+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tri.ma".
-include "Basic_2/substitution/lift.ma".
-include "Apps_2/functional/notation.ma".
-
-(* RELOCATION ***************************************************************)
-
-let rec flift d e U on U ≝ match U with
-[ TAtom I ⇒ match I with
- [ Sort _ ⇒ U
- | LRef i ⇒ #(tri … i d i (i + e) (i + e))
- | GRef _ ⇒ U
- ]
-| TPair I V T ⇒ match I with
- [ Bind2 I ⇒ ⓑ{I} (flift d e V). (flift (d+1) e T)
- | Flat2 I ⇒ ⓕ{I} (flift d e V). (flift d e T)
- ]
-].
-
-interpretation "functional relocation" 'Lift d e T = (flift d e T).
-
-(* Main properties **********************************************************)
-
-theorem flift_lift: ∀T,d,e. ⇧[d, e] T ≡ ↑[d, e] T.
-#T elim T -T
-[ * #i #d #e //
- elim (lt_or_eq_or_gt i d) #Hid normalize
- [ >(tri_lt ?????? Hid) /2 width=1/
- | /2 width=1/
- | >(tri_gt ?????? Hid) /3 width=2/
- ]
-| * /2/
-]
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem flift_inv_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ↑[d, e] T1 = T2.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
-[ #i #d #e #Hid >(tri_lt ?????? Hid) //
-| #i #d #e #Hid
- elim (le_to_or_lt_eq … Hid) -Hid #Hid
- [ >(tri_gt ?????? Hid) //
- | destruct //
- ]
-]
-qed-.
-
-(* Derived properties *******************************************************)
-
-lemma flift_join: ∀e1,e2,T. ⇧[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T.
-#e1 #e2 #T
-lapply (flift_lift T 0 (e1+e2)) #H
-elim (lift_split … H e1 e1 ? ? ?) -H // #U #H
->(flift_inv_lift … H) -H //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "functional" COMPONENT ********************************)
-
-notation "hvbox( ↑ [ d , break e ] break T )"
- non associative with precedence 55
- for @{ 'Lift $d $e $T }.
-
-notation "hvbox( [ d ← break V ] break T )"
- non associative with precedence 55
- for @{ 'Subst $V $d $T }.
-
-notation "hvbox( T1 ⇨ break T2 )"
- non associative with precedence 45
- for @{ 'SRed $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/grammar/genv.ma".
-
-(* REDUCTION AND TYPE MACHINE ***********************************************)
-
-(* machine local environment *)
-inductive xenv: Type[0] ≝
-| XAtom: xenv (* empty *)
-| XQuad: xenv → bind2 → nat → xenv → term → xenv (* entry *)
-.
-
-interpretation "atom (ext. local environment)"
- 'Star = XAtom.
-
-interpretation "environment construction (quad)"
- 'DxItem4 L I u K V = (XQuad L I u K V).
-
-(* machine stack *)
-definition stack: Type[0] ≝ list2 xenv term.
-
-(* machine status *)
-record rtm: Type[0] ≝
-{ rg: genv; (* global environment *)
- ru: nat; (* current de Bruijn's level *)
- re: xenv; (* extended local environment *)
- rs: stack; (* application stack *)
- rt: term (* code *)
-}.
-
-(* initial state *)
-definition rtm_i: genv → term → rtm ≝
- λG,T. mk_rtm G 0 (⋆) (⟠) T.
-
-(* update code *)
-definition rtm_t: rtm → term → rtm ≝
- λM,T. match M with
- [ mk_rtm G u E _ _ ⇒ mk_rtm G u E (⟠) T
- ].
-
-(* update closure *)
-definition rtm_u: rtm → xenv → term → rtm ≝
- λM,E,T. match M with
- [ mk_rtm G u _ _ _ ⇒ mk_rtm G u E (⟠) T
- ].
-
-(* get global environment *)
-definition rtm_g: rtm → genv ≝
- λM. match M with
- [ mk_rtm G _ _ _ _ ⇒ G
- ].
-
-(* get local reference level *)
-definition rtm_l: rtm → nat ≝
- λM. match M with
- [ mk_rtm _ u E _ _ ⇒ match E with
- [ XAtom ⇒ u
- | XQuad _ _ u _ _ ⇒ u
- ]
- ].
-
-(* get stack *)
-definition rtm_s: rtm → stack ≝
- λM. match M with
- [ mk_rtm _ _ _ S _ ⇒ S
- ].
-
-(* get code *)
-definition rtm_c: rtm → term ≝
- λM. match M with
- [ mk_rtm _ _ _ _ T ⇒ T
- ].
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "Apps_2/functional/rtm.ma".
-
-(* REDUCTION AND TYPE MACHINE ***********************************************)
-
-(* transitions *)
-inductive rtm_step: relation rtm ≝
-| rtm_ldrop : ∀G,u,E,I,t,F,V,S,i.
- rtm_step (mk_rtm G u (E. ④{I} {t, F, V}) S (#(i + 1)))
- (mk_rtm G u E S (#i))
-| rtm_ldelta: ∀G,u,E,t,F,V,S.
- rtm_step (mk_rtm G u (E. ④{Abbr} {t, F, V}) S (#0))
- (mk_rtm G u F S V)
-| rtm_ltype : ∀G,u,E,t,F,V,S.
- rtm_step (mk_rtm G u (E. ④{Abst} {t, F, V}) S (#0))
- (mk_rtm G u F S V)
-| rtm_gdrop : ∀G,I,V,u,E,S,p. p < |G| →
- rtm_step (mk_rtm (G. ⓑ{I} V) u E S (§p))
- (mk_rtm G u E S (§p))
-| rtm_gdelta: ∀G,V,u,E,S,p. p = |G| →
- rtm_step (mk_rtm (G. ⓓV) u E S (§p))
- (mk_rtm G u E S V)
-| rtm_gtype : ∀G,V,u,E,S,p. p = |G| →
- rtm_step (mk_rtm (G. ⓛV) u E S (§p))
- (mk_rtm G u E S V)
-| rtm_tau : ∀G,u,E,S,W,T.
- rtm_step (mk_rtm G u E S (ⓣW. T))
- (mk_rtm G u E S T)
-| rtm_appl : ∀G,u,E,S,V,T.
- rtm_step (mk_rtm G u E S (ⓐV. T))
- (mk_rtm G u E ({E, V} :: S) T)
-| rtm_beta : ∀G,u,E,F,V,S,W,T.
- rtm_step (mk_rtm G u E ({F, V} :: S) (ⓛW. T))
- (mk_rtm G u (E. ④{Abbr} {u, F, V}) S T)
-| rtm_push : ∀G,u,E,W,T.
- rtm_step (mk_rtm G u E ⟠ (ⓛW. T))
- (mk_rtm G (u + 1) (E. ④{Abst} {u, E, W}) ⟠ T)
-| rtm_theta : ∀G,u,E,S,V,T.
- rtm_step (mk_rtm G u E S (ⓓV. T))
- (mk_rtm G u (E. ④{Abbr} {u, E, V}) S T)
-.
-
-interpretation "sequential reduction (RTM)"
- 'SRed O1 O2 = (rtm_step O1 O2).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/delift_lift.ma".
-include "Apps_2/functional/lift.ma".
-
-(* CORE SUBSTITUTION ********************************************************)
-
-let rec fsubst W d U on U ≝ match U with
-[ TAtom I ⇒ match I with
- [ Sort _ ⇒ U
- | LRef i ⇒ tri … i d (#i) (↑[0, i] W) (#(i-1))
- | GRef _ ⇒ U
- ]
-| TPair I V T ⇒ match I with
- [ Bind2 I ⇒ ⓑ{I} (fsubst W d V). (fsubst W (d+1) T)
- | Flat2 I ⇒ ⓕ{I} (fsubst W d V). (fsubst W d T)
- ]
-].
-
-interpretation "functional core substitution" 'Subst V d T = (fsubst V d T).
-
-(* Main properties **********************************************************)
-
-theorem fsubst_delift: ∀K,V,T,L,d.
- ⇩[0, d] L ≡ K. ⓓV → L ⊢ T [d, 1] ≡ [d ← V] T.
-#K #V #T elim T -T
-[ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/
- elim (lt_or_eq_or_gt i d) #Hid
- [ -HLK >(tri_lt ?????? Hid) /3 width=3/
- | destruct >tri_eq /4 width=4 by tpss_strap, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *)
- | -HLK >(tri_gt ?????? Hid) /3 width=3/
- ]
-| * /3 width=1/ /4 width=1/
-]
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem fsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV →
- L ⊢ T1 [d, 1] ≡ T2 → [d ← V] T1 = T2.
-#K #V #T1 elim T1 -T1
-[ * #i #L #T2 #d #HLK #H
- [ -HLK >(delift_fwd_sort1 … H) -H //
- | elim (lt_or_eq_or_gt i d) #Hid normalize
- [ -HLK >(delift_fwd_lref1_lt … H) -H // /2 width=1/
- | destruct
- elim (delift_fwd_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0
- lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus <minus_n_n #HV2 #HVT2 destruct
- >(delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 //
- | -HLK >(delift_fwd_lref1_ge … H) -H // /2 width=1/
- ]
- | -HLK >(delift_fwd_gref1 … H) -H //
- ]
-| * #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H
- [ elim (delift_fwd_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/
- | elim (delift_fwd_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 //
- ]
-]
-qed-.
+++ /dev/null
-# 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/csuba csubc_csuba
-csubc/drop1 drop1_csubc_trans
-csubc/drop drop_csubc_trans
-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/props drop1_cons_tail
-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_cons_tail
-
-# check #############################################################
-
-lift1/fwd lifts1_flat
-lift1/fwd lifts1_nil
-lift1/fwd lifts1_cons
-lift1/props lift1_free
-lift/props thead_x_lift_y_y
-lift/props lifts_tapp
-
-# waiting ####################################################################
-
-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/pr3 nf2_pr3_unfold
-nf2/pr3 nf2_pr3_confluence
-
-nf2/props nfs2_tapp
-nf2/props nf2_appls_lref
-
-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 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_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
-
-# check #############################################################
-
-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
-pr2/fwd pr2_gen_void
-pr2/props pr2_ctail
-pr2/subst1 pr2_gen_cabbr
-
-pr3/fwd pr3_gen_abst
-pr3/fwd pr3_gen_lref
-pr3/fwd pr3_gen_void
-pr3/fwd pr3_gen_appl
-pr3/fwd pr3_gen_bind
-pr3/iso pr3_iso_appls_abbr
-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/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_flat
-pr3/props pr3_eta
-pr3/subst1 pr3_subst1
-pr3/subst1 pr3_gen_cabbr
-sn3/props sn3_cpr3_trans
-
-sn3/props sn3_shift
-sn3/props sn3_change
-sn3/props sn3_gen_def
-sn3/props sn3_cdelta
-sn3/props sn3_appl_lref
-sn3/props sn3_appl_abbr
-sn3/props sn3_appl_cast
-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_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
-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 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
-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
-
-# check ######################################################################
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrops.ma".
-
-(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
-
-definition CP1 ≝ λRR:lenv→relation term. λRS:relation term.
- ∀L,k. NF … (RR L) RS (⋆k).
-
-definition CP2 ≝ λRR:lenv→relation term. λRS:relation term.
- ∀L,K,W,i. ⇩[0,i] L ≡ K. ⓛW → NF … (RR L) RS (#i).
-
-definition CP3 ≝ λRR:lenv→relation term. λRP:lenv→predicate term.
- ∀L,V,k. RP L (ⓐ⋆k.V) → RP L V.
-
-definition CP4 ≝ λRR:lenv→relation term. λRS:relation term.
- ∀L0,L,T,T0,d,e. NF … (RR L) RS T →
- ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → NF … (RR L0) RS T0.
-
-definition CP4s ≝ λRR:lenv→relation term. λRS:relation term.
- ∀L0,L,des. ⇩*[des] L0 ≡ L →
- ∀T,T0. ⇧*[des] T ≡ T0 →
- NF … (RR L) RS T → NF … (RR L0) RS T0.
-
-(* requirements for abstract computation properties *)
-record acp (RR:lenv->relation term) (RS:relation term) (RP:lenv→predicate term) : Prop ≝
-{ cp1: CP1 RR RS;
- cp2: CP2 RR RS;
- cp3: CP3 RR RP;
- cp4: CP4 RR RS
-}.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: nf2_lift1 *)
-lemma acp_lifts: ∀RR,RS. CP4 RR RS → CP4s RR RS.
-#RR #RS #HRR #L1 #L2 #des #H elim H -L1 -L2 -des
-[ #L #T1 #T2 #H #HT1
- <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=9/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/lifts_lifts.ma".
-include "Basic_2/unfold/ldrops_ldrops.ma".
-include "Basic_2/static/aaa_lifts.ma".
-include "Basic_2/static/aaa_aaa.ma".
-include "Basic_2/computation/lsubc_ldrops.ma".
-
-(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
-
-(* Main propertis ***********************************************************)
-
-(* Basic_1: was: sc3_arity_csubc *)
-theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP.
- acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,T,A. L1 ⊢ T ÷ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
- ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 [RP] ⊑ L0 →
- ⦃L2, T0⦄ [RP] ϵ 〚A〛.
-#RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A
-[ #L #k #L0 #des #HL0 #X #H #L2 #HL20
- >(lifts_inv_sort1 … H) -H
- lapply (aacr_acr … H1RP H2RP ⓪) #HAtom
- @(s2 … HAtom … ◊) // /2 width=2/
-| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20
- lapply (aacr_acr … H1RP H2RP B) #HB
- elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b
- elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1
- >(at_mono … Hi1 … Hi0) -i1
- elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct
- elim (lsubc_ldrop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H
- elim (lsubc_inv_pair2 … H) -H *
- [ #K2 #HK20 #H destruct
- generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2
- [ elim (lift_total V0 0 (i0 +1)) #V #HV0
- elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
- @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *)
- | @(s2 … HB … ◊) // /2 width=3/
- ]
- | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
- #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b
- lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
- >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B
- elim (lift_total V2 0 (i0 +1)) #V #HV2
- @(s4 … HB … ◊ … HV2 HLK2)
- @(s7 … HB … HKV2B) //
- ]
-| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
- elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
- lapply (aacr_acr … H1RP H2RP A) #HA
- lapply (aacr_acr … H1RP H2RP B) #HB
- lapply (s1 … HB) -HB #HB
- @(s5 … HA … ◊ ◊) // /3 width=5/
-| #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
- elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
- @(aacr_abst … H1RP H2RP)
- [ lapply (aacr_acr … H1RP H2RP B) #HB
- @(s1 … HB) /2 width=5/
- | -IHB
- #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
- elim (lifts_total des3 W0) #W2 #HW02
- elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
- lapply (aaa_lifts … L2 W2 … (des @ des3) … HLWB) -HLWB /2 width=3/ #HLW2B
- @(IHA (L2. ⓛW2) … (des + 1 @ des3 + 1)) -IHA
- /2 width=3/ /3 width=5/
- ]
-| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
- elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
- /3 width=10/
-| #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
- elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
- lapply (aacr_acr … H1RP H2RP A) #HA
- lapply (s1 … HA) #H
- @(s6 … HA … ◊) /2 width=5/ /3 width=5/
-]
-qed.
-
-(* Basic_1: was: sc3_arity *)
-lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L,T,A. L ⊢ T ÷ A → ⦃L, T⦄ [RP] ϵ 〚A〛.
-/2 width=8/ qed.
-
-lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L,T,A. L ⊢ T ÷ A → RP L T.
-#RR #RS #RP #H1RP #H2RP #L #T #A #HT
-lapply (aacr_acr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=4/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/aarity.ma".
-include "Basic_2/unfold/gr2_gr2.ma".
-include "Basic_2/unfold/lifts_lift_vector.ma".
-include "Basic_2/unfold/ldrops_ldrop.ma".
-include "Basic_2/computation/acp.ma".
-
-(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
-
-(* Note: this is Girard's CR1 *)
-definition S1 ≝ λRP,C:lenv→predicate term.
- ∀L,T. C L T → RP L T.
-
-(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term.
- ∀L,Vs. all … (RP L) Vs →
- ∀T. 𝐒[T] → NF … (RR L) RS T → C L (ⒶVs.T).
-
-(* Note: this is Tait's ii *)
-definition S3 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs,V,T,W. C L (ⒶVs. ⓓV. T) → RP L W → C L (ⒶVs. ⓐV. ⓛW. T).
-
-definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i.
- C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 →
- ⇩[0, i] L ≡ K. ⓓV1 → C L (Ⓐ Vs. #i).
