#L1 #K1 #d #e #H elim H -H L1 K1 d e +[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/ +| #K1 #I #V1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X; + elim (IHLK1 … HL12) -IHLK1 HL12 /3/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct -X; + elim (tpr_inv_lift … HV12 … HWV1) -HV12 HWV1; + elim (IHLK1 … HL12) -IHLK1 HL12 /3 width=5/ +] +qed. + +(* Basic-1: was: wcpr0_drop_back *) +lemma ltpr_drop_trans: ∀L1,K1,d,e. ↓[d, e] L1 ≡ K1 → ∀K2. K1 ⇒ K2 → + ∃∃L2. ↓[d, e] L2 ≡ K2 & L1 ⇒ L2. +#L1 #K1 #d #e #H elim H -H L1 K1 d e +[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2/ +| #K1 #I #V1 #X #H + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/ +| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12 + elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/ +| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H + elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct -X; + elim (lift_total W2 d e) #V2 #HWV2 + lapply (tpr_lift … HW12 … HWV1 … HWV2) -HW12 HWV1; + elim (IHLK1 … HK12) -IHLK1 HK12 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr.ma new file mode 100644 index 000000000..fce71d15a --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr.ma @@ -0,0 +1,198 @@ +(**************************************************************************) +(* ___ *) +(* ||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/tps.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Basic-1: includes: pr0_delta1 *) +inductive tpr: relation term ≝ +| tpr_atom : ∀I. tpr (𝕒{I}) (𝕒{I}) +| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 → + tpr (𝕗{I} V1. T1) (𝕗{I} V2. T2) +| tpr_beta : ∀V1,V2,W,T1,T2. + tpr V1 V2 → tpr T1 T2 → + tpr (𝕔{Appl} V1. 𝕔{Abst} W. T1) (𝕔{Abbr} V2. T2) +| tpr_delta: ∀I,V1,V2,T1,T2,T. + tpr V1 V2 → tpr T1 T2 → ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ T → + tpr (𝕓{I} V1. T1) (𝕓{I} V2. T) +| tpr_theta: ∀V,V1,V2,W1,W2,T1,T2. + tpr V1 V2 → ↑[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 → + tpr (𝕔{Appl} V1. 𝕔{Abbr} W1. T1) (𝕔{Abbr} W2. 𝕔{Appl} V. T2) +| tpr_zeta : ∀V,T,T1,T2. ↑[0,1] T1 ≡ T → tpr T1 T2 → + tpr (𝕔{Abbr} V. T) T2 +| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (𝕔{Cast} V. T1) T2 +. + +interpretation + "context-free parallel reduction (term)" + 'PRed T1 T2 = (tpr T1 T2). + +(* Basic properties *********************************************************) + +lemma tpr_bind: ∀I,V1,V2,T1,T2. V1 ⇒ V2 → T1 ⇒ T2 → + 𝕓{I} V1. T1 ⇒ 𝕓{I} V2. T2. +/2/ qed. + +(* Basic-1: was by definition: pr0_refl *) +lemma tpr_refl: ∀T. T ⇒ T. +#T elim T -T // +#I elim I -I /2/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tpr_inv_atom1_aux: ∀U1,U2. U1 ⇒ U2 → ∀I. U1 = 𝕒{I} → U2 = 𝕒{I}. +#U1 #U2 * -U1 U2 +[ // +| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct +| #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #T #_ #_ #_ #k #H destruct +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct +| #V #T #T1 #T2 #_ #_ #k #H destruct +| #V #T1 #T2 #_ #k #H destruct +] +qed. + +(* Basic-1: was: pr0_gen_sort pr0_gen_lref *) +lemma tpr_inv_atom1: ∀I,U2. 𝕒{I} ⇒ U2 → U2 = 𝕒{I}. +/2/ qed. + +fact tpr_inv_bind1_aux: ∀U1,U2. U1 ⇒ U2 → ∀I,V1,T1. U1 = 𝕓{I} V1. T1 → + (∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 & + ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ T & + U2 = 𝕓{I} V2. T + ) ∨ + ∃∃T. ↑[0,1] T ≡ T1 & T ⇒ U2 & I = Abbr. +#U1 #U2 * -U1 U2 +[ #J #I #V #T #H destruct +| #I1 #V1 #V2 #T1 #T2 #_ #_ #I #V #T #H destruct +| #V1 #V2 #W #T1 #T2 #_ #_ #I #V #T #H destruct +| #I1 #V1 #V2 #T1 #T2 #T #HV12 #HT12 #HT2 #I0 #V0 #T0 #H destruct -I1 V1 T1 /3 width=7/ +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #I0 #V0 #T0 #H destruct +| #V #T #T1 #T2 #HT1 #HT12 #I0 #V0 #T0 #H destruct -V T /3/ +| #V #T1 #T2 #_ #I0 #V0 #T0 #H destruct +] +qed. + +lemma tpr_inv_bind1: ∀V1,T1,U2,I. 𝕓{I} V1. T1 ⇒ U2 → + (∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 & + ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ T & + U2 = 𝕓{I} V2. T + ) ∨ + ∃∃T. ↑[0,1] T ≡ T1 & T ⇒ U2 & I = Abbr. +/2/ qed. + +(* Basic-1: was pr0_gen_abbr *) +lemma tpr_inv_abbr1: ∀V1,T1,U2. 𝕓{Abbr} V1. T1 ⇒ U2 → + (∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 & + ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T & + U2 = 𝕓{Abbr} V2. T + ) ∨ + ∃∃T. ↑[0,1] T ≡ T1 & T ⇒ U2. +#V1 #T1 #U2 #H +elim (tpr_inv_bind1 … H) -H * /3 width=7/ +qed. + +fact tpr_inv_flat1_aux: ∀U1,U2. U1 ⇒ U2 → ∀I,V1,U0. U1 = 𝕗{I} V1. U0 → + ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 & + U2 = 𝕗{I} V2. T2 + | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 & + U0 = 𝕔{Abst} W. T1 & + U2 = 𝕔{Abbr} V2. T2 & I = Appl + | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 & + ↑[0,1] V2 ≡ V & + U0 = 𝕔{Abbr} W1. T1 & + U2 = 𝕔{Abbr} W2. 𝕔{Appl} V. T2 & + I = Appl + | (U0 ⇒ U2 ∧ I = Cast). +#U1 #U2 * -U1 U2 +[ #I #J #V #T #H destruct +| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct -I V1 T1 /3 width=5/ +| #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct -J V1 T /3 width=8/ +| #I #V1 #V2 #T1 #T2 #T #_ #_ #_ #J #V0 #T0 #H destruct +| #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H + destruct -J V1 T0 /3 width=12/ +| #V #T #T1 #T2 #_ #_ #J #V0 #T0 #H destruct +| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct -J V T1 /3/ +] +qed. + +lemma tpr_inv_flat1: ∀V1,U0,U2,I. 𝕗{I} V1. U0 ⇒ U2 → + ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 & + U2 = 𝕗{I} V2. T2 + | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 & + U0 = 𝕔{Abst} W. T1 & + U2 = 𝕔{Abbr} V2. T2 & I = Appl + | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 & + ↑[0,1] V2 ≡ V & + U0 = 𝕔{Abbr} W1. T1 & + U2 = 𝕔{Abbr} W2. 𝕔{Appl} V. T2 & + I = Appl + | (U0 ⇒ U2 ∧ I = Cast). +/2/ qed. + +(* Basic-1: was pr0_gen_appl *) +lemma tpr_inv_appl1: ∀V1,U0,U2. 𝕔{Appl} V1. U0 ⇒ U2 → + ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 & + U2 = 𝕔{Appl} V2. T2 + | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 & + U0 = 𝕔{Abst} W. T1 & + U2 = 𝕔{Abbr} V2. T2 + | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 & + ↑[0,1] V2 ≡ V & + U0 = 𝕔{Abbr} W1. T1 & + U2 = 𝕔{Abbr} W2. 𝕔{Appl} V. T2. +#V1 #U0 #U2 #H +elim (tpr_inv_flat1 … H) -H * /3 width=12/ #_ #H destruct +qed. + +(* Basic-1: was: pr0_gen_cast *) +lemma tpr_inv_cast1: ∀V1,T1,U2. 𝕔{Cast} V1. T1 ⇒ U2 → + (∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Cast} V2. T2) + ∨ T1 ⇒ U2. +#V1 #T1 #U2 #H +elim (tpr_inv_flat1 … H) -H * /3 width=5/ +[ #V2 #W #W1 #W2 #_ #_ #_ #_ #H destruct +| #V2 #W #W1 #W2 #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct +] +qed. + +fact tpr_inv_lref2_aux: ∀T1,T2. T1 ⇒ T2 → ∀i. T2 = #i → + ∨∨ T1 = #i + | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i & + T1 = 𝕔{Abbr} V. T + | ∃∃V,T. T ⇒ #i & T1 = 𝕔{Cast} V. T. +#T1 #T2 * -T1 T2 +[ #I #i #H destruct /2/ +| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct +| #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct +| #I #V1 #V2 #T1 #T2 #T #_ #_ #_ #i #H destruct +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct +| #V #T #T1 #T2 #HT1 #HT12 #i #H destruct /3 width=6/ +| #V #T1 #T2 #HT12 #i #H destruct /3/ +] +qed. + +lemma tpr_inv_lref2: ∀T1,i. T1 ⇒ #i → + ∨∨ T1 = #i + | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i & + T1 = 𝕔{Abbr} V. T + | ∃∃V,T. T ⇒ #i & T1 = 𝕔{Cast} V. T. +/2/ qed. + +(* Basic-1: removed theorems 3: + pr0_subst0_back pr0_subst0_fwd pr0_subst0 + Basic-1: removed local theorems: 1: pr0_delta_tau +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_lift.ma new file mode 100644 index 000000000..311a1433a --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_lift.ma @@ -0,0 +1,119 @@ +(**************************************************************************) +(* ___ *) +(* ||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/tps_lift.ma". +include "Basic-2/reduction/tpr.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Relocation properties ****************************************************) + +(* Basic-1: was: pr0_lift *) +lemma tpr_lift: ∀T1,T2. T1 ⇒ T2 → + ∀d,e,U1. ↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 → U1 ⇒ U2. +#T1 #T2 #H elim H -H T1 T2 +[ * #i #d #e #U1 #HU1 #U2 #HU2 + lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1 + [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct -U2 // + | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct -U2 // + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1; + elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/ +| #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1; + elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X; + elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct -X2 /3/ +| #I #V1 #V2 #T1 #T2 #T0 #HV12 #HT12 #HT2 #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2 + elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1; + elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct -X2; + elim (lift_total T2 (d + 1) e) #U2 #HTU2 + @tpr_delta + [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *) +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2 + elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1; + elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X; + elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct -X2; + elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct -X; + elim (lift_trans_ge … HV2 … HV3 ?) -HV2 HV3 V // /3/ +| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20 + elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct -X; + elim (lift_trans_ge … HT1 … HT3 ?) -HT1 HT3 T // /3 width=6/ +| #V #T1 #T2 #_ #IHT12 #d #e #X #HX #T #HT2 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct -X /3/ +] +qed. + +(* Basic-1: was: pr0_gen_lift *) +lemma tpr_inv_lift: ∀T1,T2. T1 ⇒ T2 → + ∀d,e,U1. ↑[d, e] U1 ≡ T1 → + ∃∃U2. ↑[d, e] U2 ≡ T2 & U1 ⇒ U2. +#T1 #T2 #H elim H -H T1 T2 +[ * #i #d #e #U1 #HU1 + [ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/ + | lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/ + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX + elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X; + elim (IHV12 … HV01) -IHV12 HV01; + elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/ +| #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX + elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X; + elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y; + elim (IHV12 … HV01) -IHV12 HV01; + elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/ +| #I #V1 #V2 #T1 #T2 #T0 #_ #_ #HT20 #IHV12 #IHT12 #d #e #X #HX + elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct -X; + elim (IHV12 … HWV1) -IHV12 HWV1 #W2 #HWV2 #HW12 + elim (IHT12 … HUT1) -IHT12 HUT1 #U2 #HUT2 #HU12 + elim (tps_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0 + @ex2_1_intro [2: /2/ |1: skip |3: /2/ ] (**) (* /3 width=5/ is slow *) +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX + elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X; + elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y; + elim (IHV12 … HV01) -IHV12 HV01 #V3 #HV32 #HV03 + elim (IHW12 … HW01) -IHW12 HW01 #W3 #HW32 #HW03 + elim (IHT12 … HT01) -IHT12 HT01 #T3 #HT32 #HT03 + elim (lift_trans_le … HV32 … HV2 ?) -HV32 HV2 V2 // #V2 #HV32 #HV2 + @ex2_1_intro [2: /3/ |1: skip |3: /2/ ] (**) (* /4 width=5/ is slow *) +| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX + elim (lift_inv_bind2 … HX) -HX #V0 #T0 #_ #HT0 #H destruct -X; + elim (lift_div_le … HT1 … HT0 ?) -HT1 HT0 T // #T #HT0 #HT1 + elim (IHT12 … HT1) -IHT12 HT1 /3 width=5/ +| #V #T1 #T2 #_ #IHT12 #d #e #X #HX + elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct -X; + elim (IHT12 … HT01) -IHT12 HT01 /3/ +] +qed. + +(* Advanced inversion lemmas ************************************************) + +fact tpr_inv_abst1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕔{Abst} V1. T1 → + ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2. +#U1 #U2 * -U1 U2 +[ #I #V #T #H destruct +| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct +| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct +| #I #V1 #V2 #T1 #T2 #T #HV12 #HT12 #HT2 #V0 #T0 #H destruct -I V1 T1; + <(tps_inv_refl_SO2 … HT2 ? ? ?) -HT2 T /2 width=5/ +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct +| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct +| #V #T1 #T2 #_ #V0 #T0 #H destruct +] +qed. + +(* Basic-1: was pr0_gen_abst *) +lemma tpr_inv_abst1: ∀V1,T1,U2. 𝕔{Abst} V1. T1 ⇒ U2 → + ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕔{Abst} V2. T2. +/2/ qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tpr.ma new file mode 100644 index 000000000..ac09a41ef --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tpr.ma @@ -0,0 +1,287 @@ +(**************************************************************************) +(* ___ *) +(* ||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/reduction/tpr_tps.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Confluence lemmas ********************************************************) + +fact tpr_conf_atom_atom: ∀I. ∃∃X. 