-
-definition S5 ≝ λRP,C:lenv→predicate term.
- ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓV. T).
-
-definition S6 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓣW. T).
-
-definition S7 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e.
- C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2.
-
-definition S7s ≝ λC:lenv→predicate term.
- ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
-
-(* properties of the abstract candidate of reducibility *)
-record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝
-{ s1: S1 RP C;
- s2: S2 RR RS RP C;
- s3: S3 RP C;
- s4: S4 RP C;
- s5: S5 RP C;
- s6: S6 RP C;
- s7: S7 C
-}.
-
-(* the abstract candidate of reducibility associated to an atomic arity *)
-let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
-λT. match A with
-[ AAtom ⇒ RP L T
-| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
- aacr RP A L0 (ⓐV0. T0)
-].
-
-interpretation
- "candidate of reducibility of an atomic arity (abstract)"
- 'InEInt RP L T A = (aacr RP A L T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: sc3_lift1 *)
-lemma acr_lifts: ∀C. S7 C → S7s C.
-#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
-[ #L #T1 #T2 #H #HT1
- <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=9/
-]
-qed.
-
-lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
- ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 →
- RP L V → RP L0 V0.
-#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV
-@acr_lifts /width=6/
-@(s7 … HRP)
-qed.
-
-(* Basic_1: was only: sns3_lifts1 *)
-lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
- ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[des] L0 ≡ L →
- all … (RP L) Vs → all … (RP L0) V0s.
-#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
-#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s
-@conj /2 width=1/ /2 width=6 by rp_lifts/
-qed.
-
-(* Basic_1: was:
- sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
-*)
-lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀A. acr RR RS RP (aacr RP A).
-#RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
-#B #A #IHB #IHA @mk_acr normalize
-[ #L #T #H
- lapply (H ? (⋆0) ? ⟠ ? ? ?) -H
- [1,3: // |2,4: skip
- | @(s2 … IHB … ◊) // /2 width=2/
- | #H @(cp3 … H1RP … 0) @(s1 … IHA) //
- ]
-| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
- elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
- lapply (s1 … IHB … HB) #HV0
- @(s2 … IHA … (V0 :: V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/
-| #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
- elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(s3 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /4 width=5/
-| #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
- elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
- >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
- elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
- elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
- elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
- >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(s4 … IHA … (V0 :: V0s) … HW12 HL02) /3 width=4/
-| #L #V1s #V2s #HV12s #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
- elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
- elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
- elim (lift_total V10 0 1) #V20 #HV120
- elim (liftv_total 0 1 V10s) #V20s #HV120s
- @(s5 … IHA … (V10 :: V10s) (V20 :: V20s)) /2 width=1/ /2 width=6 by rp_lifts/
- @(HA … (des + 1)) /2 width=1/
- [ @(s7 … IHB … HB … HV120) /2 width=1/
- | @lifts_applv //
- elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
- >(liftv_mono … HV12s … HV10s) -V1s //
- ]
-| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
- elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
- @(s6 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /3 width=4/
-| /3 width=7/
-]
-qed.
-
-lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L,W,T,A,B. RP L W → (
- ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 →
- ⦃L0, V0⦄ [RP] ϵ 〚B〛 → ⦃L0. ⓓV0, T0⦄ [RP] ϵ 〚A〛
- ) →
- ⦃L, ⓛW. T⦄ [RP] ϵ 〚②B. A〛.
-#RR #RS #RP #H1RP #H2RP #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
-lapply (aacr_acr … H1RP H2RP A) #HCA
-lapply (aacr_acr … H1RP H2RP B) #HCB
-elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
-lapply (s1 … HCB) -HCB #HCB
-@(s3 … HCA … ◊) /2 width=6 by rp_lifts/
-@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
-qed.
-
-(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
-(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Basic_1: includes: pr3_pr2 *)
-definition cprs: lenv → relation term ≝
- λL. TC … (cpr L).
-
-interpretation "context-sensitive parallel computation (term)"
- 'PRedStar L T1 T2 = (cprs L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
- ∀T2. L ⊢ T1 ➡* T2 → R T2.
-#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-axiom cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) →
- ∀T1. L ⊢ T1 ➡* T2 → R T1.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
-/2 width=1/ qed.
-
-lemma cprs_strap1: ∀L,T1,T,T2.
- L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
-
-(* Basic_1: was: pr3_step *)
-lemma cprs_strap2: ∀L,T1,T,T2.
- L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-lemma cprs_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 →
- ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡* T2.
-/3 width=3/
-qed.
-
-lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 →
- L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
-#T #T2 #_ #HT2 #IHT2
-@(cprs_strap1 … IHT2) -IHT2 /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k.
-#L #U2 #k #H @(cprs_ind … H) -U2 //
-#U2 #U #_ #HU2 #IHU2 destruct
->(cpr_inv_sort1 … HU2) -HU2 //
-qed-.
-
-(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓣW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
- ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓣW2.T2.
-#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
-#U2 #U #_ #HU2 * /3 width=3/ *
-#W #T #HW1 #HT1 #H destruct
-elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
-#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
-qed-.
-
-(* Basic_1: removed theorems 2: clear_pr3_trans pr3_cflat *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr_lift.ma".
-include "Basic_2/reducibility/cpr_cpr.ma".
-include "Basic_2/reducibility/lcpr_cpr.ma".
-include "Basic_2/computation/cprs_lcpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 →
- L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
-#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1
-[ /3 width=5/
-| #T1 #T #HT1 #_ #IHT1
- @(cprs_strap2 … IHT1) -IHT1 /2 width=1/
-]
-qed.
-
-lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
- L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
-/3 width=1/ qed.
-
-lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 →
- L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
-#L #V1 #V2 #HV12 #T1 #T2 #HT12
-lapply (lcpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 →
- ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
- U2 = ⓐV2. T2
- | ∃∃V2,W,T. L ⊢ V1 ➡* V2 &
- L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ➡* U2
- | ∃∃V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
- L ⊢ T1 ➡* ⓓV. T & L ⊢ ⓓV. ⓐV2. T ➡* U2.
-#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
-#U #U2 #_ #HU2 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
- | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/
- | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
- @or3_intro2 @(ex4_4_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
- ]
-| /4 width=8/
-| /4 width=10/
-]
-qed-.
-
-(* Main propertis ***********************************************************)
-
-(* Basic_1: was: pr3_confluence *)
-theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 →
- ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0.
-/3 width=3/ qed.
-
-(* Basic_1: was: pr3_t *)
-theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
-
-lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 →
- L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
-#I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /2 width=1/
-qed.
-
-(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT2
-@(cprs_trans … IHT2) /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/ltpr_tps.ma".
-include "Basic_2/reducibility/cpr_ltpss.ma".
-include "Basic_2/reducibility/lcpr.ma".
-include "Basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties concerning context-sensitive parallel reduction on lenv's *****)
-
-lemma ltpr_tpss_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2,d,e. L2 ⊢ T1 [d, e] ▶* T2 →
- ∃∃T. L1 ⊢ T1 [d, e] ▶* T & L1 ⊢ T ➡* T2.
-#L1 #L2 #HL12 #T1 #T2 #d #e #H @(tpss_ind … H) -T2
-[ /2 width=3/
-| #T #T2 #_ #HT2 * #T0 #HT10 #HT0
- elim (ltpr_tps_trans … HT2 … HL12) -L2 #T3 #HT3 #HT32
- @(ex2_1_intro … HT10) -T1 (**) (* explicit constructors *)
- @(cprs_strap1 … T3 …) /2 width=1/ -HT32
- @(cprs_strap1 … HT0) -HT0 /3 width=3/
-]
-qed.
-
-(* Basic_1: was just: pr3_pr0_pr2_t *)
-lemma ltpr_cpr_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2.
-#L1 #L2 #HL12 #T1 #T2 * #T #HT1
-<(ltpr_fwd_length … HL12) #HT2
-elim (ltpr_tpss_trans … HL12 … HT2) -L2 /3 width=3/
-qed.
-
-(* Basic_1: was just: pr3_pr2_pr2_t *)
-lemma lcpr_cpr_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2.
-#L1 #L2 * /3 width=7/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/cprs_cprs.ma".
-include "Basic_2/computation/lcprs_lcprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Properties exploiting context-senstive computation on local environments *)
-
-(* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma lcprs_cprs_trans: ∀L1,L2. L1 ⊢ ➡* L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-#L1 #L2 #HL12 @(lcprs_ind … HL12) -L2 // /3 width=3/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡* U2 →
- (∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
- U2 = ⓓV2. T2
- ) ∨
- ∃∃U. ⇧[0, 1] U2 ≡ U & L. ⓓV1 ⊢ T1 ➡* U.
-#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_abbr1 … HU02) -HU02 *
- [ #V #V2 #T2 #HV0 #HV2 #HT02 #H destruct
- lapply (cpr_intro … HV0 … HV2) -HV2 #HV02
- lapply (ltpr_cpr_trans (L.ⓓV0) … HT02) /2 width=1/ -V #HT02
- lapply (lcprs_cprs_trans (L. ⓓV1) … HT02) -HT02 /2 width=1/ /4 width=5/
- | -V0 #T2 #HT20 #HTU2
- elim (lift_total U2 0 1) #U0 #HU20
- lapply (cpr_lift (L.ⓓV1) … HT20 … HU20 HTU2) -T2 /2 width=1/ /4 width=5/
- ]
-| #U1 #HU01 #HTU1
- elim (lift_total U2 0 1) #U #HU2
- lapply (cpr_lift (L.ⓓV1) … HU01 … HU2 HU02) -U0 /2 width=1/ /4 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr_lift.ma".
-include "Basic_2/computation/cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr3_lift *)
-lemma cprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
- ∀T2. K ⊢ T1 ➡* T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 ➡* U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #HT12 @(cprs_ind … HT12) -T2
-[ -HLK #T2 #HT12
- <(lift_mono … HTU1 … HT12) -T1 //
-| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (cpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/
-]
-qed.
-
-(* Basic_1: was: pr3_gen_lift *)
-lemma cprs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡* U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡* T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 #HU12 @(cprs_ind … HU12) -U2 /2 width=3/
--HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1
-elim (cpr_inv_lift … HLK … HTU … HU2) -U -HLK /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tstc.ma".
-(*
-include "Basic_2/reducibility/cpr_lift.ma".
-include "Basic_2/reducibility/lcpr_cpr.ma".
-*)
-include "Basic_2/computation/cprs_cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Forward lemmas involving same top term constructor ***********************)
-(*
-lemma cpr_fwd_beta: ∀L,V,W,T,U. L ⊢ ⓐV. ⓛW. T ➡ U →
- ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⓓV. T ➡* U.
-#L #V #W #T #U #H
-elim (cpr_inv_appl1 … H) -H *
-[ #V0 #X #_ #_ #H destruct /2 width=1/
-| #V0 #W0 #T1 #T2 #HV0 #HT12 #H1 #H2 destruct
- lapply (lcpr_cpr_trans (L. ⓓV) … HT12) -HT12 /2 width=1/ /3 width=1/
-| #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma cpr_fwd_theta: ∀L,V1,V,T,U. L ⊢ ⓐV1. ⓓV. T ➡ U →
- ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1. ⓓV. T ≃ U ∨
- L ⊢ ⓓV. ⓐV2. T ➡* U.
-#L #V1 #V #T #U #H #V2 #HV12
-elim (cpr_inv_appl1 … H) -H *
-[ -HV12 #V0 #X #_ #_ #H destruct /2 width=1/
-| -HV12 #V0 #W #T1 #T2 #_ #_ #H destruct
-| #V0 #V3 #W1 #W2 #T1 #T2 #HV10 #HW12 #HT12 #HV03 #H1 #H2 destruct
- lapply (cpr_lift (L.ⓓW1) … HV12 … HV03 … HV10) -V0 -HV12 /2 width=1/ #HV23
- lapply (lcpr_cpr_trans (L. ⓓW1) … HT12) -HT12 /2 width=1/ #HT12
- /4 width=1/
-]
-qed-.
-*)
-lemma cprs_fwd_tau: ∀L,W,T,U. L ⊢ ⓣW. T ➡* U →
- ⓣW. T ≃ U ∨ L ⊢ T ➡* U.
-#L #W #T #U #H
-elim (cprs_inv_cast1 … H) -H /2 width=1/ *
-#W0 #T0 #_ #_ #H destruct /2 width=1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tstc_vector.ma".
-include "Basic_2/substitution/lift_vector.ma".
-include "Basic_2/computation/cprs_tstc.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
-
-(* Vector form of forward lemmas involving same top term constructor ********)
-(*
-lemma cpr_fwd_beta_vector: ∀L,V,W,T,U,Vs. L ⊢ ⒶVs. ⓐV. ⓛW. T ➡ U →
- ⒶVs. ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⒶVs. ⓓV. T ➡* U.
-#L #V #W #T #U * /2 width=1 by cpr_fwd_beta/
-#V0 #Vs #H
-elim (cpr_inv_appl1_simple … H ?) -H
-[ #V1 #T1 #_ #_ #H destruct /2 width=1/
-| elim Vs -Vs //
-]
-qed-.
-
-lemma cpr_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡ U →
- ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
-#L #V1s #V2s * -V1s -V2s /3 width=1/
-#V1s #V2s #V1a #V2a #HV12a * -V1s -V2s /2 width=1 by cpr_fwd_theta/ -HV12a
-#V1s #V2s #V1b #V2b #_ #_ #V #U #T #H
-elim (cpr_inv_appl1_simple … H ?) -H //
-#V0 #T0 #_ #_ #H destruct /2 width=1/
-qed-.
-*)
-
-(* Basic_1: was: pr3_iso_appls_cast *)
-lemma cprs_fwd_tau_vector: ∀L,Vs,W,T,U. L ⊢ ⒶVs. ⓣW. T ➡* U →
- ⒶVs. ⓣW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U.
-#L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/
-#V #Vs #IHVs #W #T #U #H
-elim (cprs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
-| #V0 #W0 #T0 #HV0 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
- | @or_intror
- @(cprs_trans … HU) -HU
- @(cprs_strap1 … (ⓐV0.ⓛW0.T0)) /2 width=1/ -HV0 -HT0 /3 width=1/
- ]
-| #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
- | @or_intror
- @(cprs_trans … HU) -HU
- @(cprs_strap1 … (ⓐV0.ⓓV2.T0)) /2 width=1/ -HV0 -HT0 /3 width=3/
-]
-qed-.
-
-axiom cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡* U →
- ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr.ma".
-include "Basic_2/reducibility/cnf.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …).
-
-interpretation
- "context-sensitive strong normalization (term)"
- 'SN L T = (csn L T).
-
-(* Basic eliminators ********************************************************)
-
-lemma csn_ind: ∀L. ∀R:predicate term.
- (∀T1. L ⊢ ⬇* T1 →
- (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → R T2) →
- R T1
- ) →
- ∀T. L ⊢ ⬇* T → R T.
-#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: sn3_pr2_intro *)
-lemma csn_intro: ∀L,T1.
- (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇* T2) → L ⊢ ⬇* T1.
-#L #T1 #H
-@(SN_intro … H)
-qed.
-
-(* Basic_1: was: sn3_nf2 *)
-lemma csn_cnf: ∀L,T. L ⊢ 𝐍[T] → L ⊢ ⬇* T.
-/2 width=1/ qed.
-
-lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬇* T2.
-#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csn_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
-qed.
-
-(* Basic_1: was: sn3_cast *)
-lemma csn_cast: ∀L,W. L ⊢ ⬇* W → ∀T. L ⊢ ⬇* T → L ⊢ ⬇* ⓣW. T.
-#L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpr_inv_cast1 … H1) -H1
-[ * #W0 #T0 #HLW0 #HLT0 #H destruct
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ /3 width=3/
- | -HLW0 * #H destruct /3 width=1/
- ]
-| /3 width=3/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬇* T.
-#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
-@csn_intro #T2 #HLT2 #HT2
-@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
-qed.
-
-(* Basic_1: was: sn3_gen_flat *)
-lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
-/2 width=5/ qed-.
-
-(* Basic_1: removed theorems 3: sn3_gen_cflat sn3_cflat sn3_bind *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/acp_aaa.ma".
-include "Basic_2/computation/csn_lcpr_vector.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-(* Properties concerning atomic arity assignment ****************************)
-
-lemma csn_aaa: ∀L,T,A. L ⊢ T ÷ A → L ⊢ ⬇* T.
-#L #T #A #H
-@(acp_aaa … csn_acp csn_acr … H)
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr_cpr.ma".
-include "Basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-(* Advanced forvard lemmas **************************************************)
-
-fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬇* V.