𝕒{I} ⇒ X & 𝕒{I} ⇒ X. +/2/ qed. + +fact tpr_conf_flat_flat: + ∀I,V0,V1,T0,T1,V2,T2. ( + ∀X0:term. #[X0] < #[V0] + #[T0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 → + ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0. +#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2 +elim (IH … HT01 … HT02) -HT01 HT02 /3 width=5/ +qed. + +fact tpr_conf_flat_beta: + ∀V0,V1,T1,V2,W0,U0,T2. ( + ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → + U0 ⇒ T2 → 𝕔{Abst} W0. U0 ⇒ T1 → + ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} V2. T2 ⇒ X. +#V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H +elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct -T1; +elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2 +elim (IH … HT02 … HU01) -HT02 HU01 IH /3 width=5/ +qed. + +(* basic-1: was: + pr0_cong_upsilon_refl pr0_cong_upsilon_zeta + pr0_cong_upsilon_cong pr0_cong_upsilon_delta +*) +fact tpr_conf_flat_theta: + ∀V0,V1,T1,V2,V,W0,W2,U0,U2. ( + ∀X0:term. #[X0] < #[V0] + (#[W0] + #[U0] + 1) + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → ↑[O,1] V2 ≡ V → + W0 ⇒ W2 → U0 ⇒ U2 → 𝕔{Abbr} W0. U0 ⇒ T1 → + ∃∃X. 𝕔{Appl} V1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} V. U2 ⇒ X. +#V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H +elim (IH … HV01 … HV02) -HV01 HV02 // #VV #HVV1 #HVV2 +elim (lift_total VV 0 1) #VVV #HVV +lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV +elim (tpr_inv_abbr1 … H) -H * +(* case 1: delta *) +[ -HV2 HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct -T1; + elim (IH … HW02 … HWW2) -HW02 HWW2 // #W #HW02 #HWW2 + elim (IH … HU02 … HUU02) -HU02 HUU02 IH // #U #HU2 #HUUU2 + elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1 + @ex2_1_intro + [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ] + |1:skip + |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/ + ] (**) (* /5 width=14/ is too slow *) +(* case 3: zeta *) +| -HW02 HVV HVVV #UU1 #HUU10 #HUUT1 + elim (tpr_inv_lift … HU02 … HUU10) -HU02 #UU #HUU2 #HUU1 + lapply (tw_lift … HUU10) -HUU10 #HUU10 + elim (IH … HUUT1 … HUU1) -HUUT1 HUU1 IH // -HUU10 #U #HU2 #HUUU2 + @ex2_1_intro + [2: @tpr_flat + |1: skip + |3: @tpr_zeta [2: @lift_flat |1: skip |3: @tpr_flat ] + ] /2 width=5/ (**) (* /5 width=5/ is too slow *) +] +qed. + +fact tpr_conf_flat_cast: + ∀X2,V0,V1,T0,T1. ( + ∀X0:term. #[X0] < #[V0] + #[T0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 → + ∃∃X. 𝕔{Cast} V1. T1 ⇒ X & X2 ⇒ X. +#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02 +elim (IH … HT01 … HT02) -HT01 HT02 IH /3/ +qed. + +fact tpr_conf_beta_beta: + ∀W0:term. ∀V0,V1,T0,T1,V2,T2. ( + ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 → + ∃∃X. 𝕔{Abbr} V1. T1 ⇒X & 𝕔{Abbr} V2. T2 ⇒ X. +#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02 +elim (IH … HV01 … HV02) -HV01 HV02 // +elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/ +qed. + +(* Basic-1: was: pr0_cong_delta pr0_delta_delta *) +fact tpr_conf_delta_delta: + ∀I1,V0,V1,T0,T1,TT1,V2,T2,TT2. ( + ∀X0:term. #[X0] < #[V0] + #[T0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 → + ⋆. 𝕓{I1} V1 ⊢ T1 [O, 1] ≫ TT1 → + ⋆. 𝕓{I1} V2 ⊢ T2 [O, 1] ≫ TT2 → + ∃∃X. 𝕓{I1} V1. TT1 ⇒ X & 𝕓{I1} V2. TT2 ⇒ X. +#I1 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2 +elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2 +elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2 +elim (tpr_tps_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1 +elim (tpr_tps_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2 +elim (tps_conf_eq … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2 +@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *) +qed. + +fact tpr_conf_delta_zeta: + ∀X2,V0,V1,T0,T1,TT1,T2. ( + ∀X0:term. #[X0] < #[V0] + #[T0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 → + T2 ⇒ X2 → ↑[O, 1] T2 ≡ T0 → + ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X. +#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HTX2 #HTT20 +elim (tpr_inv_lift … HT01 … HTT20) -HT01 #TT2 #HTT21 #HTT2 +lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct -T1; +lapply (tw_lift … HTT20) -HTT20 #HTT20 +elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/ +qed. + +(* Basic-1: was: pr0_upsilon_upsilon *) +fact tpr_conf_theta_theta: + ∀VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. ( + ∀X0:term. #[X0] < #[V0] + (#[W0] + #[T0] + 1) + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 → + ↑[O, 1] V1 ≡ VV1 → ↑[O, 1] V2 ≡ VV2 → + ∃∃X. 𝕔{Abbr} W1. 𝕔{Appl} VV1. T1 ⇒ X & 𝕔{Abbr} W2. 𝕔{Appl} VV2. T2 ⇒ X. +#VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2 +elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2 +elim (IH … HW01 … HW02) -HW01 HW02 // #W #HW1 #HW2 +elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2 +elim (lift_total V 0 1) #VV #HVV +lapply (tpr_lift … HV1 … HVV1 … HVV) -HV1 HVV1 #HVV1 +lapply (tpr_lift … HV2 … HVV2 … HVV) -HV2 HVV2 HVV #HVV2 +@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *) +qed. + +fact tpr_conf_zeta_zeta: + ∀V0:term. ∀X2,TT0,T0,T1,T2. ( + ∀X0:term. #[X0] < #[V0] + #[TT0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + T0 ⇒ T1 → T2 ⇒ X2 → + ↑[O, 1] T0 ≡ TT0 → ↑[O, 1] T2 ≡ TT0 → + ∃∃X. T1 ⇒ X & X2 ⇒ X. +#V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20 +lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct -T0; +lapply (tw_lift … HTT20) -HTT20 #HTT20 +elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/ +qed. + +fact tpr_conf_tau_tau: + ∀V0,T0:term. ∀X2,T1. ( + ∀X0:term. #[X0] < #[V0] + #[T0] + 1 → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + T0 ⇒ T1 → T0 ⇒ X2 → + ∃∃X. T1 ⇒ X & X2 ⇒ X. +#V0 #T0 #X2 #T1 #IH #HT01 #HT02 +elim (IH … HT01 … HT02) -HT01 HT02 IH /2/ +qed. + +(* Confluence ***************************************************************) + +fact tpr_conf_aux: + ∀Y0:term. ( + ∀X0:term. #[X0] < #[Y0] → + ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → + ∃∃X. X1 ⇒ X & X2 ⇒ X + ) → + ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 → + ∃∃X. X1 ⇒ X & X2 ⇒ X. +#Y0 #IH #X0 #X1 #X2 * -X0 X1 +[ #I1 #H1 #H2 destruct -Y0; + lapply (tpr_inv_atom1 … H1) -H1 +(* case 1: atom, atom *) + #H1 destruct -X2 // +| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0; + elim (tpr_inv_flat1 … H1) -H1 * +(* case 2: flat, flat *) + [ #V2 #T2 #HV02 #HT02 #H destruct -X2 + /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *) +(* case 3: flat, beta *) + | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct -T0 X2 I + /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *) +(* case 4: flat, theta *) + | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct -T0 X2 I + /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *) +(* case 5: flat, tau *) + | #HT02 #H destruct -I + /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *) + ] +| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0; + elim (tpr_inv_appl1 … H1) -H1 * +(* case 6: beta, flat (repeated) *) + [ #V2 #T2 #HV02 #HT02 #H destruct -X2 + @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/ +(* case 7: beta, beta *) + | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct -W0 T0 X2 + /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *) +(* case 8, beta, theta (excluded) *) + | #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct + ] +| #I1 #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct -Y0; + elim (tpr_inv_bind1 … H1) -H1 * +(* case 9: delta, delta *) + [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2 + /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *) +(* case 10: delta, zata *) + | #T2 #HT20 #HTX2 #H destruct -I1; + /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *) + ] +| #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct -Y0; + elim (tpr_inv_appl1 … H1) -H1 * +(* case 11: theta, flat (repeated) *) + [ #V2 #T2 #HV02 #HT02 #H destruct -X2 + @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/ +(* case 12: theta, beta (repeated) *) + | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct +(* case 13: theta, theta *) + | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct -W0 T0 X2 + /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *) + ] +| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct -Y0; + elim (tpr_inv_abbr1 … H1) -H1 * +(* case 14: zeta, delta (repeated) *) + [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2 + @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/ +(* case 15: zeta, zeta *) + | #T2 #HTT20 #HTX2 + /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *) + ] +| #V0 #T0 #T1 #HT01 #H1 #H2 destruct -Y0; + elim (tpr_inv_cast1 … H1) -H1 +(* case 16: tau, flat (repeated) *) + [ * #V2 #T2 #HV02 #HT02 #H destruct -X2 + @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/ +(* case 17: tau, tau *) + | #HT02 + /2 by tpr_conf_tau_tau/ + ] +] +qed. + +(* Basic-1: was: pr0_confluence *) +theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ⇒ T1 → T0 ⇒ T2 → + ∃∃T. T1 ⇒ T & T2 ⇒ T. +#T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/ +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tps.ma b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tps.ma new file mode 100644 index 000000000..ac2112c7a --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tps.ma @@ -0,0 +1,92 @@ +(**************************************************************************) +(* ___ *) +(* ||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/unfold/ltpss_ltpss.ma". +include "Basic-2/reduction/ltpr_drop.ma". + +(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************) + +(* Basic-1: was: pr0_subst1 *) +lemma tpr_tps_ltpr: ∀T1,T2. T1 ⇒ T2 → + ∀L1,d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 → + ∀L2. L1 ⇒ L2 → + ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫* U2. +#T1 #T2 #H elim H -H T1 T2 +[ #I #L1 #d #e #X #H + elim (tps_inv_atom1 … H) -H + [ #H destruct -X /2/ + | * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct -I; + elim (ltpr_drop_conf … HLK1 … HL12) -HLK1 HL12 #X #HLK2 #H + elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct -X; + elim (lift_total V2 0 (i+1)) #U2 #HVU2 + lapply (tpr_lift … HV12 … HVU1 … HVU2) -HV12 HVU1 #HU12 + @ex2_1_intro [2: @HU12 | skip | /3/ ] (**) (* /4 width=6/ is too slow *) + ] +| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct -X; + elim (IHV12 … HVW1 … HL12) -IHV12 HVW1; + elim (IHT12 … HTU1 … HL12) -IHT12 HTU1 HL12 /3 width=5/ +| #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X; + elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct -Y; + elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2 + elim (IHT12 … HTT1 (L2. 𝕓{Abst} WW) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2 + lapply (tpss_leq_repl_dx … HTT2 (L2. 𝕓{Abbr} VV2) ?) -HTT2 /3 width=5/ +| #I #V1 #V2 #T1 #T2 #U2 #HV12 #_ #HTU2 #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X; + elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2 + elim (IHT12 … HTT1 (L2. 𝕓{I} VV2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2 + elim (tpss_strip_neq … HTT2 … HTU2 ?) -HTT2 HTU2 T2 /2/ #T2 #HTT2 #HUT2 + lapply (tps_leq_repl_dx … HTT2 (L2. 𝕓{I} V2) ?) -HTT2 /2/ #HTT2 + elim (ltpss_tps_conf … HTT2 (L2. 𝕓{I} VV2) (d + 1) e ?) -HTT2 /2/ #W2 #HTTW2 #HTW2 + lapply (tpss_leq_repl_dx … HTTW2 (⋆. 𝕓{I} VV2) ?) -HTTW2 /2/ #HTTW2 + lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2 + lapply (tpss_leq_repl_dx … HTW2 (L2. 𝕓{I} VV2) ?) -HTW2 /2/ #HTW2 + lapply (tpss_trans_eq … HUT2 … HTW2) -HUT2 HTW2 /3 width=5/ +| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X; + elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct -Y; + elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2 + elim (IHW12 … HWW1 … HL12) -IHW12 HWW1 #WW2 #HWW12 #HWW2 + elim (IHT12 … HTT1 (L2. 𝕓{Abbr} WW2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2 + elim (lift_total VV2 0 1) #VV #H2VV + lapply (tpss_lift_ge … HVV2 (L2. 𝕓{Abbr} WW2) … HV2 … H2VV) -HVV2 HV2 /2/ #HVV + @ex2_1_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *) +| #V1 #TT1 #T1 #T2 #HT1 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12 + elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct -X; + elim (tps_inv_lift1_ge … HTT12 L1 … HT1 ?) -HTT12 HT1 /2/ #T2 #HT12 #HTT2 + elim (IHT12 … HT12 … HL12) -IHT12 HT12 HL12 (plus_minus_m_m e 1) /2/ +qed. + +lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → + ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V → + d ≤ i → i < d + e → + ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 & + ↓[0, i] L2 ≡ K2. 𝕓{I} V. +#L1 #L2 #d #e #H elim H -H L1 L2 d e +[ #d #e #I #K1 #V #i #H + lapply (drop_inv_atom1 … H) -H #H destruct +| #L1 #L2 #I #K1 #V #i #_ #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie + elim (drop_inv_O1 … H) -H * #Hi #HLK1 + [ -IHL12 Hie; destruct -i K1 J W; + arith_g1 // /3/ + ] +| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide + lapply (plus_S_le_to_pos … Hdi) #Hi + lapply (drop_inv_drop1 … H ?) -H // #HLK1 + elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/ +] +qed. + +(* Basic forvard lemmas *****************************************************) + +(* Basic-1: was: drop_S *) +lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 → + ↓[O, e + 1] L1 ≡ K2. +#L1 elim L1 -L1 +[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct +| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H + elim (drop_inv_O1 … H) -H * #He #H + [ -IHL1; destruct -e K2 I2 V2 /2/ + | @drop_drop >(plus_minus_m_m e 1) /2/ + ] +] +qed. + +lemma drop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1]. +#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize +[ /2/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 + >(tw_lift … HV21) -HV21 /2/ +] +qed. + +lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e. + ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|. +#L1 elim L1 -L1 +[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct +| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H + elim (drop_inv_O1 … H) -H * #He #H + [ -IHL1; destruct -e K2 I2 V2 // + | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/ + ] +] +qed. + +lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e. +#L1 elim L1 -L1 +[ #L2 #e #H >(drop_inv_atom1 … H) -H // +| #K1 #I1 #V1 #IHL1 #L2 #e #H + elim (drop_inv_O1 … H) -H * #He #H + [ -IHL1; destruct -e L2 // + | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize + >minus_le_minus_minus_comm // + ] +] +qed. + +(* Basic-1: removed theorems 49: + drop_skip_flat + cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf + drop_clear drop_clear_O drop_clear_S + clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r + clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle + getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans + getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt + getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev + drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge + getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O + getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le + getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop_drop.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop_drop.ma new file mode 100644 index 000000000..934374853 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/drop_drop.ma @@ -0,0 +1,127 @@ +(**************************************************************************) +(* ___ *) +(* ||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/lift_lift.ma". +include "Basic-2/substitution/drop.ma". + +(* DROPPING *****************************************************************) + +(* Main properties **********************************************************) + +(* Basic-1: was: drop_mono *) +theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 → + ∀L2. ↓[d, e] L ≡ L2 → L1 = L2. +#d #e #L #L1 #H elim H -H d e L L1 +[ #d #e #L2 #H + >(drop_inv_atom1 … H) -H L2 // +| #K #I #V #L2 #HL12 + <(drop_inv_refl … HL12) -HL12 L2 // +| #L #K #I #V #e #_ #IHLK #L2 #H + lapply (drop_inv_drop1 … H ?) -H /2/ +| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H + elim (drop_inv_skip1 … H ?) -H // (lift_inj … HVT1 … HVT2) -HVT1 HVT2 + >(IHLK1 … HLK2) -IHLK1 HLK2 // +] +qed. + +(* Basic-1: was: drop_conf_ge *) +theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → + ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 → + ↓[0, e2 - e1] L1 ≡ L2. +#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1 +[ #d #e #e2 #L2 #H + >(drop_inv_atom1 … H) -H L2 // +| // +| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2 + lapply (drop_inv_drop1 … H ?) -H /2/ #HL2 + minus_minus_comm /3/ (**) (* explicit constructor *) +] +qed. + +(* Basic-1: was: drop_conf_lt *) +theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → + ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 → + e2 < d1 → let d ≝ d1 - e2 - 1 in + ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 & + ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2. +#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1 +[ #d #e #e2 #K2 #I #V2 #H + lapply (drop_inv_atom1 … H) -H #H destruct +| #L #I #V #e2 #K2 #J #V2 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H + elim (lt_zero_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d + elim (drop_inv_O1 … H) -H * + [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/ + | -HL12 -HV12 #He #HLK + elim (IHL12 … HLK ?) -IHL12 HLK [ (drop_inv_atom1 … H) -H L2 /2/ +| #K #I #V #e2 #L2 #HL2 #H + lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/ +| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H + lapply (le_O_to_eq_O … H) -H #H destruct -e2; + elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0 + lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d + elim (drop_inv_O1 … H) -H * + [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/ + | -HL12 HV12 #He2 #HL2 + elim (IHL12 … HL2 ?) -IHL12 HL2 L2 + [ >minus_le_minus_minus_comm // /3/ | /2/ ] + ] +] +qed. + +(* Basic-1: was: drop_trans_ge *) +theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → + ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2. +#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L +[ #d #e #e2 #L2 #H + >(drop_inv_atom1 … H) -H L2 // +| // +| /3/ +| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2 + lapply (lt_to_le_to_lt 0 … Hde2) // #He2 + lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2 + lapply (drop_inv_drop1 … H ?) -H // #HL2 + @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *) +] +qed. + +theorem drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L. + ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 → + ↓[0, e2 + e1] L1 ≡ L2. +#e1 #e1 #e2 >commutative_plus /2 width=5/ +qed. + +(* Basic-1: was: drop_conf_rev *) +axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L → + ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma new file mode 100644 index 000000000..cccd3b03c --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma @@ -0,0 +1,257 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/term_weight.ma". + +(* RELOCATION ***************************************************************) + +(* Basic-1: includes: + lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat +*) +inductive lift: nat → nat → relation term ≝ +| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k) +| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i) +| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e)) +| lift_bind : ∀I,V1,V2,T1,T2,d,e. + lift d e V1 V2 → lift (d + 1) e T1 T2 → + lift d e (𝕓{I} V1. T1) (𝕓{I} V2. T2) +| lift_flat : ∀I,V1,V2,T1,T2,d,e. + lift d e V1 V2 → lift d e T1 T2 → + lift d e (𝕗{I} V1. T1) (𝕗{I} V2. T2) +. + +interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2). + +(* Basic properties *********************************************************) + +(* Basic-1: was: lift_lref_gt *) +lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i. +#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/ +qed. + +(* Basic-1: was: lift_r *) +lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T. +#T elim T -T +[ * #i // #d elim (lt_or_ge i d) /2/ +| * /2/ +] +qed. + +lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. +#T1 elim T1 -T1 +[ * #i /2/ #d #e elim (lt_or_ge i d) /3/ +| * #I #V1 #T1 #IHV1 #IHT1 #d #e + elim (IHV1 d e) -IHV1 #V2 #HV12 + [ elim (IHT1 (d+1) e) -IHT1 /3/ + | elim (IHT1 d e) -IHT1 /3/ + ] +] +qed. + +(* Basic-1: was: lift_free (right to left) *) +lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1. + d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → + ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2. +#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2 +[ /3/ +| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/ +| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12 + lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21 + <(arith_d1 i e2 e1) // /3/ +| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT (d2+1) … ? ? He12) /3 width = 5/ +| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT d2 … ? ? He12) /3 width = 5/ +] +qed. + +(* Basic forward lemmas *****************************************************) + +lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2]. +#d #e #T1 #T2 #H elim H -d e T1 T2; normalize // +qed. + +(* Basic inversion lemmas ***************************************************) + +fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2. +#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/ +qed. + +lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2. +/2/ qed. + +fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. +#d #e #T1 #T2 * -d e T1 T2 // +[ #i #d #e #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +] +qed. + +lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k. +/2 width=5/ qed. + +fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i → + (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). +#d #e #T1 #T2 * -d e T1 T2 +[ #k #d #e #i #H destruct +| #j #d #e #Hj #i #Hi destruct /3/ +| #j #d #e #Hj #i #Hi destruct /3/ +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +] +qed. + +lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → + (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). +/2/ qed. + +lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i. +#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // +#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd +elim (lt_refl_false … Hdd) +qed. + +lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e). +#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // +#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd +elim (lt_refl_false … Hdd) +qed. + +fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → + ∀I,V1,U1. T1 = 𝕓{I} V1.U1 → + ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & + T2 = 𝕓{I} V2. U2. +#d #e #T1 #T2 * -d e T1 T2 +[ #k #d #e #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ +| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct +] +qed. + +lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 → + ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & + T2 = 𝕓{I} V2. U2. +/2/ qed. + +fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → + ∀I,V1,U1. T1 = 𝕗{I} V1.U1 → + ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & + T2 = 𝕗{I} V2. U2. +#d #e #T1 #T2 * -d e T1 T2 +[ #k #d #e #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #i #d #e #_ #I #V1 #U1 #H destruct +| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct +| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/ +] +qed. + +lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 → + ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & + T2 = 𝕗{I} V2. U2. +/2/ qed. + +fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. +#d #e #T1 #T2 * -d e T1 T2 // +[ #i #d #e #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct +] +qed. + +(* Basic-1: was: lift_gen_sort *) +lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k. +/2 width=5/ qed. + +fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i → + (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). +#d #e #T1 #T2 * -d e T1 T2 +[ #k #d #e #i #H destruct +| #j #d #e #Hj #i #Hi destruct /3/ +| #j #d #e #Hj #i #Hi destruct le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ] + | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/ + ] + ] +| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 + lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2; + elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1 + >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/ +| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12 + lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2; + elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1 + elim (IHU … HU2 ?) /3 width = 5/ +] +qed. + +theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2. +#d #e #T #U1 #H elim H -H d e T U1 +[ #k #d #e #X #HX + lapply (lift_inv_sort1 … HX) -HX // +| #i #d #e #Hid #X #HX + lapply (lift_inv_lref1_lt … HX ?) -HX // +| #i #d #e #Hdi #X #HX + lapply (lift_inv_lref1_ge … HX ?) -HX // +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ +| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/ +] +qed. + +(* Basic-1: was: lift_free (left to right) *) +theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → + d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2. +#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ + >(lift_inv_sort1 … HT2) -HT2 // +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_ + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 + lapply (lift_inv_lref1_lt … HT2 Hid2) /2/ +| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21 + lapply (lift_inv_lref1_ge … HT2 ?) -HT2 + [ @(transitive_le … Hd21 ?) -Hd21 /2/ + | -Hd21 /2/ + ] +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; + lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10 + lapply (IHT12 … HT20 ? ?) /2/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; + lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10 + lapply (IHT12 … HT20 ? ?) /2/ +] +qed. + +(* Basic-1: was: lift_d (right to left) *) +theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 → + ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +[ #k #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_sort1 … HX) -HX /2/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ + lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2 + elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21 + lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2 + lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X; + >plus_plus_comm_23 /4/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; + elim (IHV12 … HV20 ?) -IHV12 HV20 // + elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21 + elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; + elim (IHV12 … HV20 ?) -IHV12 HV20 // + elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/ +] +qed. + +(* Basic-1: was: lift_d (left to right) *) +theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → + ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2. +#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T +[ #k #d1 #e1 #d2 #e2 #X #HX #_ + >(lift_inv_sort1 … HX) -HX /2/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded + lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e + lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e + lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2 + lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/ +| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_ + elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X; + [2: >plus_plus_comm_23] /4/ +| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded + elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X; + elim (IHV12 … HV20 ?) -IHV12 HV20 // + elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T + (plus_minus_m_m e 1) /2/ +qed. + +lemma ltps_tps1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 [d - 1, e] ≫ L2 → L2 ⊢ V1 [d - 1, e] ≫ V2 → + 0 < d → L1. 