-#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
-@csn_intro #V2 #HLV2 #HV2
-@(IH (②{I} V2. T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
-qed.
-
-(* Basic_1: was: sn3_gen_head *)
-lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬇* ②{I} V. T → L ⊢ ⬇* V.
-/2 width=5/ qed.
-
-fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬇* U →
- ∀I,V,T. U = ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
-#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
-@csn_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
-qed.
-
-(* Basic_1: was: sn3_gen_bind *)
-lemma csn_fwd_bind_dx: ∀I,L,V,T. L ⊢ ⬇* ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
-/2 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/cprs.ma".
-include "Basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-(* Properties concerning context-sensitive computation on terms *************)
-
-definition csns: lenv → predicate term ≝ λL. SN … (cprs L) (eq …).
-
-interpretation
- "context-sensitive strong normalization (term)"
- 'SNStar L T = (csns L T).
-
-(* Basic eliminators ********************************************************)
-
-lemma csns_ind: ∀L. ∀R:predicate term.
- (∀T1. L ⊢ ⬇** T1 →
- (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
- ) →
- ∀T. L ⊢ ⬇** T → R T.
-#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: sn3_intro *)
-lemma csns_intro: ∀L,T1.
- (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
-#L #T1 #H
-@(SN_intro … H)
-qed.
-
-fact csns_intro_aux: ∀L,T1.
- (∀T,T2. L ⊢ T ➡* T2 → T1 = T → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
-/4 width=3/ qed-.
-
-(* Basic_1: was: sn3_pr3_trans (old version) *)
-lemma csns_cprs_trans: ∀L,T1. L ⊢ ⬇** T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇** T2.
-#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csns_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
-qed.
-
-(* Basic_1: was: sn3_pr2_intro (old version) *)
-lemma csns_intro_cpr: ∀L,T1.
- (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) →
- L ⊢ ⬇** T1.
-#L #T1 #H
-@csns_intro_aux #T #T2 #H @(cprs_ind_dx … H) -T
-[ -H #H destruct #H
- elim (H ?) //
-| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
- elim (term_eq_dec T0 T) #HT0
- [ -HLT1 -HLT2 -H /3 width=1/
- | -IHT -HT12 /4 width=3/
- ]
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem csn_csns: ∀L,T. L ⊢ ⬇* T → L ⊢ ⬇** T.
-#L #T #H @(csn_ind … H) -T /4 width=1/
-qed.
-
-theorem csns_csn: ∀L,T. L ⊢ ⬇** T → L ⊢ ⬇* T.
-#L #T #H @(csns_ind … H) -T /4 width=1/
-qed.
-
-(* Basic_1: was: sn3_pr3_trans *)
-lemma csn_cprs_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇* T2.
-/4 width=3/ qed.
-
-(* Basic_1: was: nf2_sn3 *)
-lemma csn_cwn: ∀L,T1. L ⊢ ⬇* T1 →
- ∃∃T2. L ⊢ T1 ➡* T2 & L ⊢ 𝐍[T2].
-#L #T1 #H elim H -T1 #T1 #_ #IHT1
-elim (cnf_dec L T1)
-[ -IHT1 /2 width=3/
-| * #T2 #HLT12 #HT12
- elim (IHT1 T2 ? ?) -IHT1 // /2 width=1/ -HT12 /3 width=3/
-]
-qed.
-
-(* Main eliminators *********************************************************)
-
-lemma csn_ind_cprs: ∀L. ∀R:predicate term.
- (∀T1. L ⊢ ⬇* T1 →
- (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
- ) →
- ∀T. L ⊢ ⬇* T → R T.
-#L #R #H0 #T1 #H @(csns_ind … (csn_csns … H)) -T1 #T1 #HT1 #IHT1
-@H0 -H0 /2 width=1/ -HT1 /3 width=1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tstc_tstc.ma".
-include "Basic_2/computation/cprs_cprs.ma".
-include "Basic_2/computation/csn_lift.ma".
-include "Basic_2/computation/csn_cpr.ma".
-include "Basic_2/computation/csn_cprs.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma csn_lcpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
-#L1 #L2 #HL12 #T #H @(csn_ind_cprs … H) -T #T #_ #IHT
-@csn_intro #T0 #HLT0 #HT0
-@IHT /2 width=2/ -IHT -HT0 /2 width=3/
-qed.
-
-lemma csn_abbr: ∀L,V. L ⊢ ⬇* V → ∀T. L. ⓓV ⊢ ⬇* T → L ⊢ ⬇* ⓓV. T.
-#L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_cprs … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpr_inv_abbr1 … H1) -H1 *
-[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
- lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1
- lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ #HV1 @IHV // /2 width=1/ -HV1
- @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
- | -IHV -HLV1 * #H destruct /3 width=1/
- ]
-| -IHV -IHT -H2 #T0 #HT0 #HLT0
- lapply (csn_inv_lift … HT … HT0) -HT /2 width=3/
-]
-qed.
-
-fact csn_appl_beta_aux: ∀L,W. L ⊢ ⬇* W → ∀U. L ⊢ ⬇* U →
- ∀V,T. U = ⓓV. T → L ⊢ ⬇* ⓐV. ⓛW. T.
-#L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V #T #H destruct
-lapply (csn_fwd_pair_sn … HVT) #HV
-lapply (csn_fwd_bind_dx … HVT) #HT -HVT
-@csn_intro #X #H #H2
-elim (cpr_inv_appl1 … H) -H *
-[ #V0 #Y #HLV0 #H #H0 destruct
- elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct
- elim (eq_false_inv_beta … H2) -H2
- [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0
- @csn_abbr /2 width=3/ -HV
- @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
- | -IHW -HLW0 -HV -HT * #H #HVT0 destruct
- @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/
- ]
-| -IHW -IHVT -H2 #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct
- lapply (lcpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01
- @csn_abbr /2 width=3/ -HV
- @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
-| -IHW -IHVT -HV -HT -H2 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: sn3_beta *)
-lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* (ⓓV. T) → (**)
- L ⊢ ⬇* ⓐV. ⓛW. T.
-/2 width=3/ qed.
-
-fact csn_appl_theta_aux: ∀L,U. L ⊢ ⬇* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀V,T. U = ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
-#L #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
-lapply (csn_fwd_pair_sn … HVT) #HV
-lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
-@csn_intro #X #HL #H
-elim (cpr_inv_appl1 … HL) -HL *
-[ -HV #V0 #Y #HLV10 #HL #H0 destruct
- elim (cpr_inv_abbr1 … HL) -HL *
- [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct
- lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34
- elim (lift_total V0 0 1) #V5 #HV05
- elim (term_eq_dec (ⓓV.ⓐV2.T) (ⓓV4.ⓐV5.T3))
- [ -IHVT #H0 destruct
- elim (eq_false_inv_tpair_sn … H) -H
- [ -HLV10 -HLV34 -HV3 -HLT3 -HVT
- >(lift_inj … HV12 … HV05) -V5
- #H elim (H ?) //
- | * #_ #H elim (H ?) //
- ]
- | -H -HVT #H
- lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25
- lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3
- @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/
- ]
- | -H -IHVT #T0 #HT0 #HLT0
- @(csn_cpr_trans … (ⓐV1.T0)) /2 width=1/ -V0 -Y
- @(csn_inv_lift … 0 1 HVT) /2 width=1/
- ]
-| -HV -HV12 -HVT -IHVT -H #V0 #W0 #T0 #T1 #_ #_ #H destruct
-| -IHVT -H #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct
- lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
- lapply (lcpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01
- @csn_abbr /2 width=3/ -HV
- @(csn_lcpr_conf (L. ⓓW0)) /2 width=1/ -W1
- @(csn_cprs_trans … HVT) -HVT /2 width=1/
-]
-qed.
-
-(* Basic_1: was: sn3_appl_bind *)
-lemma csn_appl_theta: ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
- ∀L,V,T. L ⊢ ⬇* ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
-/2 width=5/ qed.
-
-(* Basic_1: was only: sn3_appl_appl *)
-lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬇* V → ∀T1.
- L ⊢ ⬇* T1 →
- (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
- 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
-#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
-@csn_intro #X #HL #H
-elim (cpr_inv_appl1_simple … HL ?) -HL //
-#V0 #T0 #HLV0 #HLT10 #H0 destruct
-elim (eq_false_inv_tpair_sn … H) -H
-[ -IHT1 #HV0
- @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
- @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
- #T2 #HLT12 #HT12
- @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
- @H2T1 -H2T1 // -HLT12 /2 width=1/
-| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
- elim (tstc_dec T1 T0) #H2T10
- [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
- #T2 #HLT02 #HT02
- @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
- | -IHT1 -H3T1 -H1T10
- @H2T1 -H2T1 /2 width=1/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/acp_cr.ma".
-include "Basic_2/computation/cprs_tstc_vector.ma".
-include "Basic_2/computation/csn_lcpr.ma".
-include "Basic_2/computation/csn_vector.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
-
-(* Advanced properties ******************************************************)
-(*
-(* Basic_1: was only: sn3_appl_appls *)
-lemma csn_appl_appls_simple_tstc: ∀L,Vs,V,T1. L ⊢ ⬇* V → L ⊢ ⬇* T1 →
- (∀T2. L ⊢ ⒶVs.T1 ➡* T2 → (ⒶVs.T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
- 𝐒[T1] → L ⊢ ⬇* ⓐV. ⒶVs. T1.
-#L *
-[ #V #T1 #HV
- @csn_appl_simple_tstc //
-| #V0 #Vs #V #T1 #HV #H1T1 #H2T1 #H3T1
- @csn_appl_simple_tstc // -HV
- [ @H2T1
-]
-qed.
-*)
-lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀V,T. L ⊢ ⬇* ⓓV. ⒶV2s. T → L ⊢ ⬇* V →
- L ⊢ ⬇* ⒶV1s. ⓓV. T.
-#L #V1s #V2s * -V1s -V2s /2 width=1/
-#V1s #V2s #V1 #V2 #HV12 #H
-generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
-elim H -V1s -V2s /2 width=3/
-#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H #HV
-lapply (csn_appl_theta … HW12 … H) -H -HW12 #H
-lapply (csn_fwd_pair_sn … H) #HW1
-lapply (csn_fwd_flat_dx … H) #H1
-@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2
-elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
-[ -H #H elim (H2 ?) -H2 //
-| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
-]
-qed.
-
-lemma csn_applv_tau: ∀L,W. L ⊢ ⬇* W →
- ∀Vs,T. L ⊢ ⬇* ⒶVs. T →
- L ⊢ ⬇* ⒶVs. ⓣW. T.
-#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
-#V #Vs #IHV #T #H1T
-lapply (csn_fwd_pair_sn … H1T) #HV
-lapply (csn_fwd_flat_dx … H1T) #H2T
-@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
-[ #X #H #H0
- elim (cprs_fwd_tau_vector … H) -H #H
- [ -H1T elim (H0 ?) -H0 //
- | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
- ]
-| -H1T elim Vs -Vs //
-]
-qed.
-(*
-theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
-@mk_acr //
-[
-|
-|
-| #L #V1 #V2 #HV12 #V #T #H #HVT
- @(csn_applv_theta … HV12) -HV12 //
- @(csn_abbr) //
-| /2 width=1/
-| @csn_lift
-]
-qed.
-*)
-axiom csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cnf_lift.ma".
-include "Basic_2/computation/acp.ma".
-include "Basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: sn3_lift *)
-lemma csn_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
- ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → L2 ⊢ ⬇* T2.
-#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
-@csn_intro #T #HLT2 #HT2
-elim (cpr_inv_lift … HL21 … HT12 … HLT2) -HLT2 #T0 #HT0 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_mono … HT0 … HT12) in HT2; -T0 /2 width=1/
-qed.
-
-(* Basic_1: was: sn3_gen_lift *)
-lemma csn_inv_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
- ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → L2 ⊢ ⬇* T2.
-#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
-@csn_intro #T #HLT2 #HT2
-elim (lift_total T d e) #T0 #HT0
-lapply (cpr_lift … HL12 … HT21 … HT0 HLT2) -HLT2 #HLT10
-@(IHT1 … HLT10) // -L1 -L2 #H destruct
->(lift_inj … HT0 … HT21) in HT2; -T0 /2 width=1/
-qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was: sn3_abbr *)
-lemma csn_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → K ⊢ ⬇* V → L ⊢ ⬇* #i.
-#L #K #V #i #HLK #HV
-@csn_intro #X #H #Hi
-elim (cpr_inv_lref1 … H) -H
-[ #H destruct elim (Hi ?) //
-| -Hi * #K0 #V0 #V1 #HLK0 #HV01 #HV1 #_
- lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
- @(csn_lift … HLK HV1) -HLK -HV1
- @(csn_cpr_trans … HV) -HV
- @(cpr_intro … HV01) -HV01 //
-]
-qed.
-
-lemma csn_abst: ∀L,W. L ⊢ ⬇* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬇* T → L ⊢ ⬇* ⓛW. T.
-#L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpr_inv_abst1 … H1 I V) -H1
-#W0 #T0 #HLW0 #HLT0 #H destruct
-elim (eq_false_inv_tpair_sn … H2) -H2
-[ /3 width=5/
-| -HLW0 * #H destruct /3 width=1/
-]
-qed.
-
-lemma csn_appl_simple: ∀L,V. L ⊢ ⬇* V → ∀T1.
- (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇* ⓐV. T2) →
- 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
-#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
-@csn_intro #X #H1 #H2
-elim (cpr_inv_appl1_simple … H1 ?) // -H1
-#V0 #T0 #HLV0 #HLT10 #H destruct
-elim (eq_false_inv_tpair_dx … H2) -H2
-[ -IHV -HT1 #HT10
- @(csn_cpr_trans … (ⓐV.T0)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // /2 width=1/
-| -HLT10 * #H #HV0 destruct
- @IHV -IHV // -HT1 /2 width=1/ -HV0
- #T2 #HLT02 #HT02
- @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
- @IHT1 -IHT1 // -HLT02 /2 width=1/
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem csn_acp: acp cpr (eq …) (csn …).
-@mk_acp
-[ /2 width=1/
-| /2 width=3/
-| /2 width=5/
-| @cnf_lift
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/computation/csn.ma".
-
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
-
-inductive csnv (L:lenv): predicate (list term) ≝
-| csnv_nil: csnv L ◊
-| csn_cons: ∀Ts,T. L ⊢ ⬇* T → csnv L Ts → csnv L (T :: Ts)
-.
-
-interpretation
- "context-sensitive strong normalization (term vector)"
- 'SN L Ts = (csnv L Ts).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact csnv_inv_cons_aux: ∀L,Ts. L ⊢ ⬇* Ts → ∀U,Us. Ts = U :: Us →
- L ⊢ ⬇* U ∧ L ⊢ ⬇* Us.
-#L #Ts * -Ts
-[ #U #Us #H destruct
-| #Ts #T #HT #HTs #U #Us #H destruct /2 width=1/
-]
-qed.
-
-lemma csnv_inv_cons: ∀L,T,Ts. L ⊢ ⬇* T :: Ts → L ⊢ ⬇* T ∧ L ⊢ ⬇* Ts.
-/2 width=3/ qed-.
-
-include "Basic_2/computation/csn_cpr.ma".
-
-lemma csn_fwd_applv: ∀L,T,Vs. L ⊢ ⬇* Ⓐ Vs. T → L ⊢ ⬇* Vs ∧ L ⊢ ⬇* T.
-#L #T #Vs elim Vs -Vs /2 width=1/
-#V #Vs #IHVs #HVs
-lapply (csn_fwd_pair_sn … HVs) #HV
-lapply (csn_fwd_flat_dx … HVs) -HVs #HVs
-elim (IHVs HVs) -IHVs -HVs /3 width=1/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/lcpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
-
-definition lcprs: relation lenv ≝ TC … lcpr.
-
-interpretation
- "context-sensitive parallel computation (environment)"
- 'CPRedStar L1 L2 = (lcprs L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lcprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
- ∀L2. L1 ⊢ ➡* L2 → R L2.
-#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lcprs_refl: ∀L. L ⊢ ➡* L.
-/2 width=1/ qed.
-
-lemma lcprs_strap1: ∀L1,L,L2.
- L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
-/2 width=3/ qed.
-
-lemma lcprs_strap2: ∀L1,L,L2.
- L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
-/2 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/lcpr_cpr.ma".
-include "Basic_2/computation/cprs.ma".