𝕓{I} V1 [d, e] ≫ L2. 𝕓{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd +>(plus_minus_m_m d 1) /2/ +qed. + +(* Basic-1: was by definition: csubst1_refl *) +lemma ltps_refl: ∀L,d,e. L [d, e] ≫ L. +#L elim L -L // +#L #I #V #IHL * /2/ * /2/ +qed. + +(* Basic inversion lemmas ***************************************************) + +fact ltps_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → e = 0 → L1 = L2. +#d #e #L1 #L2 #H elim H -H d e L1 L2 // +[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ #H + elim (plus_S_eq_O_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct -e + >(IHL12 ?) -IHL12 // >(tps_inv_refl_O2 … HV12) // +] +qed. + +lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ≫ L2 → L1 = L2. +/2/ qed. + +fact ltps_inv_atom1_aux: ∀d,e,L1,L2. + L1 [d, e] ≫ L2 → L1 = ⋆ → L2 = ⋆. +#d #e #L1 #L2 * -d e L1 L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltps_inv_atom1: ∀d,e,L2. ⋆ [d, e] ≫ L2 → L2 = ⋆. +/2 width=5/ qed. + +fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e → + ∀K1,I,V1. L1 = K1. 𝕓{I} V1 → + ∃∃K2,V2. K1 [0, e - 1] ≫ K2 & + K2 ⊢ V1 [0, e - 1] ≫ V2 & + L2 = K2. 𝕓{I} V2. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K1 #J #W1 #H destruct -L1 I V1 /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H) +] +qed. + +lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. 𝕓{I} V1 [0, e] ≫ L2 → 0 < e → + ∃∃K2,V2. K1 [0, e - 1] ≫ K2 & K2 ⊢ V1 [0, e - 1] ≫ V2 & + L2 = K2. 𝕓{I} V2. +/2 width=5/ qed. + +fact ltps_inv_tps11_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → 0 < d → + ∀I,K1,V1. L1 = K1. 𝕓{I} V1 → + ∃∃K2,V2. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L2 = K2. 𝕓{I} V2. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #I #K1 #V1 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K1 #W1 #H destruct -L1 I V1 + /2 width=5/ +] +qed. + +lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. 𝕓{I} V1 [d, e] ≫ L2 → 0 < d → + ∃∃K2,V2. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L2 = K2. 𝕓{I} V2. +/2/ qed. + +fact ltps_inv_atom2_aux: ∀d,e,L1,L2. + L1 [d, e] ≫ L2 → L2 = ⋆ → L1 = ⋆. +#d #e #L1 #L2 * -d e L1 L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma ltps_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆. +/2 width=5/ qed. + +fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e → + ∀K2,I,V2. L2 = K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & + K2 ⊢ V1 [0, e - 1] ≫ V2 & + L1 = K1. 𝕓{I} V1. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct -L2 I V2 /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H) +] +qed. + +lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ≫ K2. 𝕓{I} V2 → 0 < e → + ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & K2 ⊢ V1 [0, e - 1] ≫ V2 & + L1 = K1. 𝕓{I} V1. +/2 width=5/ qed. + +fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → 0 < d → + ∀I,K2,V2. L2 = K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L1 = K1. 𝕓{I} V1. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #I #K2 #V2 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct -L2 I V2 + /2 width=5/ +] +qed. + +lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ≫ K2. 𝕓{I} V2 → 0 < d → + ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L1 = K1. 𝕓{I} V1. +/2/ qed. + +(* Basic-1: removed theorems 27: + csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq + csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans + csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back + csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt + csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back + csubst0_snd_bind csubst0_fst_bind csubst0_both_bind + csubst1_head csubst1_flat csubst1_gen_head + csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1 + +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_drop.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_drop.ma new file mode 100644 index 000000000..dec15efaa --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_drop.ma @@ -0,0 +1,131 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ltps.ma". + +(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************) + +lemma ltps_drop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2. +#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + lapply (plus_le_weak … He12) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 + lapply (plus_le_weak … Hd1e2) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/ +] +qed. + +lemma ltps_drop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2. +#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + lapply (plus_le_weak … He12) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/ +| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 + lapply (plus_le_weak … Hd1e2) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/ +] +qed. + +lemma ltps_drop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. L2 [0, d1 + e1 - e2] ≫ L & ↓[0, e2] L1 ≡ L. +#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2; + lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK01 He21 e2 L2 plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + lapply (plus_le_weak … Hd1e2) #He2 + (drop_inv_atom1 … H) -H /2/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2; + lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK10 He21 e2 L2 plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + lapply (plus_le_weak … Hd1e2) #He2 + (drop_inv_atom1 … H) -H /2/ +| /2/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2; + lapply (drop_inv_refl … H) -H #H destruct -L2 /3/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK01 He2d1 e2 L2 (drop_inv_atom1 … H) -H /2/ +| /2/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2; + lapply (drop_inv_refl … H) -H #H destruct -L2 /3/ +| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK10 He2d1 e2 L2 arith_a2 /3/ + | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2 + [ elim (ltps_drop_conf_be … HL01 … HLK0 ? ?) -HL01 HLK0 [2,3: /2/ ] #X #H #HLK1 + elim (ltps_inv_tps21 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X; + lapply (drop_fwd_drop2 … HLK1) #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // normalize #HW01 + lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 // + | lapply (ltps_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/ + ] + ] +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 [2: /2/ ] -HL01 /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 + elim (IHVW2 … HL01) -IHVW2; + elim (IHTU2 … HL01) -IHTU2 HL01 /3 width=5/ +] +qed. + +lemma ltps_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 → + ∀L1,d1,e1. L1 [d1, e1] ≫ L0 → d1 + e1 ≤ d2 → + L1 ⊢ T2 [d2, e2] ≫ U2. +#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2 +[ // +| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2 + lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2 + lapply (ltps_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /2/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2 + @tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *) +| /3/ +] +qed. + +(* Basic-1: was: subst1_subst1 *) +lemma ltps_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 → + ∀L1,d1,e1. L1 [d1, e1] ≫ L0 → + ∃∃T. L1 ⊢ T2 [d2, e2] ≫ T & + L0 ⊢ T [d1, e1] ≫ U2. +#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2 +[ /2/ +| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 + elim (lt_or_ge i2 d1) #Hi2d1 + [ elim (ltps_drop_trans_le … HL10 … HLK0 ?) -HL10 /2/ #X #H #HLK1 + elim (ltps_inv_tps12 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X; + lapply (drop_fwd_drop2 … HLK0) -HLK0 #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // >arith_a2 /3/ + | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2 + [ elim (ltps_drop_trans_be … HL10 … HLK0 ? ?) -HL10 [2,3: /2/ ] #X #H #HLK1 + elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X; + lapply (drop_fwd_drop2 … HLK0) -HLK0 #H + elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1 + lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // normalize #HW01 + lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 // + | lapply (ltps_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /3/ + ] + ] +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2 + elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 [2: /2/ ] -HL10 /3 width=5/ +| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 + elim (IHVW2 … HL10) -IHVW2; + elim (IHTU2 … HL10) -IHTU2 HL10 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma new file mode 100644 index 000000000..7197a53d9 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps.ma @@ -0,0 +1,221 @@ +(**************************************************************************) +(* ___ *) +(* ||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/grammar/cl_weight.ma". +include "Basic-2/substitution/drop.ma". + +(* PARALLEL SUBSTITUTION ON TERMS *******************************************) + +inductive tps: nat → nat → lenv → relation term ≝ +| tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I}) +| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e → + ↓[0, i] L ≡ K. 𝕓{Abbr} V → ↑[0, i + 1] V ≡ W → tps d e L (#i) W +| tps_bind : ∀L,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 → tps (d + 1) e (L. 𝕓{I} V2) T1 T2 → + tps d e L (𝕓{I} V1. T1) (𝕓{I} V2. T2) +| tps_flat : ∀L,I,V1,V2,T1,T2,d,e. + tps d e L V1 V2 → tps d e L T1 T2 → + tps d e L (𝕗{I} V1. T1) (𝕗{I} V2. T2) +. + +interpretation "parallel substritution (term)" + 'PSubst L T1 d e T2 = (tps d e L T1 T2). + +(* Basic properties *********************************************************) + +lemma tps_leq_repl_dx: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 → + ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2. +#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e +[ // +| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12 + elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // /2/ +| /4/ +| /3/ +] +qed. + +lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T. +#T elim T -T // +#I elim I -I /2/ +qed. + +lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → + L ⊢ T1 [d2, e2] ≫ T2. +#L #T1 #T2 #d1 #e1 #H elim H -H L T1 T2 d1 e1 +[ // +| #L #K #V #W #i #d1 #e1 #Hid1 #Hide1 #HLK #HVW #d2 #e2 #Hd12 #Hde12 + lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2 + lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 /2/ +| /4/ +| /4/ +] +qed. + +lemma tps_weak_top: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ // +| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW + lapply (drop_fwd_drop2_length … HLK) #Hi + lapply (le_to_lt_to_lt … Hdi Hi) #Hd + lapply (plus_minus_m_m_comm (|L|) d ?) /2/ +| normalize /2/ +| /2/ +] +qed. + +lemma tps_weak_all: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2. +#L #T1 #T2 #d #e #HT12 +lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tps_weak_top … HT12) // +qed. + +lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e → + ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ /2/ +| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde + elim (lt_or_ge i j) + [ -Hide Hjde; + >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/ + | -Hdi Hdj; #Hid + generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *) + >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/ + ] +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2 + elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *) + -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2 + lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /3 width=5/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide + elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 // + -Hdi Hide /3 width=5/ +] +qed. + +(* Basic inversion lemmas ***************************************************) + +fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} → + T2 = 𝕒{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2 & + I = LRef i. +#L #T1 #T2 #d #e * -L T1 T2 d e +[ #L #I #d #e #J #H destruct -I /2/ +| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct -I /3 width=8/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct +] +qed. + +lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 → + T2 = 𝕒{I} ∨ + ∃∃K,V,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2 & + I = LRef i. +/2/ qed. + + +(* Basic-1: was: subst1_gen_sort *) +lemma tps_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫ T2 → T2 = ⋆k. +#L #T2 #k #d #e #H +elim (tps_inv_atom1 … H) -H // +* #K #V #i #_ #_ #_ #_ #H destruct +qed. + +(* Basic-1: was: subst1_gen_lref *) +lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 → + T2 = #i ∨ + ∃∃K,V. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V & + ↑[O, i + 1] V ≡ T2. +#L #T2 #i #d #e #H +elim (tps_inv_atom1 … H) -H /2/ +* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct -i /3/ +qed. + +fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → + ∀I,V1,T1. U1 = 𝕓{I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & + L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 & + U2 = 𝕓{I} V2. T2. +#d #e #L #U1 #U2 * -d e L U1 U2 +[ #L #k #d #e #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ +| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct +] +qed. + +lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & + L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫ T2 & + U2 = 𝕓{I} V2. T2. +/2/ qed. + +fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → + ∀I,V1,T1. U1 = 𝕗{I} V1. T1 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 & + U2 = 𝕗{I} V2. T2. +#d #e #L #U1 #U2 * -d e L U1 U2 +[ #L #k #d #e #I #V1 #T1 #H destruct +| #L #K #V #W #i #d #e #_ #_ #_ #_ #I #V1 #T1 #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct +| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/ +] +qed. + +lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 & + U2 = 𝕗{I} V2. T2. +/2/ qed. + +fact tps_inv_refl_O2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 0 → T1 = T2. +#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e +[ // +| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct -e; + lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi Hide (le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK + lapply (drop_mono … HLK0 … HLK) #H destruct +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK + >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 /2/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK + >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 // +] +qed. + +lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫ T2 → + ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2. +/2 width=8/ qed. + +(* Relocation properties ****************************************************) + +(* Basic-1: was: subst1_lift_lt *) +lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 → + ∀L,U1,U2,d,e. ↓[d, e] L ≡ K → + ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 → + dt + et ≤ d → + L ⊢ U1 [dt, et] ≫ U2. +#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et +[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 H2 // +| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd + lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid + lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1; + elim (lift_trans_ge … HVW … HVU2 ?) -HVW HVU2 W // (lift_mono … HVY … HVW) -HVY HVW Y #H destruct -X /2/ +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2; + @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *) +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2; + /3 width=6/ +] +qed. + +(* Basic-1: was: subst1_lift_ge *) +lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 → + ∀L,U1,U2,d,e. ↓[d, e] L ≡ K → + ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 → + d ≤ dt → + L ⊢ U1 [dt + e, et] ≫ U2. +#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et +[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ + >(lift_mono … H1 … H2) -H1 H2 // +| #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt + lapply (transitive_le … Hddt … Hdti) -Hddt #Hid + lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1; + lapply (lift_trans_be … HVW … HWU2 ? ?) -HVW HWU2 W // [ /2/ ] >plus_plus_comm_23 #HVU2 + lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid /3/ +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt + elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2; + @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *) +| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt + elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 + elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2; + /3 width=5/ +] +qed. + +(* Basic-1: was: subst1_gen_lift_lt *) +lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 → + ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 → + dt + et ≤ d → + ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et +[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/ + | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/ + ] +| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd + lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid + lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1; + elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #U #HKL #_ #HUV + elim (lift_trans_le … HUV … HVW ?) -HUV HVW V // >arith_a2 // -Hid /3/ +| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd + elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X; + elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2 + elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK Hdetd (**) (* /3 width=5/ is too slow *) + /3 width=5/ +| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd + elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X; + elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // + elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/ +] +qed. + +(* Basic-1: was: subst1_gen_lift_ge *) +lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 → + ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 → + d + e ≤ dt → + ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et +[ #L * #i #dt #et #K #d #e #_ #T1 #H #_ + [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/ + | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/ + ] +| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt + lapply (transitive_le … Hdedt … Hdti) #Hdei + lapply (plus_le_weak … Hdedt) -Hdedt #Hedt + lapply (plus_le_weak … Hdei) #Hei + lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1; + lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV + elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02 + @ex2_1_intro + [2: @tps_subst [3: /2/ |5,6: // |1,2: skip |4: @arith5 // ] + |1: skip + | // + ] (**) (* explicitc constructors *) +| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd + elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X; + lapply (plus_le_weak … Hdetd) #Hedt + elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2 + elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ] + IHV12 // >IHT12 // +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX + elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X + >IHV12 // >IHT12 // +] +qed. +(* + Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?) + (subst0 i v t1 (lift h d u2)) -> + (le (plus d h) i) -> + (EX u1 | (subst0 (minus i h) v u1 u2) & + t1 = (lift h d u1) + ). + + + Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?) + (subst0 i v t1 (lift h d u2)) -> + (le d i) -> (lt i (plus d h)) -> + (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)). +*) + +lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 → + ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 → + d ≤ dt → dt ≤ d + e → d + e ≤ dt + et → + ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet +elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2 +lapply (tps_weak … HU1 d e ? ?) -HU1 // (lift_mono … HVT1 … HVT2) -HVT1 HVT2 /2/ + ] +| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V1) ?) -HT02 /2/ #HT02 + elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2 + elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2 + lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1 /2/ + lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V) ?) -HT2 /3 width=5/ +| #L #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV01 … HV02) -IHV01 HV02; + elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/ +] +qed. + +(* Basic-1: was: subst1_confluence_neq *) +theorem tps_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫ T1 → + ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 → + (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → + ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫ T. +#L1 #T0 #T1 #d1 #e1 #H elim H -H L1 T0 T1 d1 e1 +[ /2/ +| #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2 + elim (tps_inv_lref1 … H1) -H1 + [ #H destruct -T2 /4/ + | -HLK1 HVT1 * #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded + [ -Hd1 Hde2; + lapply (transitive_le … Hded Hd2) -Hded Hd2 #H + lapply (lt_to_le_to_lt … Hde1 H) -Hde1 H #H + elim (lt_refl_false … H) + | -Hd2 Hde1; + lapply (transitive_le … Hded Hd1) -Hded Hd1 #H + lapply (lt_to_le_to_lt … Hde2 H) -Hde2 H #H + elim (lt_refl_false … H) + ] + ] +| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV01 … HV02 H) -IHV01 HV02 #V #HV1 #HV2 + elim (IHT01 … HT02 ?) -IHT01 HT02 + [ -H #T #HT1 #HT2 + lapply (tps_leq_repl_dx … HT1 (L2. 𝕓{I} V) ?) -HT1 /2/ + lapply (tps_leq_repl_dx … HT2 (L1. 𝕓{I} V) ?) -HT2 /3 width=5/ + | -HV1 HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %) elim H -H #H + [ @or_introl | @or_intror ] /2 by monotonic_le_plus_l/ (**) (* /3/ is too slow *) + ] +| #L1 #I #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV01 … HV02 H) -IHV01 HV02; + elim (IHT01 … HT02 H) -IHT01 HT02 H /3 width=5/ +] +qed. + +(* Note: the constant 1 comes from tps_subst *) +(* Basic-1: was: subst1_trans *) +theorem tps_trans_ge: ∀L,T1,T0,d,e. L ⊢ T1 [d, e] ≫ T0 → + ∀T2. L ⊢ T0 [d, 1] ≫ T2 → 1 ≤ e → + L ⊢ T1 [d, e] ≫ T2. +#L #T1 #T0 #d #e #H elim H -L T1 T0 d e +[ #L #I #d #e #T2 #H #He + elim (tps_inv_atom1 … H) -H + [ #H destruct -T2 // + | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct -I; + lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2/ + ] +| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He + lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2/ #HVT2 + <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2/ +| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (tps_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct -X; + lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V0) ?) -HT02 /2/ #HT02 + lapply (IHT10 … HT02 He) -IHT10 HT02 #HT12 + lapply (tps_leq_repl_dx … HT12 (L. 𝕓{I} V2) ?) -HT12 /3/ +| #L #I #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He + elim (tps_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct -X /3/ +] +qed. + +theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 → + ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫ T2. +#L #T1 #T0 #d1 #e1 #H elim H -L T1 T0 d1 e1 +[ /2/ +| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1 + lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1 + lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2; normalize /2/ -Hde2i1 #HWT2 + <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4/ +| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + lapply (tps_leq_repl_dx … HT02 (L. 𝕓{I} V0) ?) -HT02 /2/ #HT02 + elim (IHV10 … HV02 ?) -IHV10 HV02 // #V + elim (IHT10 … HT02 ?) -IHT10 HT02 [2: /2/ ] #T #HT1 #HT2 + lapply (tps_leq_repl_dx … HT1 (L. 𝕓{I} V) ?) -HT1; + lapply (tps_leq_repl_dx … HT2 (L. 𝕓{I} V2) ?) -HT2 /3 width=6/ +| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1 + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV10 … HV02 ?) -IHV10 HV02 // + elim (IHT10 … HT02 ?) -IHT10 HT02 // /3 width=6/ +] +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss.ma new file mode 100644 index 000000000..4f2062a14 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss.ma @@ -0,0 +1,107 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ltps.ma". +include "Basic-2/unfold/tpss.ma". + +(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************) + +definition ltpss: nat → nat → relation lenv ≝ + λd,e. TC … (ltps d e). + +interpretation "partial unfold (local environment)" + 'PSubstStar L1 d e L2 = (ltpss d e L1 L2). + +(* Basic eliminators ********************************************************) + +lemma ltpss_ind: ∀d,e,L1. ∀R: lenv → Prop. R L1 → + (∀L,L2. L1 [d, e] ≫* L → L [d, e] ≫ L2 → R L → R L2) → + ∀L2. L1 [d, e] ≫* L2 → R L2. +#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) // +qed. + +(* Basic properties *********************************************************) + +lemma ltpss_strap: ∀L1,L,L2,d,e. + L1 [d, e] ≫ L → L [d, e] ≫* L2 → L1 [d, e] ≫* L2. +/2/ qed. + +lemma ltpss_refl: ∀L,d,e. L [d, e] ≫* L. +/2/ qed. + +(* Basic inversion lemmas ***************************************************) + +lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ≫* L2 → L1 = L2. +#d #L1 #L2 #H @(ltpss_ind … H) -L2 // +#L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 // +qed. + +lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ≫* L2 → L2 = ⋆. +#d #e #L2 #H @(ltpss_ind … H) -L2 // +#L #L2 #_ #HL2 #IHL destruct -L +>(ltps_inv_atom1 … HL2) -HL2 // +qed. +(* +fact ltps_inv_atom2_aux: ∀d,e,L1,L2. + L1 [d, e] ≫ L2 → L2 = ⋆ → L1 = ⋆. +#d #e #L1 #L2 * -d e L1 L2 +[ // +| #L #I #V #H destruct +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H destruct +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct +] +qed. + +lemma drop_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆. +/2 width=5/ qed. + +fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e → + ∀K2,I,V2. L2 = K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & + K2 ⊢ V1 [0, e - 1] ≫ V2 & + L1 = K1. 𝕓{I} V1. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #_ #K1 #I #V1 #H destruct +| #L1 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct -L2 I V2 /2 width=5/ +| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H) +] +qed. + +lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ≫ K2. 𝕓{I} V2 → 0 < e → + ∃∃K1,V1. K1 [0, e - 1] ≫ K2 & K2 ⊢ V1 [0, e - 1] ≫ V2 & + L1 = K1. 𝕓{I} V1. +/2 width=5/ qed. + +fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → 0 < d → + ∀I,K2,V2. L2 = K2. 𝕓{I} V2 → + ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L1 = K1. 𝕓{I} V1. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #I #K2 #V2 #H destruct +| #L #I #V #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct -L2 I V2 + /2 width=5/ +] +qed. + +lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ≫ K2. 𝕓{I} V2 → 0 < d → + ∃∃K1,V1. K1 [d - 1, e] ≫ K2 & + K2 ⊢ V1 [d - 1, e] ≫ V2 & + L1 = K1. 𝕓{I} V1. +/2/ qed. + +*) diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_drop.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_drop.ma new file mode 100644 index 000000000..4b5088638 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_drop.ma @@ -0,0 +1,132 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ltps.ma". + +(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************) + +lemma ltps_drop_conf_ge: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2. +#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H // +| // +| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12 + lapply (plus_le_weak … He12) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 + lapply (plus_le_weak … Hd1e2) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK01 … HK0L2 ?) -IHK01 HK0L2 /2/ +] +qed. + +lemma ltps_drop_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫ L0 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → + d1 + e1 ≤ e2 → ↓[0, e2] L1 ≡ L2. +#L1 #L0 #d1 #e1 #H elim H -H L1 L0 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H // +| // +| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12 + lapply (plus_le_weak … He12) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/ +| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 + lapply (plus_le_weak … Hd1e2) #He2 + lapply (drop_inv_drop1 … H ?) -H // #HK0L2 + lapply (IHK10 … HK0L2 ?) -IHK10 HK0L2 /2/ +] +qed. + +lemma ltps_drop_conf_be: ∀L0,L1,d1,e1. L0 [d1, e1] ≫ L1 → + ∀L2,e2. ↓[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 → + ∃∃L. L2 [0, d1 + e1 - e2] ≫ L & ↓[0, e2] L1 ≡ L. +#L0 #L1 #d1 #e1 #H elim H -H L0 L1 d1 e1 +[ #d1 #e1 #L2 #e2 #H >(drop_inv_atom1 … H) -H /2/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2; + lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK01 He21 e2 L2 plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1 + lapply (plus_le_weak … Hd1e2) #He2 + (drop_inv_atom1 … H) -H /2/ +| normalize #L #I #V #L2 #e2 #HL2 #_ #He2 + lapply (le_n_O_to_eq … He2) -He2 #H destruct -e2; + lapply (drop_inv_refl … HL2) -HL2 #H destruct -L2 /2/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK10 He21 e2 L2 plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1 + lapply (plus_le_weak … Hd1e2) #He2 + (drop_inv_atom1 … H) -H /2/ +| /2/ +| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2; + lapply (drop_inv_refl … H) -H #H destruct -L2 /3/ +| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK01 He2d1 e2 L2 (drop_inv_atom1 … H) -H /2/ +| /2/ +| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2 + lapply (le_n_O_to_eq … He2) -He2 #He2 destruct -e2; + lapply (drop_inv_refl … H) -H #H destruct -L2 /3/ +| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1 + lapply (drop_inv_O1 … H) -H * * #He2 #HK0L2 + [ destruct -IHK10 He2d1 e2 L2 (plus_minus_m_m e 1) /2/ +qed. + +lemma ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e. + L1 [d, e] ≫* L2 → L2 ⊢ V1 [d, e] ≫* V2 → + L1. 𝕓{I} V1 [d + 1, e] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #H @(tpss_ind … H) -V2 +[ /2/ +| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2/ +] +qed. + +lemma ltpss_tpss1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 [d - 1, e] ≫* L2 → L2 ⊢ V1 [d - 1, e] ≫* V2 → + 0 < d → L1. 𝕓{I} V1 [d, e] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd +>(plus_minus_m_m d 1) /2/ +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_tpss.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_tpss.ma new file mode 100644 index 000000000..b8e425e63 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/ltpss_tpss.ma @@ -0,0 +1,169 @@ +(**************************************************************************) +(* ___ *) +(* ||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/unfold/tpss_ltps.ma". +include "Basic-2/unfold/ltpss.ma". + +(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************) + +(* Properties concerning partial unfold on terms ****************************) + +lemma ltpss_tpss_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 → + ∀T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫* U2 → + d1 + e1 ≤ d2 → L1 ⊢ T2 [d2, e2] ≫* U2. +#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0 // +#L #L0 #_ #HL0 #IHL #T2 #U2 #d2 #e2 #HTU2 #Hde1d2 +lapply (ltps_tpss_trans_ge … HL0 HTU2) -HL0 HTU2 /2/ +qed. + +lemma ltpss_tps_trans_ge: ∀L1,L0,d1,e1. L1 [d1, e1] ≫* L0 → + ∀T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 → + d1 + e1 ≤ d2 → L1 ⊢ T2 [d2, e2] ≫* U2. +#L1 #L0 #d1 #e1 #HL10 #T2 #U2 #d2 #e2 #HTU2 #Hde1d2 +@(ltpss_tpss_trans_ge … HL10 … Hde1d2) /2/ (**) (* /3 width=6/ is too slow *) +qed. + +lemma ltpss_tpss_trans_eq: ∀L0,L1,d,e. L0 [d, e] ≫* L1 → + ∀T2,U2. L1 ⊢ T2 [d, e] ≫* U2 → L0 ⊢ T2 [d, e] ≫* U2. +#L0 #L1 #d #e #H @(ltpss_ind … H) -L1 +[ /2/ +| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2 + lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 HTU2 /2/ +] +qed. + +lemma ltpss_tps_trans_eq: ∀L0,L1,d,e. L0 [d, e] ≫* L1 → + ∀T2,U2. L1 ⊢ T2 [d, e] ≫ U2 → L0 ⊢ T2 [d, e] ≫* U2. +/3/ qed. + +lemma ltpss_tpss_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 → + L0 ⊢ T2 [d2, e2] ≫* U2 → L0 [d1, e1] ≫* L1 → + L1 ⊢ T2 [d2, e2] ≫* U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #H @(ltpss_ind … H) -L1 +[ // +| -HTU2 #L #L1 #_ #HL1 #IHL + lapply (ltps_tpss_conf_ge … HL1 IHL) -HL1 IHL // +] +qed. + +lemma ltpss_tps_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. d1 + e1 ≤ d2 → + L0 ⊢ T2 [d2, e2] ≫ U2 → L0 [d1, e1] ≫* L1 → + L1 ⊢ T2 [d2, e2] ≫* U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HTU2 #HL01 +@(ltpss_tpss_conf_ge … Hde1d2 … HL01) /2/ (**) (* /3 width=6/ is too slow *) +qed. + +lemma ltpss_tpss_conf_eq: ∀L0,L1,T2,U2,d,e. + L0 ⊢ T2 [d, e] ≫* U2 → L0 [d, e] ≫* L1 → + ∃∃T. L1 ⊢ T2 [d, e] ≫* T & L1 ⊢ U2 [d, e] ≫* T. +#L0 #L1 #T2 #U2 #d #e #HTU2 #H @(ltpss_ind … H) -L1 +[ /2/ +| -HTU2 #L #L1 #_ #HL1 * #W2 #HTW2 #HUW2 + elim (ltps_tpss_conf … HL1 HTW2) -HTW2 #T #HT2 #HW2T + elim (ltps_tpss_conf … HL1 HUW2) -HL1 HUW2 #U #HU2 #HW2U + elim (tpss_conf_eq … HW2T … HW2U) -HW2T HW2U #V #HTV #HUV + lapply (tpss_trans_eq … HT2 … HTV) -T; + lapply (tpss_trans_eq … HU2 … HUV) -U /2/ +] +qed. + +lemma ltpss_tps_conf_eq: ∀L0,L1,T2,U2,d,e. + L0 ⊢ T2 [d, e] ≫ U2 → L0 [d, e] ≫* L1 → + ∃∃T. L1 ⊢ T2 [d, e] ≫* T & L1 ⊢ U2 [d, e] ≫* T. +/3/ qed. + +lemma ltpss_tpss_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫* T2 → + ∀L2,ds,es. L1 [ds, es] ≫* L2 → + ∃∃T. L2 ⊢ T1 [d, e] ≫* T & L1 ⊢ T2 [ds, es] ≫* T. +#L1 #T1 #T2 #d #e #HT12 #L2 #ds #es #H @(ltpss_ind … H) -L2 +[ /3/ +| #L #L2 #HL1 #HL2 * #T #HT1 #HT2 + elim (ltps_tpss_conf … HL2 HT1) -HT1 #T0 #HT10 #HT0 + lapply (ltps_tpss_trans_eq … HL2 … HT0) -HL2 HT0 #HT0 + lapply (ltpss_tpss_trans_eq … HL1 … HT0) -HL1 HT0 #HT0 + lapply (tpss_trans_eq … HT2 … HT0) -T /2/ +] +qed. + +lemma ltpss_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 → + ∀L2,ds,es. L1 [ds, es] ≫* L2 → + ∃∃T. L2 ⊢ T1 [d, e] ≫* T & L1 ⊢ T2 [ds, es] ≫* T. +/3/ qed. + +(* Advanced properties ******************************************************) + +lemma ltpss_tps2: ∀L1,L2,I,e. + L1 [0, e] ≫* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ≫ V2 → + L1. 𝕓{I} V1 [0, e + 1] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #e #H @(ltpss_ind … H) -L2 +[ /3/ +| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12 + elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2 + lapply (IHL … HV1) -IHL HV1 #HL1 + @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e. + L1 [0, e - 1] ≫* L2 → L2 ⊢ V1 [0, e - 1] ≫ V2 → + 0 < e → L1. 𝕓{I} V1 [0, e] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He +>(plus_minus_m_m e 1) /2/ +qed. + +lemma ltpss_tps1: ∀L1,L2,I,d,e. L1 [d, e] ≫* L2 → + ∀V1,V2. L2 ⊢ V1 [d, e] ≫ V2 → + L1. 𝕓{I} V1 [d + 1, e] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2 +[ /3/ +| #L #L2 #_ #HL2 #IHL #V1 #V2 #HV12 + elim (ltps_tps_trans … HV12 … HL2) -HV12 #V #HV1 #HV2 + lapply (IHL … HV1) -IHL HV1 #HL1 + @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e. + L1 [d - 1, e] ≫* L2 → L2 ⊢ V1 [d - 1, e] ≫ V2 → + 0 < d → L1. 𝕓{I} V1 [d, e] ≫* L2. 𝕓{I} V2. +#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd +>(plus_minus_m_m d 1) /2/ +qed. + +(* Advanced forward lemmas **************************************************) + +lemma ltpss_fwd_tpsa21: ∀e,K1,I,V1,L2. 0 < e → K1. 𝕓{I} V1 [0, e] ≫* L2 → + ∃∃K2,V2. K1 [0, e - 1] ≫* K2 & K1 ⊢ V1 [0, e - 1] ≫* V2 & + L2 = K2. 𝕓{I} V2. +#e #K1 #I #V1 #L2 #He #H @(ltpss_ind … H) -L2 +[ /2 width=5/ +| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -L; + elim (ltps_inv_tps21 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H + lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2 + lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/ +] +qed. + +lemma ltpss_fwd_tpss11: ∀d,e,I,K1,V1,L2. 0 < d → K1. 𝕓{I} V1 [d, e] ≫* L2 → + ∃∃K2,V2. K1 [d - 1, e] ≫* K2 & + K1 ⊢ V1 [d - 1, e] ≫* V2 & + L2 = K2. 𝕓{I} V2. +#d #e #K1 #I #V1 #L2 #Hd #H @(ltpss_ind … H) -L2 +[ /2 width=5/ +| #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct -L; + elim (ltps_inv_tps11 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H + lapply (ltps_tps_trans_eq … HV2 … HK2) -HV2 #HV2 + lapply (ltpss_tpss_trans_eq … HK1 … HV2) -HV2 #HV2 /3 width=5/ +] +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss.ma new file mode 100644 index 000000000..6b63c4e1b --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss.ma @@ -0,0 +1,137 @@ +(**************************************************************************) +(* ___ *) +(* ||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/tps.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +definition tpss: nat → nat → lenv → relation term ≝ + λd,e,L. TC … (tps d e L). + +interpretation "partial unfold (term)" + 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2). + +(* Basic eliminators ********************************************************) + +lemma tpss_ind: ∀d,e,L,T1. ∀R: term → Prop. R T1 → + (∀T,T2. L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫ T2 → R T → R T2) → + ∀T2. L ⊢ T1 [d, e] ≫* T2 → R T2. +#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) // +qed. + +(* Basic properties *********************************************************) + +lemma tpss_strap: ∀L,T1,T,T2,d,e. + L ⊢ T1 [d, e] ≫ T → L ⊢ T [d, e] ≫* T2 → L ⊢ T1 [d, e] ≫* T2. +/2/ qed. + +lemma tpss_leq_repl_dx: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫* T2 → + ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫* T2. +/3/ qed. + +lemma tpss_refl: ∀d,e,L,T. L ⊢ T [d, e] ≫* T. +/2/ qed. + +lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 [d, e] ≫* V2 → + ∀I,T1,T2. L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫* T2 → + L ⊢ 𝕓{I} V1. T1 [d, e] ≫* 𝕓{I} V2. T2. +#L #V1 #V2 #d #e #HV12 elim HV12 -HV12 V2 +[ #V2 #HV12 #I #T1 #T2 #HT12 elim HT12 -HT12 T2 + [ /3 width=5/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #I #T1 #T2 #HT12 + lapply (tpss_leq_repl_dx … HT12 (L. 