-include "Basic_2/computation/lcprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
-
-(* Advanced properties ******************************************************)
-
-lemma lcprs_pair_dx: ∀I,L1,L2. L1 ⊢ ➡ L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
- L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
-#I #L1 #L2 #HL12 #V1 #V2 #H @(cprs_ind … H) -V2
-/3 width=1/ /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/lcprs_cprs.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
-
-(* Main properties **********************************************************)
-
-theorem lcprs_trans: ∀L1,L. L1 ⊢ ➡* L → ∀L2. L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
-/2 width=3/ qed.
-
-lemma lcprs_pair: ∀L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
- ∀I. L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
-#L1 #L2 #H @(lcprs_ind … H) -L2 /2 width=1/
-#L #L2 #_ #HL2 #IHL1 #V1 #V2 #HV12 #I
-@(lcprs_trans … (L.ⓑ{I}V1)) /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/aaa.ma".
-include "Basic_2/computation/acp_cr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
-
-inductive lsubc (RP:lenv→predicate term): relation lenv ≝
-| lsubc_atom: lsubc RP (⋆) (⋆)
-| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ÷ A →
- lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW)
-.
-
-interpretation
- "local environment refinement (abstract candidates of reducibility)"
- 'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#RP #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: csubc_gen_sort_r *)
-lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
-/2 width=4/ qed-.
-
-fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
- K1 [RP] ⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-#RP #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
-]
-qed.
-
-(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
- (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
- K1 [RP] ⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
-
-fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
-#RP #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: csubc_gen_sort_l *)
-lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
-/2 width=4/ qed-.
-
-fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
- K1 [RP] ⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-#RP #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
-]
-qed.
-
-(* Basic_1: was: csubc_gen_head_l *)
-lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. ⓑ{I} W →
- (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
- K1 [RP] ⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: csubc_refl *)
-lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
-#RP #L elim L -L // /2 width=1/
-qed.
-
-(* Basic_1: removed theorems 2: csubc_clear_conf csubc_getl_conf *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/aaa_lift.ma".
-include "Basic_2/computation/acp_cr.ma".
-include "Basic_2/computation/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
-
-(* Properties concerning basic local environment slicing ********************)
-
-(* Basic_1: was: csubc_drop_conf_O *)
-(* Note: the constant (0) can not be generalized *)
-lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 [RP] ⊑ K2.
-#RP #L1 #L2 #H elim H -L1 -L2
-[ #X #e #H
- >(ldrop_inv_atom1 … H) -H /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #X #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H destruct
- [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/
- | elim (IHL12 … H) -L2 /3 width=3/
- ]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H destruct
- [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/
- | elim (IHL12 … H) -L2 /3 width=3/
- ]
-qed-.
-
-(* Basic_1: was: csubc_drop_conf_rev *)
-lemma ldrop_lsubc_trans: ∀RR,RS,RP.
- acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
- ∃∃L2. L1 [RP] ⊑ L2 & ⇩[d, e] L2 ≡ K2.
-#RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H
- >(lsubc_inv_atom1 … H) -H /2 width=3/
-| #L1 #I #V1 #X #H
- elim (lsubc_inv_pair1 … H) -H *
- [ #K1 #HLK1 #H destruct /3 width=3/
- | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/
- ]
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
- elim (IHLK1 … HK12) -K1 /3 width=5/
-| #L1 #K1 #I #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H
- elim (lsubc_inv_pair1 … H) -H *
- [ #K2 #HK12 #H destruct
- elim (IHLK1 … HK12) -K1 /3 width=5/
- | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct
- elim (IHLK1 … HK12) #K #HL1K #HK2
- lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA
- lapply (s7 … HA … HV2 … HLK1 HV21) -HV2
- elim (lift_total W2 d e) /4 width=9/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic_2/computation/lsubc_ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
-
-(* Properties concerning generic local environment slicing ******************)
-
-(* Basic_1: was: csubc_drop1_conf_rev *)
-lemma ldrops_lsubc_trans: ∀RR,RS,RP.
- acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
- ∃∃L2. L1 [RP] ⊑ L2 & ⇩*[des] L2 ≡ K2.
-#RR #RS #RP #Hacp #Hacr #L1 #K1 #des #H elim H -L1 -K1 -des
-[ /2 width=3/
-| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12
- elim (ldrop_lsubc_trans … Hacp Hacr … HLK1 … HK12) -Hacp -Hacr -K1 #K #HLK #HK2
- elim (IHL … HLK) -IHL -HLK /3 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/lsuba.ma".
-include "Basic_2/computation/acp_aaa.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
-
-(* properties concerning lenv refinement for atomic arity assignment ********)
-
-lemma lsubc_lsuba: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
- ∀L1,L2. L1 ÷⊑ L2 → L1 [RP] ⊑ L2.
-#RR #RS #RP #H1RP #H2RP #L1 #L2 #H elim H -L1 -L2
-// /2 width=1/ /3 width=4/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
-
-definition cpc: lenv → relation term ≝
- λL,T1,T2. L ⊢ T1 ➡ T2 ∨ L ⊢ T2 ➡ T1.
-
-interpretation
- "context-sensitive parallel conversion (term)"
- 'PConv L T1 T2 = (cpc L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpc_refl: ∀L,T. L ⊢ T ⬌ T.
-/2 width=1/ qed.
-
-lemma cpc_sym: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → L ⊢ T2 ⬌ T1.
-#L #T1 #T2 * /2 width=1/
-qed.
-
-lemma cpc_cpr: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
-#L #T1 #T2 * /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/conversion/cpc.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
-
-(* Main properties **********************************************************)
-
-theorem cpc_conf: ∀L,T0,T1,T2. L ⊢ T0 ⬌ T1 → L ⊢ T0 ⬌ T2 →
- ∃∃T. L ⊢ T1 ⬌ T & L ⊢ T2 ⬌ T.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/conversion/cpc.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
-
-definition cpcs: lenv → relation term ≝
- λL. TC … (cpc L).
-
-interpretation "context-sensitive parallel equivalence (term)"
- 'PConvStar L T1 T2 = (cpcs L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
- (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
- ∀T2. L ⊢ T1 ⬌* T2 → R T2.
-#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: pc3_refl *)
-lemma cpcs_refl: ∀L,T. L ⊢ T ⬌* T.
-/2 width=1/ qed.
-
-lemma cpcs_strap1: ∀L,T1,T,T2.
- L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
-/2 width=3/ qed.
-
-lemma cpcs_strap2: ∀L,T1,T,T2.
- L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
-/2 width=3/ qed.
-
-(* Basic_1: removed theorems 1: clear_pc3_trans *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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
- * Support for abstract candidates of reducibility closed: 2012 January 27
- * Confluence of context-sensitive parallel reduction closed: 2011 September 21
- * Confluence of context-free parallel reduction closed: 2011 September 6
- * Specification started: 2011 April 17
- * - Patience on me to gain peace and perfection! -
- * [ suggested invocation to start formal specifications with ]
- *)
-
-include "Ground_2/star.ma".
-include "Basic_2/notation.ma".
-
-(* ATOMIC ARITY *************************************************************)
-
-inductive aarity: Type[0] ≝
- | AAtom: aarity (* atomic aarity construction *)
- | APair: aarity → aarity → aarity (* binary aarity construction *)
-.
-
-interpretation "aarity construction (atomic)"
- 'Item0 = AAtom.
-
-interpretation "aarity construction (binary)"
- 'SnItem2 A1 A2 = (APair A1 A2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma discr_apair_xy_x: ∀A,B. ②B. A = B → False.
-#A #B elim B -B
-[ #H destruct
-| #Y #X #IHY #_ #H destruct
- -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
- /2 width=1/
-]
-qed-.
-
-lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → False.
-#B #A elim A -A
-[ #H destruct
-| #Y #X #_ #IHX #H destruct
- -H (**) (* destruct: the destucted equality is not erased *)
- /2 width=1/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
-#A1 elim A1 -A1
-[ #A2 elim A2 -A2 /2 width=1/
- #B2 #A2 #_ #_ @or_intror #H destruct
-| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
- [ -IHB1 -IHA1 @or_intror #H destruct
- | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
- [ #H destruct elim (IHA1 A2) -IHA1
- [ #H destruct /2 width=1/
- | #HA12 @or_intror #H destruct /2 width=1/
- ]
- | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
- ]
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 (ⓑ{I} V. T)
-].
-
-interpretation "shift (closure)" 'Append L T = (shift L T).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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→predicate term.
- (∀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. ⓑ{I} V, T] < #[K, ②{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 width=2/ | skip ]
-qed.
-
-(* Basic_1: removed theorems 6:
- flt_thead_sx flt_thead_dx flt_arith0 flt_arith1 flt_arith2 flt_trans
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-include "Basic_2/grammar/term.ma".
-
-(* GLOBAL ENVIRONMENTS ******************************************************)
-
-(* global environments *)
-definition genv ≝ list2 bind2 term.
-
-interpretation "sort (global environment)"
- 'Star = (nil2 bind2 term).
-
-interpretation "environment construction (binary)"
- 'DxItem2 L I T = (cons2 bind2 term I T L).
-
-interpretation "environment binding construction (binary)"
- 'DxBind2 L I T = (cons2 bind2 term I T L).
-
-interpretation "abbreviation (global environment)"
- 'DxAbbr L T = (cons2 bind2 term Abbr T L).
-
-interpretation "abstraction (global environment)"
- 'DxAbst L T = (cons2 bind2 term Abst T L).
-
-(* Basic properties *********************************************************)
-
-axiom genv_eq_dec: ∀T1,T2:genv. Decidable (T1 = T2).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "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 *)
- | GRef: nat → item0 (* reference by position: 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] ≝
- | Bind2: bind2 → item2 (* binding item *)
- | Flat2: flat2 → item2 (* non-binding item *)
-.
-
-coercion Bind2.
-
-coercion Flat2.
-
-(* Basic properties *********************************************************)
-
-axiom item0_eq_dec: ∀I1,I2:item0. Decidable (I1 = I2).
-
-(* Basic_1: was: bind_dec *)
-axiom bind2_eq_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
-
-(* Basic_1: was: flat_dec *)
-axiom flat2_eq_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
-
-(* Basic_1: was: kind_dec *)
-axiom item2_eq_dec: ∀I1,I2:item2. Decidable (I1 = I2).
-
-(* Basic_1: removed theorems 21:
- 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
- not_abbr_abst bind_dec_not
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 construction (binary)"
- 'DxItem2 L I T = (LPair L I T).
-
-interpretation "environment binding construction (binary)"
- 'DxBind2 L I T = (LPair L I T).
-
-interpretation "abbreviation (local environment)"
- 'DxAbbr L T = (LPair L Abbr T).
-
-interpretation "abstraction (local environment)"
- 'DxAbst L T = (LPair L Abst T).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 REFINEMENT FOR SUBSTITUTION ****************************)
-
-inductive lsubs: nat → nat → relation lenv ≝
-| lsubs_sort: ∀d,e. lsubs d e (⋆) (⋆)
-| lsubs_OO: ∀L1,L2. lsubs 0 0 L1 L2
-| lsubs_abbr: ∀L1,L2,V,e. lsubs 0 e L1 L2 →
- lsubs 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
-| lsubs_abst: ∀L1,L2,I,V1,V2,e. lsubs 0 e L1 L2 →
- lsubs 0 (e + 1) (L1. ⓛV1) (L2.ⓑ{I} V2)
-| lsubs_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
- lsubs d e L1 L2 → lsubs (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
-.
-
-interpretation
- "local environment refinement (substitution)"
- 'SubEq L1 d e L2 = (lsubs d e L1 L2).
-
-definition lsubs_conf: ∀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_lsubs_conf: ∀S,R. lsubs_conf S R → lsubs_conf S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -s2
-[ /3 width=5/
-| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
- lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
-]
-qed.
-
-lemma lsubs_bind_eq: ∀L1,L2,e. L1 [0, e] ≼ L2 → ∀I,V.
- L1. ⓑ{I} V [0, e + 1] ≼ L2.ⓑ{I} V.
-#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
-qed.
-
-lemma lsubs_refl: ∀d,e,L. L [d, e] ≼ L.
-#d elim d -d
-[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
-| #d #IHd #e #L elim L -L // /2 width=1/
-]
-qed.
-
-lemma lsubs_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≼ L2 → 0 < d →
- ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 [d, e] ≼ L2. ⓑ{I2} V2.
-
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic forward lemmas ***************************************************)
-
-fact lsubs_fwd_length_full1_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- d = 0 → e = |L1| → |L1| ≤ |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubs_fwd_length_full1: ∀L1,L2. L1 [0, |L1|] ≼ L2 → |L1| ≤ |L2|.
-/2 width=5/ qed-.
-
-fact lsubs_fwd_length_full2_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- d = 0 → e = |L2| → |L2| ≤ |L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
-[ //
-| /2 width=1/
-| /3 width=1/
-| /3 width=1/
-| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma lsubs_fwd_length_full2: ∀L1,L2. L1 [0, |L2|] ≼ L2 → |L2| ≤ |L1|.
-/2 width=5/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "term construction (atomic)"
- 'Item0 I = (TAtom I).
-
-interpretation "term construction (binary)"
- 'SnItem2 I T1 T2 = (TPair I T1 T2).
-
-interpretation "term binding construction (binary)"
- 'SnBind2 I T1 T2 = (TPair (Bind2 I) T1 T2).
-
-interpretation "term flat construction (binary)"
- 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
-
-interpretation "sort (term)"
- 'Star k = (TAtom (Sort k)).
-
-interpretation "local reference (term)"
- 'LRef i = (TAtom (LRef i)).
-
-interpretation "global reference (term)"
- 'GRef p = (TAtom (GRef p)).
-
-interpretation "abbreviation (term)"
- 'SnAbbr T1 T2 = (TPair (Bind2 Abbr) T1 T2).
-
-interpretation "abstraction (term)"
- 'SnAbst T1 T2 = (TPair (Bind2 Abst) T1 T2).
-
-interpretation "application (term)"
- 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
-
-interpretation "native type annotation (term)"
- 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: term_dec *)
-axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → False.
-#I #T #V elim V -V
-[ #J #H destruct
-| #J #W #U #IHW #_ #H destruct
- -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
- /2 width=1/
-]
-qed-.
-
-(* Basic_1: was: thead_x_y_y *)
-lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → False.
-#I #V #T elim T -T
-[ #J #H destruct
-| #J #W #U #_ #IHU #H destruct
- -H (**) (* destruct: the destucted equality is not erased *)
- /2 width=1/
-]
-qed-.
-
-lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I} V2. T2 → False) →
- (V1 = V2 → False) ∨ (V1 = V2 ∧ (T1 = T2 → False)).
-#I #V1 #T1 #V2 #T2 #H
-elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct
-@or_intror @conj // #HT12 destruct /2 width=1/
-qed-.
-
-lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I} V2. T2 → False) →
- (T1 = T2 → False) ∨ (T1 = T2 ∧ (V1 = V2 → False)).
-#I #V1 #T1 #V2 #T2 #H
-elim (term_eq_dec T1 T2) /3 width=1/ #HT12 destruct
-@or_intror @conj // #HT12 destruct /2 width=1/
-qed-.
-
-lemma eq_false_inv_beta: ∀V1,V2,W1,W2,T1,T2.
- (ⓐV1. ⓛW1. T1 = ⓐV2. ⓛW2 .T2 →False) →
- (W1 = W2 → False) ∨
- (W1 = W2 ∧ (ⓓV1. T1 = ⓓV2. T2 → False)).
-#V1 #V2 #W1 #W2 #T1 #T2 #H
-elim (eq_false_inv_tpair_sn … H) -H
-[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/
- #H destruct @or_intror @conj // #H destruct /2 width=1/
-| * #H1 #H2 destruct
- elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/
- * #H #HT12 destruct
- @or_intror @conj // #H destruct /2 width=1/
-]
-qed.
-
-(* Basic_1: removed theorems 3:
- not_void_abst not_abbr_void not_abst_void
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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: predicate term ≝
- | simple_atom: ∀I. simple (⓪{I})
- | simple_flat: ∀I,V,T. simple (ⓕ{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 = ⓑ{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[ⓑ{I} V. T] → False.
-/2 width=6/ qed-.
-
-lemma simple_inv_pair: ∀I,V,T. 𝐒[②{I}V.T] → ∃J. I = Flat2 J.
-* /2 width=2/ #I #V #T #H
-elim (simple_inv_bind … H)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-include "Basic_2/grammar/term.ma".
-
-(* TERMS ********************************************************************)
-
-let rec applv Vs T on Vs ≝
- match Vs with
- [ nil ⇒ T
- | cons hd tl ⇒ ⓐhd. (applv tl T)
- ].
-
-interpretation "application o vevtor (term)"
- 'SnApplV Vs T = (applv Vs T).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *********************************************************)
-
-(* Basic_1: was: tweight_lt *)
-lemma tw_pos: ∀T. 1 ≤ #[T].
-#T elim T -T //
-qed.
-
-(* Basic eliminators ********************************************************)
-
-axiom tw_wf_ind: ∀R:predicate term.
- (∀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
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* SAME HEAD TERM FORMS *****************************************************)
-
-inductive tshf: relation term ≝
- | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
- | tshf_abst: ∀V1,V2,T1,T2. tshf (ⓛV1. T1) (ⓛV2. T2)
- | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒[T1] → 𝐒[T2] →
- tshf (ⓐV1. T1) (ⓐV2. T2)
-.
-
-interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1.
-#T1 #T2 #H elim H -T1 -T2 /2 width=1/
-qed.
-
-lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2.
-#T1 #T2 #H elim H -T1 -T2 // /2 width=1/
-qed.
-
-lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
-/3 width=2/ qed.
-
-lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒[T1] → 𝐒[T2].
-#T1 #T2 #H elim H -T1 -T2 //
-#V1 #V2 #T1 #T2 #H
-elim (simple_inv_bind … H)
-qed. (**) (* remove from index *)
-
-lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒[T2] → 𝐒[T1].
-/3 width=3/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓑ{I}W1.U1 →
- ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
-#T1 #T2 * -T1 -T2
-[ #J #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/
-| #V1 #V2 #T1 #T2 #H_ #_ #_ #I #W1 #U1 #H destruct
-]
-qed.
-
-lemma tshf_inv_bind1: ∀I,W1,U1,T2. ⓑ{I}W1.U1 ≈ T2 →
- ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
-/2 width=5/ qed-.
-
-fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
- I = Appl & T2 = ⓐW2. U2.
-#T1 #T2 * -T1 -T2
-[ #J #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
-| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
-]
-qed.
-
-lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
- ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
- I = Appl & T2 = ⓐW2. U2.
-/2 width=4/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* SAME TOP TERM CONSTRUCTOR ************************************************)
-
-inductive tstc: relation term ≝
- | tstc_atom: ∀I. tstc (⓪{I}) (⓪{I})
- | tstc_pair: ∀I,V1,V2,T1,T2. tstc (②{I} V1. T1) (②{I} V2. T2)
-.
-
-interpretation "same top constructor (term)" 'Iso T1 T2 = (tstc T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tstc_inv_atom1_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I}.
-#T1 #T2 * -T1 -T2 //
-#J #V1 #V2 #T1 #T2 #I #H destruct
-qed.
-
-(* Basic_1: was: iso_gen_sort iso_gen_lref *)
-lemma tstc_inv_atom1: ∀I,T2. ⓪{I} ≃ T2 → T2 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tstc_inv_pair1_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W1,U1. T1 = ②{I}W1.U1 →
- ∃∃W2,U2. T2 = ②{I}W2. U2.
-#T1 #T2 * -T1 -T2
-[ #J #I #W1 #U1 #H destruct
-| #J #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/
-]
-qed.
-
-(* Basic_1: was: iso_gen_head *)
-lemma tstc_inv_pair1: ∀I,W1,U1,T2. ②{I}W1.U1 ≃ T2 →
- ∃∃W2,U2. T2 = ②{I}W2. U2.
-/2 width=5/ qed-.
-
-fact tstc_inv_atom2_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T2 = ⓪{I} → T1 = ⓪{I}.
-#T1 #T2 * -T1 -T2 //
-#J #V1 #V2 #T1 #T2 #I #H destruct
-qed.
-
-lemma tstc_inv_atom2: ∀I,T1. T1 ≃ ⓪{I} → T1 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tstc_inv_pair2_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W2,U2. T2 = ②{I}W2.U2 →
- ∃∃W1,U1. T1 = ②{I}W1. U1.
-#T1 #T2 * -T1 -T2
-[ #J #I #W2 #U2 #H destruct
-| #J #V1 #V2 #T1 #T2 #I #W2 #U2 #H destruct /2 width=3/
-]
-qed.
-
-lemma tstc_inv_pair2: ∀I,T1,W2,U2. T1 ≃ ②{I}W2.U2 →
- ∃∃W1,U1. T1 = ②{I}W1. U1.
-/2 width=5/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: iso_refl *)
-lemma tstc_refl: ∀T. T ≃ T.
-#T elim T -T //
-qed.
-
-lemma tstc_sym: ∀T1,T2. T1 ≃ T2 → T2 ≃ T1.
-#T1 #T2 #H elim H -T1 -T2 //
-qed.
-
-lemma tstc_dec: ∀T1,T2. Decidable (T1 ≃ T2).
-* #I1 [2: #V1 #T1 ] * #I2 [2,4: #V2 #T2 ]
-[ elim (item2_eq_dec I1 I2) #HI12
- [ destruct /2 width=1/
- | @or_intror #H
- elim (tstc_inv_pair1 … H) -H #V #T #H destruct /2 width=1/
- ]
-| @or_intror #H
- lapply (tstc_inv_atom1 … H) -H #H destruct
-| @or_intror #H
- lapply (tstc_inv_atom2 … H) -H #H destruct
-| elim (item0_eq_dec I1 I2) #HI12
- [ destruct /2 width=1/
- | @or_intror #H
- lapply (tstc_inv_atom2 … H) -H #H destruct /2 width=1/
- ]
-]
-qed.
-
-lemma simple_tstc_repl_dx: ∀T1,T2. T1 ≃ T2 → 𝐒[T1] → 𝐒[T2].
-#T1 #T2 * -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #H
-elim (simple_inv_pair … H) -H #J #H destruct //
-qed. (**) (* remove from index *)
-
-lemma simple_tstc_repl_sn: ∀T1,T2. T1 ≃ T2 → 𝐒[T2] → 𝐒[T1].
-/3 width=3/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tstc.ma".
-
-(* SAME TOP TERM CONSTRUCTOR ************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: iso_trans *)
-theorem tstc_trans: ∀T1,T. T1 ≃ T → ∀T2. T ≃ T2 → T1 ≃ T2.
-#T1 #T * -T1 -T //
-#I #V1 #V #T1 #T #X #H
-elim (tstc_inv_pair1 … H) -H #V2 #T2 #H destruct //
-qed.
-
-theorem tstc_canc_sn: ∀T,T1. T ≃ T1 → ∀T2. T ≃ T2 → T1 ≃ T2.
-/3 width=3/ qed.
-
-theorem tstc_canc_dx: ∀T1,T. T1 ≃ T → ∀T2. T2 ≃ T → T1 ≃ T2.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/grammar/tstc.ma".
-
-(* SAME TOP TERM CONSTRUCTOR ************************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
-lemma tstc_inv_bind_appls_simple: ∀I,Vs,V2,T1,T2. ⒶVs.T1 ≃ ⓑ{I} V2. T2 →
- 𝐒[T1] → False.
-#I #Vs #V2 #T1 #T2 #H
-elim (tstc_inv_pair2 … H) -H #V0 #T0
-elim Vs -Vs normalize
-[ #H destruct #H
- @(simple_inv_bind … H)
-| #V #Vs #_ #H destruct
-]
-qed.
-
+++ /dev/null
-NAMING CONVENTIONS FOR METAVARIABLES
-
-A,B : arity
-C,D : candidate of reducibility
-E,F : RTM environment
-G : global environment
-H : reserved: transient premise
-IH : reserved: inductive premise
-I,J : item
-K,L : local environment
-M,N : reserved: future use
-O :
-P,Q : reserved: future use
-R : generic predicate (relation)
-S : RTM stack
-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
-p,q : global reference position
-t,u : local reference position level (de Bruijn's)
-x,y,z : reserved: transient objet denoted by a small letter
-
-NAMING CONVENTIONS FOR CONSTRUCTORS
-
-0: atomic
-2: binary
-
-A: application to vector
-
-a: application
-b: binder
-d: abbreviation
-f: flat
-l: abstraction
-t: native type annotation
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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( ⓪ )"
- non associative with precedence 90
- for @{ 'Item0 }.
-
-notation "hvbox( ⓪ { I } )"
- non associative with precedence 90
- for @{ 'Item0 $I }.
-
-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 i )"
- non associative with precedence 90
- for @{ 'LRef $i }.
-
-notation "hvbox( § term 90 p )"
- non associative with precedence 90
- for @{ 'GRef $p }.
-
-notation "hvbox( ② term 90 T1 . break term 90 T )"
- non associative with precedence 90
- for @{ 'SnItem2 $T1 $T }.
-
-notation "hvbox( ② { I } break term 90 T1 . break term 90 T )"
- non associative with precedence 90
- for @{ 'SnItem2 $I $T1 $T }.
-
-notation "hvbox( ⓑ { I } break term 90 T1 . break term 90 T )"
- non associative with precedence 90
- for @{ 'SnBind2 $I $T1 $T }.
-
-notation "hvbox( ⓕ { I } break term 90 T1 . break term 90 T )"
- non associative with precedence 90
- for @{ 'SnFlat2 $I $T1 $T }.
-
-notation "hvbox( ⓓ term 90 T1 . break term 90 T2 )"
- non associative with precedence 90
- for @{ 'SnAbbr $T1 $T2 }.
-
-notation "hvbox( ⓛ term 90 T1 . break term 90 T2 )"
- non associative with precedence 90
- for @{ 'SnAbst $T1 $T2 }.
-
-notation "hvbox( ⓐ term 90 T1 . break term 90 T2 )"
- non associative with precedence 90
- for @{ 'SnAppl $T1 $T2 }.
-
-notation "hvbox( ⓣ term 90 T1 . break term 90 T2 )"
- non associative with precedence 90
- for @{ 'SnCast $T1 $T2 }.
-
-notation "hvbox( Ⓐ term 90 T1 . break term 90 T )"
- non associative with precedence 90
- for @{ 'SnApplV $T1 $T }.
-
-notation > "hvbox( T . break ②{ I } break term 47 T1 )"
- non associative with precedence 46
- for @{ 'DxBind2 $T $I $T1 }.
-
-notation "hvbox( T . break ⓑ { I } break term 90 T1 )"
- non associative with precedence 89
- for @{ 'DxBind2 $T $I $T1 }.
-
-notation "hvbox( T1 . break ⓓ T2 )"
- left associative with precedence 48
- for @{ 'DxAbbr $T1 $T2 }.
-
-notation "hvbox( T1 . break ⓛ T2 )"
- left associative with precedence 49
- for @{ 'DxAbst $T1 $T2 }.
-
-notation "hvbox( T . break ④ { I } break { T1 , break T2 , break T3 } )"
- non associative with precedence 47
- for @{ 'DxItem4 $T $I $T1 $T2 $T3 }.
-
-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( L ⊢ break term 90 T1 ≈ break T2 )"
- non associative with precedence 45
- for @{ 'Hom $L $T1 $T2 }.
-
-notation "hvbox( T1 ≃ break T2 )"
- non associative with precedence 45
- for @{ 'Iso $T1 $T2 }.
-
-notation "hvbox( T1 break [ d , break e ] ≼ break T2 )"
- non associative with precedence 45
- for @{ 'SubEq $T1 $d $e $T2 }.
-
-(* Substitution *************************************************************)
-
-notation "hvbox( L ⊢ break [ d , break e ] break 𝐑 [ T ] )"
- non associative with precedence 45
- for @{ 'Reducible $L $d $e $T }.
-
-notation "hvbox( L ⊢ break [ d , break e ] break 𝐈 [ T ] )"
- non associative with precedence 45
- for @{ 'NotReducible $L $d $e $T }.
-
-notation "hvbox( L ⊢ break [ d , break e ] break 𝐍 [ T ] )"
- non associative with precedence 45
- for @{ 'Normal $L $d $e $T }.
-
-notation "hvbox( ⇧ [ d , break e ] break T1 ≡ break T2 )"
- non associative with precedence 45
- for @{ 'RLift $d $e $T1 $T2 }.
-
-notation "hvbox( ⇩ [ e ] break L1 ≡ break L2 )"
- non associative with precedence 45
- for @{ 'RDrop $e $L1 $L2 }.
-
-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 f ≡ break T2 )"
- non associative with precedence 45
- for @{ 'RAt $T1 $f $T2 }.
-
-notation "hvbox( T1 ▭ break T2 ≡ break T )"
- non associative with precedence 45
- for @{ 'RMinus $T1 $T2 $T }.
-
-notation "hvbox( ⇧ * [ e ] break T1 ≡ break T2 )"
- non associative with precedence 45
- for @{ 'RLiftStar $e $T1 $T2 }.
-
-notation "hvbox( ⇩ * [ e ] break L1 ≡ break L2 )"
- non associative with precedence 45
- for @{ 'RDropStar $e $L1 $L2 }.
-
-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 }.
-
-notation "hvbox( T1 break [ d , break e ] ≡ break T2 )"
- non associative with precedence 45
- for @{ 'TSubst $T1 $d $e $T2 }.
-
-notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≡ break T2 )"
- non associative with precedence 45
- for @{ 'TSubst $L $T1 $d $e $T2 }.
-
-(* Static Typing ************************************************************)
-
-notation "hvbox( L ⊢ break term 90 T ÷ break A )"
- non associative with precedence 45
- for @{ 'AtomicArity $L $T $A }.
-
-notation "hvbox( T1 ÷ ⊑ break T2 )"
- non associative with precedence 45
- for @{ 'CrSubEqA $T1 $T2 }.
-
-(* Reducibility *************************************************************)
-
-notation "hvbox( 𝐑 [ T ] )"
- non associative with precedence 45
- for @{ 'Reducible $T }.
-
-notation "hvbox( L ⊢ break 𝐑 [ T ] )"
- non associative with precedence 45
- for @{ 'Reducible $L $T }.
-
-notation "hvbox( 𝐈 [ T ] )"
- non associative with precedence 45
- for @{ 'NotReducible $T }.
-
-notation "hvbox( L ⊢ break 𝐈 [ T ] )"
- non associative with precedence 45
- for @{ 'NotReducible $L $T }.
-
-notation "hvbox( 𝐍 [ T ] )"
- non associative with precedence 45
- for @{ 'Normal $T }.
-
-notation "hvbox( L ⊢ break 𝐍 [ T ] )"
- non associative with precedence 45
- for @{ 'Normal $L $T }.
-
-notation "hvbox( 𝐖𝐇𝐑 [ T ] )"
- non associative with precedence 45
- for @{ 'WHdReducible $T }.
-
-notation "hvbox( L ⊢ break 𝐖𝐇𝐑 [ T ] )"
- non associative with precedence 45
- for @{ 'WHdReducible $L $T }.
-
-notation "hvbox( 𝐖𝐇𝐈 [ T ] )"
- non associative with precedence 45
- for @{ 'NotWHdReducible $T }.
-
-notation "hvbox( L ⊢ break 𝐖𝐇𝐈 [ T ] )"
- non associative with precedence 45
- for @{ 'NotWHdReducible $L $T }.
-
-notation "hvbox( 𝐖𝐇𝐍 [ T ] )"
- non associative with precedence 45
- for @{ 'WHdNormal $T }.
-
-notation "hvbox( L ⊢ break 𝐖𝐇𝐍 [ T ] )"
- non associative with precedence 45
- for @{ 'WHdNormal $L $T }.
-
-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 }.
-
-notation "hvbox( ⬇ * T )"
- non associative with precedence 45
- for @{ 'SN $T }.
-
-notation "hvbox( L ⊢ ⬇ * T )"
- non associative with precedence 45
- for @{ 'SN $L $T }.
-
-notation "hvbox( L ⊢ ⬇ * * T )"
- non associative with precedence 45
- for @{ 'SNStar $L $T }.
-
-notation "hvbox( ⦃ L, break T ⦄ break [ R ] ϵ break 〚 A 〛 )"
- non associative with precedence 45
- for @{ 'InEInt $R $L $T $A }.
-
-notation "hvbox( T1 break [ R ] ⊑ break T2 )"
- non associative with precedence 45
- for @{ 'CrSubEq $T1 $R $T2 }.
-
-(* Conversion ***************************************************************)
-
-notation "hvbox( L ⊢ break term 90 T1 ⬌ break T2 )"
- non associative with precedence 45
- for @{ 'PConv $L $T1 $T2 }.
-
-(* Equivalence **************************************************************)
-
-notation "hvbox( L ⊢ break term 90 T1 ⬌* break T2 )"
- non associative with precedence 45
- for @{ 'PConvStar $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
-
-interpretation
- "context-sensitive normality (term)"
- 'Normal L T = (cnf L T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: nf2_sort *)
-lemma cnf_sort: ∀L,k. L ⊢ 𝐍[⋆k].