𝕓{I} V) ?) -HT12 /2/ #HT12 + lapply (IHV … HT12) -IHV HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e. + L ⊢ V1 [d, e] ≫ * V2 → L ⊢ T1 [d, e] ≫* T2 → + L ⊢ 𝕗{I} V1. T1 [d, e] ≫* 𝕗{I} V2. T2. +#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -HV12 V2 +[ #V2 #HV12 #HT12 elim HT12 -HT12 T2 + [ /3/ + | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *) + ] +| #V #V2 #_ #HV12 #IHV #HT12 + lapply (IHV … HT12) -IHV HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *) +] +qed. + +lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫* T2 → + ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → + L ⊢ T1 [d2, e2] ≫* T2. +#L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -H T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 Hd21 Hde12 /2/ +] +qed. + +lemma tpss_weak_top: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 [d, |L| - d] ≫* T2. +#L #T1 #T2 #d #e #H @(tpss_ind … H) -H T2 +[ // +| #T #T2 #_ #HT12 #IHT + lapply (tps_weak_top … HT12) -HT12 /2/ +] +qed. + +lemma tpss_weak_all: ∀L,T1,T2,d,e. + L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 [0, |L|] ≫* T2. +#L #T1 #T2 #d #e #HT12 +lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12 +lapply (tpss_weak_top … HT12) // +qed. + +(* Basic inversion lemmas ***************************************************) + +(* Note: this can be derived from tpss_inv_atom1 *) +lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫* T2 → T2 = ⋆k. +#L #T2 #k #d #e #H @(tpss_ind … H) -H T2 +[ // +| #T #T2 #_ #HT2 #IHT destruct -T + >(tps_inv_sort1 … HT2) -HT2 // +] +qed. + +lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫* U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫* V2 & + L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫* T2 & + U2 = 𝕓{I} V2. T2. +#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -H U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -U; + elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H + lapply (tpss_leq_repl_dx … HT1 (L. 𝕓{I} V2) ?) -HT1 /3 width=5/ +] +qed. + +lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫* U2 → + ∃∃V2,T2. L ⊢ V1 [d, e] ≫* V2 & L ⊢ T1 [d, e] ≫* T2 & + U2 = 𝕗{I} V2. T2. +#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -H U2 +[ /2 width=5/ +| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -U; + elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/ +] +qed. + +lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫* T2 → T1 = T2. +#L #T1 #T2 #d #H @(tpss_ind … H) -H T2 +[ // +| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 // +] +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_lift.ma new file mode 100644 index 000000000..f28ff3896 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_lift.ma @@ -0,0 +1,131 @@ +(**************************************************************************) +(* ___ *) +(* ||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/tps_lift.ma". +include "Basic-2/unfold/tpss.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +(* Advanced properties ******************************************************) + +lemma tpss_subst: ∀L,K,V,U1,i,d,e. + d ≤ i → i < d + e → + ↓[0, i] L ≡ K. 𝕓{Abbr} V → K ⊢ V [0, d + e - i - 1] ≫* U1 → + ∀U2. ↑[0, i + 1] U1 ≡ U2 → L ⊢ #i [d, e] ≫* U2. +#L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -H U1 +[ /3/ +| #U #U1 #_ #HU1 #IHU #U2 #HU12 + elim (lift_total U 0 (i+1)) #U0 #HU0 + lapply (IHU … HU0) -IHU #H + lapply (drop_fwd_drop2 … HLK) -HLK #HLK + lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 HLK HU0 HU12 // normalize #HU02 + lapply (tps_weak … HU02 d e ? ?) -HU02 [ >arith_i2 // | /2/ | /2/ ] +] +qed. + +(* Advanced inverion lemmas *************************************************) + +lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫* T2 → + T2 = 𝕒{I} ∨ + ∃∃K,V1,V2,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V1 & + K ⊢ V1 [0, d + e - i - 1] ≫* V2 & + ↑[O, i + 1] V2 ≡ T2 & + I = LRef i. +#L #T2 #I #d #e #H @(tpss_ind … H) -H T2 +[ /2/ +| #T #T2 #_ #HT2 * + [ #H destruct -T; + elim (tps_inv_atom1 … HT2) -HT2 [ /2/ | * /3 width=10/ ] + | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI + lapply (drop_fwd_drop2 … HLK) #H + elim (tps_inv_lift1_up … HT2 … H … HVT ? ? ?) normalize -HT2 H HVT [2,3,4: /2/ ] #V2 (lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 HLK HTU HTU2 /2/ +] +qed. + +lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫* T2 → + ∀L,U1,d,e. d ≤ dt → ↓[d, e] L ≡ K → + ↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 → + L ⊢ U1 [dt + e, et] ≫* U2. +#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -H T2 +[ #U2 #H >(lift_mono … HTU1 … H) -H // +| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2 + elim (lift_total T d e) #U #HTU + lapply (IHT … HTU) -IHT #HU1 + lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 HLK HTU HTU2 /2/ +] +qed. + +lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 → + ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 → + dt + et ≤ d → + ∃∃T2. K ⊢ T1 [dt, et] ≫* T2 & ↑[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -H U2 +[ /2/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 HLK HTU /3/ +] +qed. + +lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫* U2 → + ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 → + d + e ≤ dt → + ∃∃T2. K ⊢ T1 [dt - e, et] ≫* T2 & ↑[d, e] T2 ≡ U2. +#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -H U2 +[ /2/ +| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU + elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 HLK HTU /3/ +] +qed. + +lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e. + L ⊢ U1 [d, e] ≫* U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2. +#L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -H U2 // +#U #U2 #_ #HU2 #IHU destruct -U1 +<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 HTU1 // +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_ltps.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_ltps.ma new file mode 100644 index 000000000..645d9d08d --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_ltps.ma @@ -0,0 +1,94 @@ +(**************************************************************************) +(* ___ *) +(* ||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/ltps_tps.ma". +include "Basic-2/unfold/tpss_tpss.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +(* Properties concerning parallel substitution on local environments ********) + +lemma ltps_tpss_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. + d1 + e1 ≤ d2 → L0 [d1, e1] ≫ L1 → + L0 ⊢ T2 [d2, e2] ≫* U2 → L1 ⊢ T2 [d2, e2] ≫* U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HL01 #H @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltps_tps_conf_ge … HU2 … HL01 ?) -HU2 HL01 /2/ +qed. + +lemma ltps_tpss_conf: ∀L0,L1,T2,U2,d1,e1,d2,e2. + L0 [d1, e1] ≫ L1 → L0 ⊢ T2 [d2, e2] ≫* U2 → + ∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L1 ⊢ U2 [d1, e1] ≫* T. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #HL01 #H @(tpss_ind … H) -U2 +[ /3/ +| #U #U2 #_ #HU2 * #T #HT2 #HUT + elim (ltps_tps_conf … HU2 … HL01) -HU2 HL01 #W #HUW #HU2W + elim (tpss_strip_eq … HUT … HUW) -U + /3 width=5 by ex2_1_intro, step, tpss_strap/ (**) (* just /3 width=5/ is too slow *) +] +qed. + +lemma ltps_tpss_trans_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2. + d1 + e1 ≤ d2 → L1 [d1, e1] ≫ L0 → + L0 ⊢ T2 [d2, e2] ≫* U2 → L1 ⊢ T2 [d2, e2] ≫* U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HL10 #H @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU +lapply (ltps_tps_trans_ge … HU2 … HL10 ?) -HU2 HL10 /2/ +qed. + +lemma ltps_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 → + L1 [d1, e1] ≫ L0 → L0 ⊢ T2 [d2, e2] ≫* U2 → + ∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L0 ⊢ T [d1, e1] ≫* U2. +#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2 +[ /3/ +| #U #U2 #_ #HU2 * #T #HT2 #HTU + elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2 + elim (ltps_tps_trans … HTU … HL10) -HTU HL10 #W #HTW #HWU + @(ex2_1_intro … W) /2/ (**) (* /3 width=5/ does not work as in ltps_tpss_conf *) +] +qed. + +fact ltps_tps_trans_eq_aux: ∀Y1,X2,L1,T2,U2,d,e. + L1 ⊢ T2 [d, e] ≫ U2 → ∀L0. L0 [d, e] ≫ L1 → + Y1 = L1 → X2 = T2 → L0 ⊢ T2 [d, e] ≫* U2. +#Y1 #X2 @(cw_wf_ind … Y1 X2) -Y1 X2 #Y1 #X2 #IH +#L1 #T2 #U2 #d #e * -L1 T2 U2 d e +[ // +| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L0 #HL10 #H1 #H2 destruct -Y1 X2; + lapply (drop_fwd_lw … HLK1) normalize #H1 + elim (ltps_drop_trans_be … HL10 … HLK1 ? ?) -HL10 HLK1 [2,3: /2/ ] #X #H #HLK0 + elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K0 #V0 #HK01 #HV01 #H destruct -X; + lapply (tps_fwd_tw … HV01) #H2 + lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 [ /2/ ] -H2 #H + lapply (IH … HV01 … HK01 ? ?) -IH HV01 HK01 [1,3: // |2,4: skip | /2/ | /3 width=6/ ] +| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct -Y1 X2; + lapply (tps_leq_repl_dx … HT12 (L. 𝕓{I} V1) ?) -HT12 /2/ #HT12 + lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3,5: // |2,4: skip ] #HV12 + lapply (IH … HT12 (L0. 𝕓{I} V1) ? ? ?) -IH HT12 [1,3,5: /2/ |2,4: skip | normalize // ] -HL0 #HT12 + lapply (tpss_leq_repl_dx … HT12 (L0. 𝕓{I} V2) ?) -HT12 /2/ +| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct -Y1 X2; + lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3,5: // |2,4: skip ] + lapply (IH … HT12 … HL0 ? ?) -IH HT12 [1,3,5: // |2,4: skip ] -HL0 /2/ +] +qed. + +lemma ltps_tps_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 [d, e] ≫ U2 → + ∀L0. L0 [d, e] ≫ L1 → L0 ⊢ T2 [d, e] ≫* U2. +/2 width=5/ qed. + +lemma ltps_tpss_trans_eq: ∀L0,L1,T2,U2,d,e. L0 [d, e] ≫ L1 → + L1 ⊢ T2 [d, e] ≫* U2 → L0 ⊢ T2 [d, e] ≫* U2. +#L0 #L1 #T2 #U2 #d #e #HL01 #H @(tpss_ind … H) -U2 // +#U #U2 #_ #HU2 #IHU @(tpss_trans_eq … IHU) /2/ +qed. diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_tpss.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_tpss.ma new file mode 100644 index 000000000..a84928453 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/tpss_tpss.ma @@ -0,0 +1,93 @@ +(**************************************************************************) +(* ___ *) +(* ||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/tps_tps.ma". +include "Basic-2/unfold/tpss_lift.ma". + +(* PARTIAL UNFOLD ON TERMS **************************************************) + +(* Advanced properties ******************************************************) + +lemma tpss_tps: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫* T2 → L ⊢ T1 [d, 1] ≫ T2. +#L #T1 #T2 #d #H @(tpss_ind … H) -H T2 // +#T #T2 #_ #HT2 #IHT1 +lapply (tps_trans_ge … IHT1 … HT2 ?) // +qed. + +lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 → + ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → + ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T2 [d1, e1] ≫* T. +/3/ qed. + +lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 → + ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 → + (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) → + ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫* T. +/3/ qed. + +lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 → + ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2. +/3/ qed. + +lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 → + ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 → + ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫ T2. +/3/ qed. + +lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 → + ∀i. d ≤ i → i ≤ d + e → + ∃∃T. L ⊢ T1 [d, i - d] ≫* T & L ⊢ T [i, d + e - i] ≫* T2. +#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -H T2 +[ /2/ +| #T #T2 #_ #HT12 * #T3 #HT13 #HT3 + elim (tps_split_up … HT12 … Hdi Hide) -HT12 Hide #T0 #HT0 #HT02 + elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: (commutative_plus p) (le_O_to_eq_O … H) -H // +| #b #IHb #c elim c -c // + #c #_ #a #Hcb + lapply (le_S_S_to_le … Hcb) -Hcb #Hcb + minus_plus @eq_f2 /2/ +qed. + +lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b. +#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2/ +qed. + +lemma arith_c1: ∀a,b,c1. a + c1 - (b + c1) = a - b. +// qed. + +lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b. +#x #a #b #c1 >plus_plus_comm_23 // +qed. + +lemma arith_d1: ∀a,b,c1. c1 ≤ b → a + c1 + (b - c1) = a + b. +/2/ qed. + +lemma arith_e2: ∀a,c1,c2. a ≤ c1 → c1 + c2 - (c1 - a + c2) = a. +/3/ qed. + +lemma arith_f1: ∀a,b,c1. a + b ≤ c1 → c1 - (c1 - a - b) = a + b. +#a #b #c1 #H >minus_plus minus_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l // +qed. diff --git a/matita/matita/contribs/lambda_delta/Ground_2/list.ma b/matita/matita/contribs/lambda_delta/Ground_2/list.ma new file mode 100644 index 000000000..1b64bacce --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/list.ma @@ -0,0 +1,34 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "Ground-2/arith.ma". +include "Ground-2/notation.ma". + +(* LISTS ********************************************************************) + +inductive list (A:Type[0]) : Type[0] := + | nil : list A + | cons: A -> list A -> list A. + +interpretation "nil (list)" 'Nil = (nil ?). + +interpretation "cons (list)" 'Cons hd tl = (cons ? hd tl). + +let rec append A (l1: list A) l2 on l1 ≝ + match l1 with + [ nil ⇒ l2 + | cons hd tl ⇒ hd :: append A tl l2 + ]. + +interpretation "append (list)" 'Append l1 l2 = (append ? l1 l2). diff --git a/matita/matita/contribs/lambda_delta/Ground_2/notation.ma b/matita/matita/contribs/lambda_delta/Ground_2/notation.ma new file mode 100644 index 000000000..dab6b4982 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/notation.