-#L #k #X #H
->(cpr_inv_sort1 … H) //
-qed.
-
-axiom cnf_dec: ∀L,T1. L ⊢ 𝐍[T1] ∨
- ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → False).
-
-(* Basic_1: removed theorems 1: nf2_abst_shift *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr_lift.ma".
-include "Basic_2/reducibility/cnf.ma".
-
-(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was only: nf2_csort_lref *)
-lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍[#i].
-#L #i #HLK #X #H
-elim (cpr_inv_lref1 … H) // *
-#K0 #V0 #V1 #HLK0 #_ #_ #_
-lapply (ldrop_mono … HLK … HLK0) -L #H destruct
-qed.
-
-(* Basic_1: was: nf2_lref_abst *)
-lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍[#i].
-#L #K #V #i #HLK #X #H
-elim (cpr_inv_lref1 … H) // *
-#K0 #V0 #V1 #HLK0 #_ #_ #_
-lapply (ldrop_mono … HLK … HLK0) -L #H destruct
-qed.
-
-(* Basic_1: was: nf2_abst *)
-lemma cnf_abst: ∀I,L,V,W,T. L ⊢ 𝐍[W] → L. ⓑ{I} V ⊢ 𝐍[T] → L ⊢ 𝐍[ⓛW.T].
-#I #L #V #W #T #HW #HT #X #H
-elim (cpr_inv_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct
->(HW … HW0) -W >(HT … HT0) -T //
-qed.
-
-(* Basic_1: was only: nf2_appl_lref *)
-lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍[V] → L ⊢ 𝐍[T] → 𝐒[T] → L ⊢ 𝐍[ⓐV.T].
-#L #V #T #HV #HT #HS #X #H
-elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct
->(HV … HV0) -V >(HT … HT0) -T //
-qed.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: nf2_lift *)
-lemma cnf_lift: ∀L0,L,T,T0,d,e.
- L ⊢ 𝐍[T] → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍[T0].
-#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
-elim (cpr_inv_lift … HL0 … HT0 … H) -L0 #T1 #HT10 #HT1
->(HLT … HT1) in HT0; -L #HT0
->(lift_mono … HT10 … HT0) -T1 -X //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/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 *********************************************************)
-
-lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T [d, e] ▶* T2 → L ⊢ T1 ➡ T2.
-/4 width=3/ qed-.
-
-(* Basic_1: was by definition: pr2_free *)
-lemma cpr_pr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
-/2 width=3/ 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 width=1/ 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 ⊢ ⓕ{I} V1. T1 ➡ ⓕ{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 ⊢ ⓣV. T1 ➡ T2.
-#L #V #T1 #T2 * /3 width=3/
-qed.
-
-(* Note: it does not hold replacing |L1| with |L2| *)
-(* Basic_1: was only: pr2_change *)
-lemma cpr_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡ T2.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
-lapply (tpss_lsubs_conf … HT2 … HL12) -HT2 -HL12 /3 width=4/
-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_abbr *)
-lemma cpr_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡ U2 →
- (∃∃V,V2,T2. V1 ➡ V & L ⊢ V [O, |L|] ▶* V2 &
- L. ⓓV ⊢ T1 ➡ T2 &
- U2 = ⓓV2. T2
- ) ∨
- ∃∃T. ⇧[0,1] T ≡ T1 & L ⊢ T ➡ U2.
-#L #V1 #T1 #Y * #X #H1 #H2
-elim (tpr_inv_abbr1 … H1) -H1 *
-[ #V #T0 #T #HV1 #HT10 #HT0 #H destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tps_lsubs_conf … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
- lapply (tps_weak_all … HT0) -HT0 #HT0
- lapply (tpss_lsubs_conf … HT2 (L. ⓓV) ?) -HT2 /2 width=1/ #HT2
- lapply (tpss_weak_all … HT2) -HT2 #HT2
- lapply (tpss_strap … HT0 HT2) -T /4 width=7/
-| /4 width=5/
-]
-qed-.
-
-(* Basic_1: was: pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓣV1. T1 ➡ U2 → (
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U2 = ⓣV2. T2
- ) ∨ L ⊢ T1 ➡ U2.
-#L #V1 #T1 #U2 * #X #H #HU2
-elim (tpr_inv_cast1 … H) -H /3 width=3/
-* #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
-qed-.
-
-(* Basic_1: removed theorems 4:
- 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
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr_tpr.ma".
-include "Basic_2/reducibility/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 ⊢ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
-@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /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. ⓑ{I} V2 ⊢ T1 ➡ T2 →
- L ⊢ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
-elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
-lapply (tpss_lsubs_conf … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
-lapply (tpss_tps … HT0) -HT0 #HT0
-@ex2_1_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
-qed.
-
-(* Basic_1: was only: pr2_head_1 *)
-lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
- L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
-* /2 width=1/ /3 width=1/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @ T1 ➡ L @ T2.
-#L elim L -L
-[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
-| normalize /3 width=1/
-].
-qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: pr2_confluence *)
-theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
- ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
-#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
-elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
-elim (tpr_tpss_ltpr ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
-elim (tpr_tpss_ltpr ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
-elim (tpss_conf_eq … HU1 … HU2) -U /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_lift.ma".
-include "Basic_2/reducibility/tpr_lift.ma".
-include "Basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
- ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 [0, |L| - i - 1] ▶* W1 →
- ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
-#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
-lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
-@ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_1: was: pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
- T2 = #i ∨
- ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ V1 [0, |L| - i - 1] ▶* T1 &
- ⇧[0, i + 1] T1 ≡ T2 &
- i < |L|.
-#L #T2 #i * #X #H
->(tpr_inv_atom1 … H) -H #H
-elim (tpss_inv_lref1 … H) -H /2 width=1/
-* /3 width=6/
-qed-.
-
-(* Basic_1: was: pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
-#L #V1 #T1 #Y * #X #H1 #H2 #I #W
-elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
-elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
-lapply (tpss_lsubs_conf … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
-qed-.
-
-(* Basic_1: was pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
- U0 = ⓛW. T1 &
- U2 = ⓓV2. T2
- | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓW1. T1 &
- U2 = ⓓW2. ⓐV. T2.
-#L #V1 #U0 #Y * #X #H1 #H2
-elim (tpr_inv_appl1 … H1) -H1 *
-[ #V #U #HV1 #HU0 #H destruct
- elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
-| #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
-| #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
- elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
- elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
- elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
- lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
-]
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒[T1] →
- ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
- U = ⓐV2. T2.
-#L #V1 #T1 #U #H #HT1
-elim (cpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: pr2_lift *)
-lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
- K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
-elim (lt_or_ge (|K|) d) #HKd
-[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
-| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
-]
-qed.
-
-(* Basic_1: was: pr2_gen_lift *)
-lemma cpr_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
- ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
- ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
-elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
-elim (lt_or_ge (|L|) d) #HLd
-[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
-| elim (lt_or_ge (|L|) (d + e)) #HLde
- [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=2/ ]
- | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr_tpss.ma".
-include "Basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning parallel unfold on terms ***************************)
-
-(* Note: we could invoke tpss_weak_all instead of ltpr_fwd_length *)
-(* Basic_1: was only: pr2_subst1 *)
-lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
-#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
-elim (tpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
-elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
-#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
->(ltpr_fwd_length … HL12) in HT2; #HT2
-elim (tpr_tpss_ltpr … HL12 … HT2) -L1 /3 width=3/
-qed.
-
-lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
- ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
-elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
-elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
-lapply (tpss_weak_all … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_tpss.ma".
-include "Basic_2/reducibility/cpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-(* Properties concerning partial unfold on local environments ***************)
-
-lemma ltpss_cpr_trans: ∀L1,L2,d,e. L1 [d, e] ▶* L2 →
- ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
-#L1 #L2 #d #e #HL12 #T1 #T2 *
-lapply (ltpss_weak_all … HL12)
-<(ltpss_fwd_length … HL12) -HL12 /3 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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.ma".
-include "Basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************)
-
-definition lcpr: relation lenv ≝
- λL1,L2. ∃∃L. L1 ➡ L & L [0, |L|] ▶* L2
-.
-
-interpretation
- "context-sensitive parallel reduction (environment)"
- 'CPRed L1 L2 = (lcpr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lcpr_refl: ∀L. L ⊢ ➡ L.
-/2 width=3/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lcpr_inv_atom1: ∀L2. ⋆ ⊢ ➡ L2 → L2 = ⋆.
-#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_inv_atom1 … HL2) -HL2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/cpr.ma".
-include "Basic_2/reducibility/lcpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************)
-
-(* Advanced properties ****************************************************)
-
-lemma lcpr_pair: ∀L1,L2. L1 ⊢ ➡ L2 → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
- ∀I. L1. ⓑ{I} V1 ⊢ ➡ L2. ⓑ{I} V2.
-#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
-<(ltpss_fwd_length … HL2) /4 width=5/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-inductive ltpr: relation lenv ≝
-| ltpr_stom: ltpr (⋆) (⋆)
-| ltpr_pair: ∀K1,K2,I,V1,V2.
- ltpr K1 K2 → V1 ➡ V2 → ltpr (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) (*ⓑ*)
-.
-
-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 width=1/
-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 width=3/ qed-.
-
-fact ltpr_inv_pair1_aux: ∀L1,L2. L1 ➡ L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-#L1 #L2 * -L1 -L2
-[ #K1 #I #V1 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
-]
-qed.
-
-(* Basic_1: was: wcpr0_gen_head *)
-lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
- ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3/ 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 width=3/ qed-.
-
-fact ltpr_inv_pair2_aux: ∀L1,L2. L1 ➡ L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-#L1 #L2 * -L1 -L2
-[ #K2 #I #V2 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
-]
-qed.
-
-lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
-#L1 #L2 #H elim H -L1 -L2 normalize //
-qed-.
-
-(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr_lift.ma".
-include "Basic_2/reducibility/ltpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Basic_1: was: wcpr0_drop *)
-lemma ltpr_ldrop_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 -L1 -K1 -d -e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
-| #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
- elim (IHLK1 … HL12) -L1 /3 width=3/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (tpr_inv_lift … HV12 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5/
-]
-qed.
-
-(* Basic_1: was: wcpr0_drop_back *)
-lemma ldrop_ltpr_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 -L1 -K1 -d -e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
-| #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) -K1 /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
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (tpr_lift … HW12 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1 /3 width=5/
-]
-qed.
-
-fact ltpr_ldrop_trans_O1_aux: ∀L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. L1 ➡ L2 →
- d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
-#L2 #K2 #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #X #H >(ltpr_inv_atom2 … H) -H /2 width=3/
-| #K2 #I #V2 #X #H
- elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
-| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
- elim (ltpr_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
-| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
- >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpr_ldrop_trans_O1: ∀L1,L2. L1 ➡ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
-/2 width=5/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-(* Properties concerning parallel substitution on terms *********************)
-
-lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 [d, e] ▶ T2 → ∀L1. L1 ➡ L2 →
- ∃∃T. L1 ⊢ T1 [d, e] ▶ T & T ➡ T2.
-#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
-[ /2 width=3/
-| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
- elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
- elim (lift_total V1 0 (i+1)) #W1 #HVW1
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
-| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
-| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L2 /3 width=5/
-]
-qed.
-
-lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶ T2 → ∀L2. L1 ➡ L2 →
- ∃∃T. L2 ⊢ T1 [d, e] ▶ T & T2 ➡ T.
-#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
-[ /2 width=3/
-| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
- elim (lift_total V2 0 (i+1)) #W2 #HVW2
- lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
-| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
- elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
-| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
- elim (IHV12 … HL12) -IHV12
- elim (IHT12 … HL12) -L1 /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/trf.ma".
-
-(* CONTEXT-FREE IRREDUCIBLE TERMS *******************************************)
-
-definition tif: predicate term ≝ λT. 𝐑[T] → False.
-
-interpretation "context-free irreducibility (term)"
- 'NotReducible T = (tif T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tif_inv_abbr: ∀V,T. 𝐈[ⓓV.T] → False.
-/2 width=1/ qed-.
-
-lemma tif_inv_abst: ∀V,T. 𝐈[ⓛV.T] → 𝐈[V] ∧ 𝐈[T].
-/4 width=1/ qed-.
-
-lemma tif_inv_appl: ∀V,T. 𝐈[ⓐV.T] → ∧∧ 𝐈[V] & 𝐈[T] & 𝐒[T].
-#V #T #HVT @and3_intro /3 width=1/
-generalize in match HVT; -HVT elim T -T //
-* // * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
-qed-.
-
-lemma tif_inv_cast: ∀V,T. 𝐈[ⓣV.T] → False.
-/2 width=1/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tif_atom: ∀I. 𝐈[⓪{I}].
-/2 width=4/ qed.
-
-lemma tif_abst: ∀V,T. 𝐈[V] → 𝐈[T] → 𝐈[ⓛV.T].
-#V #T #HV #HT #H
-elim (trf_inv_abst … H) -H /2 width=1/
-qed.
-
-lemma tif_appl: ∀V,T. 𝐈[V] → 𝐈[T] → 𝐒[T] → 𝐈[ⓐV.T].
-#V #T #HV #HT #S #H
-elim (trf_inv_appl … H) -H /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr.ma".
-
-(* CONTEXT-FREE NORMAL TERMS ************************************************)
-
-definition tnf: predicate term ≝ NF … tpr (eq …).
-
-interpretation
- "context-free normality (term)"
- 'Normal T = (tnf T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tnf_inv_abst: ∀V,T. 𝐍[ⓛV.T] → 𝐍[V] ∧ 𝐍[T].
-#V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓛV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓛV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
-]
-qed-.
-
-lemma tnf_inv_appl: ∀V,T. 𝐍[ⓐV.T] → ∧∧ 𝐍[V] & 𝐍[T] & 𝐒[T].
-#V1 #T1 #HVT1 @and3_intro
-[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
-| generalize in match HVT1; -HVT1 elim T1 -T1 * // * #W1 #U1 #_ #_ #H
- [ elim (lift_total V1 0 1) #V2 #HV12
- lapply (H (ⓓW1.ⓐV2.U1) ?) -H /2 width=3/ -HV12 #H destruct
- | lapply (H (ⓓV1.U1) ?) -H /2 width=1/ #H destruct
-]
-qed-.
-
-lemma tnf_inv_abbr: ∀V,T. 𝐍[ⓓV.T] → False.
-#V #T #H elim (is_lift_dec T 0 1)
-[ * #U #HTU
- lapply (H U ?) -H /2 width=3/ #H destruct
- elim (lift_inv_pair_xy_y … HTU)
-| #HT
- elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
- lapply (H (ⓓV.T2) ?) -H /2 width=3/ -HT2 #H destruct /3 width=2/
-]
-qed.
-
-lemma tnf_inv_cast: ∀V,T. 𝐍[ⓣV.T] → False.
-#V #T #H lapply (H T ?) -H /2 width=1/ #H
-@(discr_tpair_xy_y … H)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tif.ma".
-include "Basic_2/reducibility/tnf.ma".
-
-(* CONTEXT-FREE NORMAL TERMS ************************************************)
-
-(* Main properties properties ***********************************************)
-
-lemma tpr_tif_eq: ∀T1,T2. T1 ➡ T2 → 𝐈[T1] → T1 = T2.
-#T1 #T2 #H elim H -T1 -T2
-[ //
-| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
- [ elim (tif_inv_appl … H) -H #HV1 #HT1 #_
- >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
- | elim (tif_inv_cast … H)
- ]
-| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
- elim (tif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #H
- [ -HT2 -IHV1 -IHT1 elim (tif_inv_abbr … H)
- | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
- elim (tif_inv_abst … H) -H #HV1 #HT1
- >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
- ]
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
- elim (tif_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-| #V1 #T1 #T2 #T #_ #_ #_ #H
- elim (tif_inv_abbr … H)
-| #V1 #T1 #T #_ #_ #H
- elim (tif_inv_cast … H)
-]
-qed.
-
-theorem tif_tnf: ∀T1. 𝐈[T1] → 𝐍[T1].
-/2 width=1/ qed.
-
-(* Note: this property is unusual *)
-lemma tnf_trf_false: ∀T1. 𝐑[T1] → 𝐍[T1] → False.