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************) + +(* Lists ********************************************************************) + +notation "hvbox( hd break :: tl )" + right associative with precedence 47 + for @{'Cons $hd $tl}. + +notation "hvbox( l1 break @ l2)" + right associative with precedence 47 + for @{'Append $l1 $l2 }. diff --git a/matita/matita/contribs/lambda_delta/Ground_2/star.ma b/matita/matita/contribs/lambda_delta/Ground_2/star.ma new file mode 100644 index 000000000..baed9b78e --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/star.ma @@ -0,0 +1,109 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "basics/star.ma". +include "Ground-2/xoa_props.ma". + +(* PROPERTIES of RELATIONS **************************************************) + +definition confluent: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. + ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & R1 a2 a. + +definition transitive: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2. + ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & R1 a a2. + +lemma TC_strip1: ∀A,R1,R2. confluent A R1 R2 → + ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & TC … R1 a2 a. +#A #R1 #R2 #HR12 #a0 #a1 #H elim H -H a1 +[ #a1 #Ha01 #a2 #Ha02 + elim (HR12 … Ha01 … Ha02) -HR12 a0 /3/ +| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 + elim (IHa0 … Ha02) -IHa0 Ha02 a0 #a0 #Ha0 #Ha20 + elim (HR12 … Ha1 … Ha0) -HR12 a /4/ +] +qed. + +lemma TC_strip2: ∀A,R1,R2. confluent A R1 R2 → + ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 → + ∃∃a. TC … R2 a1 a & R1 a2 a. +#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2 +[ #a2 #Ha02 #a1 #Ha01 + elim (HR12 … Ha01 … Ha02) -HR12 a0 /3/ +| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01 + elim (IHa0 … Ha01) -IHa0 Ha01 a0 #a0 #Ha10 #Ha0 + elim (HR12 … Ha0 … Ha2) -HR12 a /4/ +] +qed. + +lemma TC_confluent: ∀A,R1,R2. + confluent A R1 R2 → confluent A (TC … R1) (TC … R2). +#A #R1 #R2 #HR12 #a0 #a1 #H elim H -H a1 +[ #a1 #Ha01 #a2 #Ha02 + elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 a0 /3/ +| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 + elim (IHa0 … Ha02) -IHa0 Ha02 a0 #a0 #Ha0 #Ha20 + elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 a /4/ +] +qed. + +lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2. + R a1 a → TC … R a a2 → TC … R a1 a2. +/3/ qed. + +lemma TC_strap1: ∀A,R1,R2. transitive A R1 R2 → + ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 → + ∃∃a. R2 a1 a & TC … R1 a a2. +#A #R1 #R2 #HR12 #a1 #a0 #H elim H -H a0 +[ #a0 #Ha10 #a2 #Ha02 + elim (HR12 … Ha10 … Ha02) -HR12 a0 /3/ +| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 + elim (HR12 … Ha0 … Ha02) -HR12 a0 #a0 #Ha0 #Ha02 + elim (IHa … Ha0) -a /4/ +] +qed. + +lemma TC_strap2: ∀A,R1,R2. transitive A R1 R2 → + ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 → + ∃∃a. TC … R2 a1 a & R1 a a2. +#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2 +[ #a2 #Ha02 #a1 #Ha10 + elim (HR12 … Ha10 … Ha02) -HR12 a0 /3/ +| #a #a2 #_ #Ha02 #IHa #a1 #Ha10 + elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0 + elim (HR12 … Ha0 … Ha02) -HR12 a /4/ +] +qed. + +lemma TC_transitive: ∀A,R1,R2. + transitive A R1 R2 → transitive A (TC … R1) (TC … R2). +#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0 +[ #a0 #Ha10 #a2 #Ha02 + elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 a0 /3/ +| #a #a0 #_ #Ha0 #IHa #a2 #Ha02 + elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 a0 #a0 #Ha0 #Ha02 + elim (IHa … Ha0) -a /4/ +] +qed. + +lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R). +/2/ qed. + +lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:A→Prop. + P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) → + ∀a2. TC … R a1 a2 → P a2. +#A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -Ha12 a2 /3/ +qed. diff --git a/matita/matita/contribs/lambda_delta/Ground_2/xoa.conf.xml b/matita/matita/contribs/lambda_delta/Ground_2/xoa.conf.xml new file mode 100644 index 000000000..f749fe18a --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/xoa.conf.xml @@ -0,0 +1,36 @@ + + +
+ $(MATITA_RT_BASE_DIR) + +
+
+ contribs/lambda-delta/Ground-2 + xoa + xoa_notation + basics/pts.ma + 2 1 + 2 2 + 3 1 + 3 2 + 3 3 + 4 2 + 4 3 + 4 4 + 5 3 + 5 4 + 6 4 + 6 6 + 7 6 + 3 + 4 + +
+ diff --git a/matita/matita/contribs/lambda_delta/Ground_2/xoa.ma b/matita/matita/contribs/lambda_delta/Ground_2/xoa.ma new file mode 100644 index 000000000..2b7af4503 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/xoa.ma @@ -0,0 +1,143 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* This file was generated by xoa.native: do not edit *********************) + +include "basics/pts.ma". + +(* multiple existental quantifier (2, 1) *) + +inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝ + | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ? +. + +interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1). + +(* multiple existental quantifier (2, 2) *) + +inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝ + | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ? +. + +interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1). + +(* multiple existental quantifier (3, 1) *) + +inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝ + | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2). + +(* multiple existental quantifier (3, 2) *) + +inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝ + | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2). + +(* multiple existental quantifier (3, 3) *) + +inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝ + | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2). + +(* multiple existental quantifier (4, 2) *) + +inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝ + | ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 3) *) + +inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝ + | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (4, 4) *) + +inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝ + | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3). + +(* multiple existental quantifier (5, 3) *) + +inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝ + | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4). + +(* multiple existental quantifier (5, 4) *) + +inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝ + | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? 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 ≝ + | ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (6, 6) *) + +inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝ + | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5). + +(* multiple existental quantifier (7, 6) *) + +inductive ex7_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝ + | ex7_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → P6 x0 x1 x2 x3 x4 x5 → ex7_6 ? ? ? ? ? ? ? ? ? ? ? ? ? +. + +interpretation "multiple existental quantifier (7, 6)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6). + +(* multiple disjunction connective (3) *) + +inductive or3 (P0,P1,P2:Prop) : Prop ≝ + | or3_intro0: P0 → or3 ? ? ? + | or3_intro1: P1 → or3 ? ? ? + | or3_intro2: P2 → or3 ? ? ? +. + +interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2). + +(* multiple disjunction connective (4) *) + +inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝ + | or4_intro0: P0 → or4 ? ? ? ? + | or4_intro1: P1 → or4 ? ? ? ? + | or4_intro2: P2 → or4 ? ? ? ? + | or4_intro3: P3 → or4 ? ? ? ? +. + +interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3). + diff --git a/matita/matita/contribs/lambda_delta/Ground_2/xoa_notation.ma b/matita/matita/contribs/lambda_delta/Ground_2/xoa_notation.ma new file mode 100644 index 000000000..47e3268b4 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/xoa_notation.ma @@ -0,0 +1,158 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* This file was generated by xoa.native: do not edit *********************) + +(* multiple existental quantifier (2, 1) *) + +notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }. + +notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }. + +(* multiple existental quantifier (2, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }. + +(* multiple existental quantifier (3, 1) *) + +notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }. + +notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }. + +(* multiple existental quantifier (3, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }. + +(* multiple existental quantifier (3, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }. + +(* multiple existental quantifier (4, 2) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }. + +(* multiple existental quantifier (4, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }. + +(* multiple existental quantifier (4, 4) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)" + non associative with precedence 20 + for @{ 'Ex (λ${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.$P3) }. + +(* multiple existental quantifier (5, 3) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }. + +(* multiple existental quantifier (5, 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)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)" + non associative with precedence 20 + for @{ 'Ex (λ${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 (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)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P5) }. + +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)" + 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.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }. + +(* multiple existental quantifier (6, 6) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)" + 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.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }. + +(* multiple existental quantifier (7, 6) *) + +notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P6) }. + +notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)" + non associative with precedence 20 + for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P6) }. + +(* multiple disjunction connective (3) *) + +notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)" + non associative with precedence 30 + for @{ 'Or $P0 $P1 $P2 }. + +(* multiple disjunction connective (4) *) + +notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)" + non associative with precedence 30 + for @{ 'Or $P0 $P1 $P2 $P3 }. + diff --git a/matita/matita/contribs/lambda_delta/Ground_2/xoa_props.ma b/matita/matita/contribs/lambda_delta/Ground_2/xoa_props.ma new file mode 100644 index 000000000..f1ed781c3 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Ground_2/xoa_props.ma @@ -0,0 +1,20 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "Ground-2/xoa_notation.ma". +include "Ground-2/xoa.ma". + +lemma ex2_1_comm: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0. +#A0 #P0 #P1 * /2/ +qed. diff --git a/matita/matita/contribs/lambda_delta/Makefile b/matita/matita/contribs/lambda_delta/Makefile new file mode 100644 index 000000000..77f49615d --- /dev/null +++ b/matita/matita/contribs/lambda_delta/Makefile @@ -0,0 +1,60 @@ +H = @ +XOA_DIR = ../../../components/binaries/xoa +XOA = xoa.native + +CONF = Ground-2/xoa.conf.xml +TARGETS = Ground-2/xoa_natation.ma Ground-2/xoa.ma + +PACKAGES = Ground-2 Basic-2 + +all: + +# xoa ######################################################################## + +xoa: $(TARGETS) + +$(TARGETS): $(CONF) + @echo " EXEC $(XOA) $(CONF)" + $(H)MATITA_RT_BASE_DIR=../.. $(XOA_DIR)/$(XOA) $(CONF) + +# stats ###################################################################### + +stats: $(PACKAGES:%=%.stats) + +%.stats: MAS = $(shell find $* -name "*.ma") + +%.stats: CHARS = $(shell cat $(MAS) | wc -c) + +%.stats: + @printf '\e[1;40;37m' + @printf '%-15s %-42s' 'Statistics for:' $* + @printf '\e[0m\n' + @printf '\e[1;40;35m' + @printf '%-8s %6i' Chars $(CHARS) + @printf ' %-8s %5i' Lines `cat $(MAS) | wc -l` + @printf ' %-6s %3i' Pages `echo $$(($(CHARS) / 5120))` + @printf ' %-10s' '' + @printf '\e[0m\n' + @printf '\e[1;40;36m' + @printf '%-8s %6i' Sources `ls $(MAS) | wc -l` + @printf ' %-40s' '' +# @printf ' %-8s %5i' Objs `ls *.vo | wc -l` +# @printf ' %-6s %3i' Files `ls *.v | wc -l` + @printf '\e[0m\n' + @printf '\e[1;40;32m' + @printf '%-8s %6i' Theorems `grep theorem $(MAS) | wc -l` + @printf ' %-8s %5i' Lemmas `grep lemma $(MAS) | wc -l` + @printf ' %-6s %3i' Facts `grep fact $(MAS) | wc -l` + @printf ' %-6s %3i' Proofs `grep qed $(MAS) | wc -l` + @printf '\e[0m\n' + @printf '\e[1;40;33m' + @printf '%-8s %6i' Defs `grep "definition\|let rec\|inductive\|record" $(MAS) | wc -l` + @printf ' %-40s' '' +# @printf ' %-8s %5i' Local `grep "Local" *.v | wc -l` + @printf '\e[0m\n' + @printf '\e[1;40;31m' + @printf '%-8s %6i' Axioms `grep axiom $(MAS) | wc -l` + @printf ' %-8s %5i' Comments `grep "(\*[^*:]*$$" $(MAS) | wc -l` + @printf ' %-6s %3i' Marks `grep "(\*\*)" $(MAS) | wc -l` + @printf ' %-10s' '' + @printf '\e[0m\n' diff --git a/matita/matita/contribs/lambda_delta/replace.sh b/matita/matita/contribs/lambda_delta/replace.sh new file mode 100644 index 000000000..6caaacdb8 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/replace.sh @@ -0,0 +1,8 @@ +#!/bin/sh +for V in `cat Make`; do + echo ${V}; sed "s/$1/$2/g" ${V} > ${V}.new + if diff ${V} ${V}.new > /dev/null; + then rm -f ${V}.new; else mv -f ${V}.new ${V}; fi +done + +unset V diff --git a/matita/matita/contribs/lambda_delta/root b/matita/matita/contribs/lambda_delta/root new file mode 100644 index 000000000..c41bf7380 --- /dev/null +++ b/matita/matita/contribs/lambda_delta/root @@ -0,0 +1 @@ +baseuri=cic:/matita/lambda_delta/