-#T1 #H elim H -T1
-[ #V #T #_ #IHV #H elim (tnf_inv_abst … H) -H /2 width=1/
-| #V #T #_ #IHT #H elim (tnf_inv_abst … H) -H /2 width=1/
-| #V #T #_ #IHV #H elim (tnf_inv_appl … H) -H /2 width=1/
-| #V #T #_ #IHV #H elim (tnf_inv_appl … H) -H /2 width=1/
-| #V #T #H elim (tnf_inv_abbr … H)
-| #V #T #H elim (tnf_inv_cast … H)
-| #V #W #T #H elim (tnf_inv_appl … H) -H #_ #_ #H
- elim (simple_inv_bind … H)
-]
-qed.
-
-theorem tnf_tif: ∀T1. 𝐍[T1] → 𝐈[T1].
-/2 width=3/ qed.
-
-lemma tnf_abst: ∀V,T. 𝐍[V] → 𝐍[T] → 𝐍[ⓛV.T].
-/4 width=1/ qed.
-
-lemma tnf_appl: ∀V,T. 𝐍[V] → 𝐍[T] → 𝐒[T] → 𝐍[ⓐV.T].
-/4 width=1/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 (⓪{I}) (⓪{I})
-| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| tpr_beta : ∀V1,V2,W,T1,T2.
- tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛW. T1) (ⓓV2. T2)
-| tpr_delta: ∀I,V1,V2,T1,T2,T.
- tpr V1 V2 → tpr T1 T2 → ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T →
- tpr (ⓑ{I} V1. T1) (ⓑ{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 (ⓐV1. ⓓW1. T1) (ⓓW2. ⓐV. T2)
-| tpr_zeta : ∀V,T,T1,T2. ⇧[0,1] T1 ≡ T → tpr T1 T2 → tpr (ⓓV. T) T2
-| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓣ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 →
- ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
-/2 width=3/ qed.
-
-(* Basic_1: was by definition: pr0_refl *)
-lemma tpr_refl: ∀T. T ➡ T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{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. ⓪{I} ➡ U2 → U2 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
- (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
- ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T &
- U2 = ⓑ{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 /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 /3 width=3/
-| #V #T1 #T2 #_ #I0 #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_bind1: ∀V1,T1,U2,I. ⓑ{I} V1. T1 ➡ U2 →
- (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
- ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T &
- U2 = ⓑ{I} V2. T
- ) ∨
- ∃∃T. ⇧[0,1] T ≡ T1 & T ➡ U2 & I = Abbr.
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_abbr *)
-lemma tpr_inv_abbr1: ∀V1,T1,U2. ⓓV1. T1 ➡ U2 →
- (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
- ⋆. ⓓV2 ⊢ T2 [0, 1] ▶ T &
- U2 = ⓓ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 = ⓕ{I} V1. U0 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛW. T1 &
- U2 = ⓓV2. T2 & I = Appl
- | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓW1. T1 &
- U2 = ⓓW2. ⓐ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 /3 width=5/
-| #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /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 /3 width=12/
-| #V #T #T1 #T2 #_ #_ #J #V0 #T0 #H destruct
-| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛW. T1 &
- U2 = ⓓV2. T2 & I = Appl
- | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓW1. T1 &
- U2 = ⓓW2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_appl *)
-lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛW. T1 &
- U2 = ⓓV2. T2
- | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓW1. T1 &
- U2 = ⓓW2. ⓐV. T2.
-#V1 #U0 #U2 #H
-elim (tpr_inv_flat1 … H) -H * /3 width=12/ #_ #H destruct
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒[T1] →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
- U = ⓐV2. T2.
-#V1 #T1 #U #H #HT1
-elim (tpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Basic_1: was: pr0_gen_cast *)
-lemma tpr_inv_cast1: ∀V1,T1,U2. ⓣV1. T1 ➡ U2 →
- (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓣ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 = ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓣV. T.
-#T1 #T2 * -T1 -T2
-[ #I #i #H destruct /2 width=1/
-| #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 width=4/
-]
-qed.
-
-lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
- ∨∨ T1 = #i
- | ∃∃V,T,T0. ⇧[O,1] T0 ≡ T & T0 ➡ #i &
- T1 = ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓣV. T.
-/2 width=3/ qed-.
-
-(* Basic_1: removed theorems 3:
- pr0_subst0_back pr0_subst0_fwd pr0_subst0
- Basic_1: removed local theorems: 1: pr0_delta_tau
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/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 -T1 -T2
-[ * #i #d #e #U1 #HU1 #U2 #HU2
- lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
- [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
- | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
- | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
- ]
-| #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
- elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
-| #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
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
-| #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
- elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
- elim (lift_total T2 (d + 1) e) #U2 #HTU2
- @tpr_delta
- [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/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
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
- elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
- elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
- elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
-| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20
- elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct
- elim (lift_trans_ge … 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 /3 width=4/
-]
-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 -T1 -T2
-[ * #i #d #e #U1 #HU1
- [ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct /2 width=3/
- | lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … HU1) -HU1 #H destruct /2 width=3/
- ]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /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
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1
- elim (IHT12 … HT01) -T1 /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
- elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
- elim (IHT12 … HUT1) -T1 #U2 #HUT2 #HU12
- elim (tps_inv_lift1_le … HT20 … HUT2 ?) -T2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
- @ex2_1_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /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
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
- elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
- elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
- elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
- elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
- @ex2_1_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /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
- elim (lift_div_le … HT1 … HT0 ?) -T // #T #HT0 #HT1
- elim (IHT12 … HT1) -T1 /3 width=5/
-| #V #T1 #T2 #_ #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
- elim (IHT12 … HT01) -T1 /3 width=3/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tpr_inv_abst1_aux: ∀U1,U2. U1 ➡ U2 → ∀V1,T1. U1 = ⓛV1. T1 →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ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
- <(tps_inv_refl_SO2 … 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. ⓛV1. T1 ➡ U2 →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛV2. T2.
-/2 width=3/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/tpr_tpss.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Confluence lemmas ********************************************************)
-
-fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
-/2 width=3/ 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. ⓕ{I} V1. T1 ➡ T0 & ⓕ{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 -IH // /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 → ⓛW0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓ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
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /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 → ⓓW0. U0 ➡ T1 →
- ∃∃X. ⓐV1. T1 ➡ X & ⓓW2. ⓐ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 /2 width=1/ #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
- elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
- elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
- elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -UU2 #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 /2 width=1/ -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. ⓣV1. T1 ➡ X & X2 ➡ X.
-#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=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. ⓓV1. T1 ➡X & ⓓV2. T2 ➡ X.
-#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /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 →
- ⋆. ⓑ{I1} V1 ⊢ T1 [O, 1] ▶ TT1 →
- ⋆. ⓑ{I1} V2 ⊢ T2 [O, 1] ▶ TT2 →
- ∃∃X. ⓑ{I1} V1. TT1 ➡ X & ⓑ{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) -T1 #U1 #TTU1 #HTU1
-elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
-elim (tps_conf_eq … HTU1 … HTU2) -T #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 → ⋆. ⓓV1 ⊢ T1 [O,1] ▶ TT1 →
- T2 ➡ X2 → ⇧[O, 1] T2 ≡ T0 →
- ∃∃X. ⓓ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
-lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HTX2 … HTT2) -HTX2 -HTT2 -IH // /3 width=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. ⓓW1. ⓐVV1. T1 ➡ X & ⓓW2. ⓐ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 /2 width=1/ #V #HV1 #HV2
-elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
-elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
-elim (lift_total V 0 1) #VV #HVV
-lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
-lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -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
-lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HT01 … HTX2) -HT01 -HTX2 -IH // /2 width=3/
-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 width=3/
-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
- lapply (tpr_inv_atom1 … H1) -H1
-(* case 1: atom, atom *)
- #H1 destruct //
-| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
- elim (tpr_inv_flat1 … H1) -H1 *
-(* case 2: flat, flat *)
- [ #V2 #T2 #HV02 #HT02 #H destruct
- /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
- /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
- /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
-(* case 5: flat, tau *)
- | #HT02 #H destruct
- /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
- ]
-| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
- elim (tpr_inv_appl1 … H1) -H1 *
-(* case 6: beta, flat (repeated) *)
- [ #V2 #T2 #HV02 #HT02 #H destruct
- @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
-(* case 7: beta, beta *)
- | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct
- /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
- elim (tpr_inv_bind1 … H1) -H1 *
-(* case 9: delta, delta *)
- [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
- /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
-(* case 10: delta, zata *)
- | #T2 #HT20 #HTX2 #H destruct
- /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
- elim (tpr_inv_appl1 … H1) -H1 *
-(* case 11: theta, flat (repeated) *)
- [ #V2 #T2 #HV02 #HT02 #H destruct
- @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
- /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
- ]
-| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct
- elim (tpr_inv_abbr1 … H1) -H1 *
-(* case 14: zeta, delta (repeated) *)
- [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
- @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
- elim (tpr_inv_cast1 … H1) -H1
-(* case 16: tau, flat (repeated) *)
- [ * #V2 #T2 #HV02 #HT02 #H destruct
- @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
-(* case 17: tau, tau *)
- | #HT02
- /3 width=5 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.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/reducibility/ltpr_ldrop.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Unfold properties ********************************************************)
-
-(* 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 -T1 -T2
-[ #I #L1 #d #e #X #H
- elim (tps_inv_atom1 … H) -H
- [ #H destruct /2 width=3/
- | * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 #HU12
- @ex2_1_intro [2: @HU12 | skip | /3 width=4/ ] (**) (* /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
- elim (IHV12 … HVW1 … HL12) -V1
- elim (IHT12 … HTU1 … HL12) -T1 -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
- elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- lapply (tpss_lsubs_conf … HTT2 (L2. ⓓ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
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHT12 … HTT1 (L2. ⓑ{I} VV2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- elim (tpss_strip_neq … HTT2 … HTU2 ?) -T2 /2 width=1/ #T2 #HTT2 #HUT2
- lapply (tps_lsubs_conf … HTT2 (L2. ⓑ{I} V2) ?) -HTT2 /2 width=1/ #HTT2
- elim (ltpss_tps_conf … HTT2 (L2. ⓑ{I} VV2) (d + 1) e ?) -HTT2 /2 width=1/ #W2 #HTTW2 #HTW2
- lapply (tpss_lsubs_conf … HTTW2 (⋆. ⓑ{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
- lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2
- lapply (tpss_lsubs_conf … HTW2 (L2. ⓑ{I} VV2) ?) -HTW2 /2 width=1/ #HTW2
- lapply (tpss_trans_eq … HUT2 … HTW2) -T2 /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
- elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
- elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
- elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
- elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
- elim (lift_total VV2 0 1) #VV #H2VV
- lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #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 #HTT1 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct
- elim (tps_inv_lift1_ge … HTT12 L1 … HTT1 ?) -TT1 /2 width=1/ #T2 #HT12 #HTT2
- elim (IHT12 … HT12 … HL12) -T1 -HL12 <minus_plus_m_m /3 width=3/
-| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
- elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
- elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
-]
-qed.
-
-lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
- ⋆. ⓑ{I} V1 ⊢ T1 [0, 1] ▶ U1 →
- ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ U2.
-#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
-elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ /3 width=3/
-qed.
-
-lemma tpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
-#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
-[ /2 width=3/
-| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
- elim (tpr_tps_ltpr … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
- lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* CONTEXT-FREE REDUCIBLE TERMS *********************************************)
-
-(* reducible terms *)
-inductive trf: predicate term ≝
-| trf_abst_sn: ∀V,T. trf V → trf (ⓛV. T)
-| trf_abst_dx: ∀V,T. trf T → trf (ⓛV. T)
-| trf_appl_sn: ∀V,T. trf V → trf (ⓐV. T)
-| trf_appl_dx: ∀V,T. trf T → trf (ⓐV. T)
-| trf_abbr : ∀V,T. trf (ⓓV. T)
-| trf_cast : ∀V,T. trf (ⓣV. T)
-| trf_beta : ∀V,W,T. trf (ⓐV. ⓛW. T)
-.
-
-interpretation
- "context-free reducibility (term)"
- 'Reducible T = (trf T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact trf_inv_atom_aux: ∀I,T. 𝐑[T] → T = ⓪{I} → False.
-#I #T * -T
-[ #V #T #_ #H destruct
-| #V #T #_ #H destruct
-| #V #T #_ #H destruct
-| #V #T #_ #H destruct
-| #V #T #H destruct
-| #V #T #H destruct
-| #V #W #T #H destruct
-]
-qed.
-
-lemma trf_inv_atom: ∀I. 𝐑[⓪{I}] → False.
-/2 width=4/ qed-.
-
-fact trf_inv_abst_aux: ∀W,U,T. 𝐑[T] → T = ⓛW. U → 𝐑[W] ∨ 𝐑[U].
-#W #U #T * -T
-[ #V #T #HV #H destruct /2 width=1/
-| #V #T #HT #H destruct /2 width=1/
-| #V #T #_ #H destruct
-| #V #T #_ #H destruct
-| #V #T #H destruct
-| #V #T #H destruct
-| #V #W0 #T #H destruct
-]
-qed.
-
-lemma trf_inv_abst: ∀V,T. 𝐑[ⓛV.T] → 𝐑[V] ∨ 𝐑[T].
-/2 width=3/ qed-.
-
-fact trf_inv_appl_aux: ∀W,U,T. 𝐑[T] → T = ⓐW. U →
- ∨∨ 𝐑[W] | 𝐑[U] | (𝐒[U] → False).
-#W #U #T * -T
-[ #V #T #_ #H destruct
-| #V #T #_ #H destruct
-| #V #T #HV #H destruct /2 width=1/
-| #V #T #HT #H destruct /2 width=1/
-| #V #T #H destruct
-| #V #T #H destruct
-| #V #W0 #T #H destruct
- @or3_intro2 #H elim (simple_inv_bind … H)
-]
-qed.
-
-lemma trf_inv_appl: ∀W,U. 𝐑[ⓐW.U] → ∨∨ 𝐑[W] | 𝐑[U] | (𝐒[U] → False).
-/2 width=3/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/tshf.ma".
-include "Basic_2/reducibility/tpr.ma".
-
-(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************)
-
-definition twhnf: predicate term ≝ NF … tpr tshf.
-
-interpretation
- "context-free weak head normality (term)"
- 'WHdNormal T = (twhnf T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma twhnf_inv_tshf: ∀T. 𝐖𝐇𝐍[T] → T ≈ T.
-normalize /2 width=1/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tpr_tshf: ∀T1,T2. T1 ➡ T2 → T1 ≈ T1 → T1 ≈ T2.
-#T1 #T2 #H elim H -T1 -T2 //
-[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
- lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2
- lapply (simple_tshf_repl_dx … HUT2 HT1) /2 width=1/
-| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
- elim (simple_inv_bind … H)
-| #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H destruct //
-| #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
- elim (simple_inv_bind … H)
-| #V #T #T1 #T2 #_ #_ #_ #H
- elim (tshf_inv_bind1 … H) -H #W2 #U2 #H destruct
-| #V #T1 #T2 #_ #_ #H
- elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
-]
-qed.
-
-lemma twhnf_tshf: ∀T. T ≈ T → 𝐖𝐇𝐍[T].
-/2 width=1/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/aarity.ma".
-include "Basic_2/substitution/ldrop.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-inductive aaa: lenv → term → predicate aarity ≝
-| aaa_sort: ∀L,k. aaa L (⋆k) ⓪
-| aaa_lref: ∀I,L,K,V,B,i. ⇩[0, i] L ≡ K. ⓑ{I} V → aaa K V B → aaa L (#i) B
-| aaa_abbr: ∀L,V,T,B,A.
- aaa L V B → aaa (L. ⓓV) T A → aaa L (ⓓV. T) A
-| aaa_abst: ∀L,V,T,B,A.
- aaa L V B → aaa (L. ⓛV) T A → aaa L (ⓛV. T) (②B. A)
-| aaa_appl: ∀L,V,T,B,A. aaa L V B → aaa L T (②B. A) → aaa L (ⓐV. T) A
-| aaa_cast: ∀L,V,T,A. aaa L V A → aaa L T A → aaa L (ⓣV. T) A
-.
-
-interpretation "atomic arity assignment (term)"
- 'AtomicArity L T A = (aaa L T A).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact aaa_inv_sort_aux: ∀L,T,A. L ⊢ T ÷ A → ∀k. T = ⋆k → A = ⓪.
-#L #T #A * -L -T -A
-[ //
-| #I #L #K #V #B #i #_ #_ #k #H destruct
-| #L #V #T #B #A #_ #_ #k #H destruct
-| #L #V #T #B #A #_ #_ #k #H destruct
-| #L #V #T #B #A #_ #_ #k #H destruct
-| #L #V #T #A #_ #_ #k #H destruct
-]
-qed.
-
-lemma aaa_inv_sort: ∀L,A,k. L ⊢ ⋆k ÷ A → A = ⓪.
-/2 width=5/ qed-.
-
-fact aaa_inv_lref_aux: ∀L,T,A. L ⊢ T ÷ A → ∀i. T = #i →
- ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ÷ A.
-#L #T #A * -L -T -A
-[ #L #k #i #H destruct
-| #I #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5/
-| #L #V #T #B #A #_ #_ #i #H destruct
-| #L #V #T #B #A #_ #_ #i #H destruct
-| #L #V #T #B #A #_ #_ #i #H destruct
-| #L #V #T #A #_ #_ #i #H destruct
-]
-qed.
-
-lemma aaa_inv_lref: ∀L,A,i. L ⊢ #i ÷ A →
- ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ÷ A.
-/2 width=3/ qed-.
-
-fact aaa_inv_abbr_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓓW. U →
- ∃∃B. L ⊢ W ÷ B & L. ⓓW ⊢ U ÷ A.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=2/
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #A #_ #_ #W #U #H destruct
-]
-qed.
-
-lemma aaa_inv_abbr: ∀L,V,T,A. L ⊢ ⓓV. T ÷ A →
- ∃∃B. L ⊢ V ÷ B & L. ⓓV ⊢ T ÷ A.
-/2 width=3/ qed-.
-
-fact aaa_inv_abst_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓛW. U →
- ∃∃B1,B2. L ⊢ W ÷ B1 & L. ⓛW ⊢ U ÷ B2 & A = ②B1. B2.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=5/
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #A #_ #_ #W #U #H destruct
-]
-qed.
-
-lemma aaa_inv_abst: ∀L,W,T,A. L ⊢ ⓛW. T ÷ A →
- ∃∃B1,B2. L ⊢ W ÷ B1 & L. ⓛW ⊢ T ÷ B2 & A = ②B1. B2.
-/2 width=3/ qed-.
-
-fact aaa_inv_appl_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓐW. U →
- ∃∃B. L ⊢ W ÷ B & L ⊢ U ÷ ②B. A.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3/
-| #L #V #T #A #_ #_ #W #U #H destruct
-]
-qed.
-
-lemma aaa_inv_appl: ∀L,V,T,A. L ⊢ ⓐV. T ÷ A →
- ∃∃B. L ⊢ V ÷ B & L ⊢ T ÷ ②B. A.
-/2 width=3/ qed-.
-
-fact aaa_inv_cast_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓣW. U →
- L ⊢ W ÷ A ∧ L ⊢ U ÷ A.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #A #HV #HT #W #U #H destruct /2 width=1/
-]
-qed.
-
-lemma aaa_inv_cast: ∀L,W,T,A. L ⊢ ⓣW. T ÷ A →
- L ⊢ W ÷ A ∧ L ⊢ T ÷ A.
-/2 width=3/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop_ldrop.ma".
-include "Basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Main properties **********************************************************)
-
-theorem aaa_mono: ∀L,T,A1. L ⊢ T ÷ A1 → ∀A2. L ⊢ T ÷ A2 → A1 = A2.
-#L #T #A1 #H elim H -L -T -A1
-[ #L #k #A2 #H
- >(aaa_inv_sort … H) -H //
-| #I1 #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H
- elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2
- lapply (ldrop_mono … HLK1 … HLK2) -L #H destruct /2 width=1/
-| #L #V #T #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
- elim (aaa_inv_abbr … H) -H /2 width=1/
-| #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H
- elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1/
-| #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
- elim (aaa_inv_appl … H) -H #B2 #_ #HA2
- lapply (IHA1 … HA2) -L #H destruct //
-| #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H
- elim (aaa_inv_cast … H) -H /2 width=1/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop_ldrop.ma".
-include "Basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties concerning basic relocation ***********************************)
-
-lemma aaa_lift: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
- ∀T2. ⇧[d, e] T1 ≡ T2 → L2 ⊢ T2 ÷ A.
-#L1 #T1 #A #H elim H -L1 -T1 -A
-[ #L1 #k #L2 #d #e #_ #T2 #H
- >(lift_inv_sort1 … H) -H //
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H
- elim (lift_inv_lref1 … H) -H * #Hid #H destruct
- [ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=8/
- | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
- ]
-| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=4/
-| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=4/
-| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=4/
-| #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=4/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrops.ma".
-include "Basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties concerning generic relocation *********************************)
-
-lemma aaa_lifts: ∀L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
- L1 ⊢ T1 ÷ A → L2 ⊢ T2 ÷ A.
-#L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des
-[ #L #T1 #H #HT1
- <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1
- elim (lifts_inv_cons … H) -H /3 width=9/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/aaa.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
-
-inductive lsuba: relation lenv ≝
-| lsuba_atom: lsuba (⋆) (⋆)
-| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ÷ A → L2 ⊢ W ÷ A →
- lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
-.
-
-interpretation
- "local environment refinement (atomic arity assigment)"
- 'CrSubEqA L1 L2 = (lsuba L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ÷⊑ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsuba_inv_atom1: ∀L2. ⋆ ÷⊑ L2 → L2 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-#L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
-]
-qed.
-
-lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ÷⊑ L2 →
- (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
- L2 = K2. ⓛW & I = Abbr.
-/2 width=3/ qed-.
-
-fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ÷⊑ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
-]
-qed.
-
-lemma lsubc_inv_atom2: ∀L1. L1 ÷⊑ ⋆ → L1 = ⋆.
-/2 width=3/ qed-.
-
-fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-#L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
-| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
-]
-qed.
-
-lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ÷⊑ K2. ⓑ{I} W →
- (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
- L1 = K1. ⓓV & I = Abst.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsuba_refl: ∀L. L ÷⊑ L.
-#L elim L -L // /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/aaa_aaa.ma".
-include "Basic_2/static/lsuba_ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
-
-(* Properties concerning atomic arity assignment ****************************)
-
-lemma lsuba_aaa_conf: ∀L1,V,A. L1 ⊢ V ÷ A → ∀L2. L1 ÷⊑ L2 → L2 ⊢ V ÷ A.
-#L1 #V #A #H elim H -L1 -V -A
-[ //
-| #I #L1 #K1 #V1 #B #i #HLK1 #HV1B #IHV1 #L2 #HL12
- elim (lsuba_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
- elim (lsuba_inv_pair1 … H) -H * #K2
- [ #HK12 #H destruct /3 width=5/
- | #V2 #A1 #HV1A1 #HV2 #_ #H1 #H2 destruct
- >(aaa_mono … HV1B … HV1A1) -B -HV1A1 /2 width=5/
- ]
-| /4 width=2/
-| /4 width=1/
-| /3 width=3/
-| /3 width=1/
-]
-qed.
-
-lemma lsuba_aaa_trans: ∀L2,V,A. L2 ⊢ V ÷ A → ∀L1. L1 ÷⊑ L2 → L1 ⊢ V ÷ A.
-#L2 #V #A #H elim H -L2 -V -A
-[ //
-| #I #L2 #K2 #V2 #B #i #HLK2 #HV2B #IHV2 #L1 #HL12
- elim (lsuba_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsuba_inv_pair2 … H) -H * #K1
- [ #HK12 #H destruct /3 width=5/
- | #V1 #A1 #HV1 #HV2A1 #_ #H1 #H2 destruct
- >(aaa_mono … HV2B … HV2A1) -B -HV2A1 /2 width=5/
- ]
-| /4 width=2/
-| /4 width=1/
-| /3 width=3/
-| /3 width=1/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/lsuba.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
-
-(* Properties concerning basic local environment slicing ********************)
-
-(* Note: the constant 0 cannot be generalized *)
-
-lemma lsuba_ldrop_O1_conf: ∀L1,L2. L1 ÷⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. K1 ÷⊑ K2 & ⇩[0, e] L2 ≡ K2.
-#L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
- [ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK1) -L1 /3 width=3/
- ]
-]
-qed-.
-
-lemma lsuba_ldrop_O1_trans: ∀L1,L2. L1 ÷⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. K1 ÷⊑ K2 & ⇩[0, e] L1 ≡ K1.
-#L1 #L2 #H elim H -L1 -L2
-[ /2 width=3/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
- [ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
- | elim (IHL12 … HLK2) -L2 /3 width=3/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/static/lsuba_aaa.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
-
-(* Main properties **********************************************************)
-
-theorem lsuba_trans: ∀L1,L. L1 ÷⊑ L → ∀L2. L ÷⊑ L2 → L1 ÷⊑ L2.
-#L1 #L #H elim H -L1 -L
-[ #X #H >(lsuba_inv_atom1 … H) -H //
-| #I #L1 #L #V #HL1 #IHL1 #X #H
- elim (lsuba_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=1/
- | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/
- ]
-| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H
- elim (lsuba_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=5/
- | #V #A2 #_ #_ #_ #_ #H destruct
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* 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 *)
-}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/genv.ma".
-
-(* GLOBAL ENVIRONMENT SLICING ***********************************************)
-
-inductive gdrop (e:nat): relation genv ≝
-| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆)
-| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G
-| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2
-.
-
-interpretation "global slicing"
- 'RDrop e G1 G2 = (gdrop e G1 G2).
-
-(* basic inversion lemmas ***************************************************)
-
-lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
-#G1 #G2 #e * -G1 -G2 //
-[ #G #H >H -H >commutative_plus #H
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ #H2
- lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-]
-qed-.
-
-lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
-#G1 #G2 #e * -G1 -G2 //
-[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ normalize #H2
- <(injective_plus_l … H2) in H1; -H2 #H
- elim (lt_refl_false … H)
-]
-qed-.
-
-fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
- e < |G1| → ⇩[e] G1 ≡ G2.
-#I #G #G1 #G2 #V #e * -G -G2
-[ #G #H1 #H destruct #H2
- lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
- elim (lt_refl_false … H)
-| #J #G #G2 #W #_ #HG2 #H destruct //
-]
-qed.
-
-lemma gdrop_inv_lt: ∀I,G1,G2,V,e.
- ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
-/2 width=5/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
-#e #G1 elim G1 -G1 /3 width=2/
-#I #V #G1 * #G2 #HG12
-elim (lt_or_eq_or_gt e (|G1|)) #He
-[ /3 width=2/
-| destruct /3 width=2/
-| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/gdrop.ma".
-
-(* GLOBAL ENVIRONMENT SLICING ***********************************************)
-
-(* Main properties **********************************************************)
-
-theorem gdrop_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
-#G #G1 #e #H elim H -G -G1
-[ #G #He #G2 #H
- >(gdrop_inv_gt … H He) -H -He //
-| #G #He #G2 #H
- >(gdrop_inv_eq … H He) -H -He //
-| #I #G #G1 #V #He #_ #IHG1 #G2 #H
- lapply (gdrop_inv_lt … H He) -H -He /2 width=1/
-]
-qed-.
-
-lemma gdrop_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
-#G1 #G2 #e
-elim (gdrop_total e G1) #G #HG1
-elim (genv_eq_dec G G2) #HG2
-[ destruct /2 width=1/
-| @or_intror #HG12
- lapply (gdrop_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/lsubs.ma".
-include "Basic_2/substitution/lift.ma".
-
-(* LOCAL ENVIRONMENT SLICING ************************************************)
-
-(* Basic_1: includes: drop_skip_bind *)
-inductive ldrop: nat → nat → relation lenv ≝
-| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
-| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2
-| ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
- ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 →
- ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| //
-| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed-.
-
-fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ //
-| #L #I #V #H destruct
-| #L1 #L2 #I #V #e #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-(* Basic_1: was: drop_gen_sort *)
-lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed-.
-
-fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. ⓑ{I} V →
- (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
- (0 < e ∧ ⇩[d, e - 1] K ≡ L2).
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #K #I #V #H destruct
-| #L #I #V #_ #K #J #W #HX destruct /3 width=1/
-| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 →
- (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
- (0 < e ∧ ⇩[0, e - 1] K ≡ L2).
-/2 width=3/ qed-.
-
-(* Basic_1: was: drop_gen_drop *)
-lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
- ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
-#e #K #I #V #L2 #H #He
-elim (ldrop_inv_O1 … H) -H * // #H destruct
-elim (lt_refl_false … He)
-qed-.
-
-fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
- L2 = K2. ⓑ{I} V2.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K #V #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/
-]
-qed.
-
-(* Basic_1: was: drop_gen_skip_l *)
-lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#d #e #L1 #L2 * -d -e -L1 -L2
-[ #d #e #_ #I #K #V #H destruct
-| #L #I #V #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/
-]
-qed.
-
-(* Basic_1: was: drop_gen_skip_r *)
-lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was by definition: drop_refl *)
-lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
-#L elim L -L //
-qed.
-
-lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
- ⇩[0, e - 1] L1 ≡ L2 → 0 < e → ⇩[0, e] L1. ⓑ{I} V ≡ L2.
-#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- ∀K1,V,i. ⇩[0, i] L1 ≡ K1. ⓓV →
- d ≤ i → i < d + e →
- ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
- ⇩[0, i] L2 ≡ K2. ⓓV.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #K1 #V #i #H
- lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #L1 #L2 #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie destruct
- <minus_n_O <minus_plus_m_m // /2 width=3/
- | -HL12
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /4 width=3/
- ]
-| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie -Hi destruct
- | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/
- ]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- elim (le_inv_plus_l … Hdi) #Hdim #Hi
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/
-]
-qed.
-
-(* Basic forvard lemmas *****************************************************)
-
-(* Basic_1: was: drop_S *)
-lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
- ⇩[O, e + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H
- [ -IHL1 destruct /2 width=1/
- | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
- ]
-]
-qed-.
-
-lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
-[ /2 width=3/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
- >(tw_lift … HV21) -HV21 /2 width=1/
-]
-qed-.
-
-lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
- ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H
- [ -IHL1 destruct //
- | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/
- ]
-]
-qed-.
-
-lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 elim L1 -L1
-[ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
-| #K1 #I1 #V1 #IHL1 #L2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #H
- [ -IHL1 destruct //
- | 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
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop_mono *)
-theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
- ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
-#d #e #L #L1 #H elim H -d -e -L -L1
-[ #d #e #L2 #H
- >(ldrop_inv_atom1 … H) -L2 //
-| #K #I #V #L2 #HL12
- <(ldrop_inv_refl … HL12) -L2 //
-| #L #K #I #V #e #_ #IHLK #L2 #H
- lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/
-| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
- elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
- >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
- >(IHLK1 … HLK2) -IHLK1 -HLK2 //
-]
-qed-.
-
-(* Basic_1: was: drop_conf_ge *)
-theorem ldrop_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 -d1 -e1 -L -L1
-[ #d #e #e2 #L2 #H
- >(ldrop_inv_atom1 … H) -L2 //
-| //
-| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2
- <minus_plus >minus_minus_comm /3 width=1/
-| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
- lapply (transitive_le 1 … Hdee2) // #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2
- lapply (transitive_le (1 + e) … Hdee2) // #Hee2
- @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *)
-]
-qed.
-
-(* Basic_1: was: drop_conf_lt *)
-theorem ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
- ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 &
- ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
-#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
-[ #d #e #e2 #K2 #I #V2 #H
- lapply (ldrop_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 (ldrop_inv_O1 … H) -H *
- [ -IHL12 -He2d #H1 #H2 destruct /2 width=5/
- | -HL12 -HV12 #He #HLK
- elim (IHL12 … HLK ?) -IHL12 -HLK [ <minus_minus /3 width=5/ | /2 width=1/ ] (**) (* a bit slow *)
- ]
-]
-qed.
-
-(* Basic_1: was: drop_trans_le *)
-theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
- ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 →
- ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
-[ #d #e #e2 #L2 #H
- >(ldrop_inv_atom1 … H) -L2 /2 width=3/
-| #K #I #V #e2 #L2 #HL2 #H
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
-| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
- lapply (le_n_O_to_eq … H) -H #H destruct
- elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
- elim (ldrop_inv_O1 … H) -H *
- [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/
- | -HL12 -HV12 #He2 #HL2
- elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
- ]
-]
-qed.
-
-(* Basic_1: was: drop_trans_ge *)
-theorem ldrop_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 -d1 -e1 -L1 -L
-[ #d #e #e2 #L2 #H
- >(ldrop_inv_atom1 … H) -H -L2 //
-| //
-| /3 width=1/
-| #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 (ldrop_inv_ldrop1 … H ?) -H // #HL2
- @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *)
-]
-qed.
-
-theorem ldrop_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 ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
- ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/term_simple.ma".
-
-(* BASIC TERM 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_gref : ∀p,d,e. lift d e (§p) (§p)
-| lift_bind : ∀I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 → lift (d + 1) e T1 T2 →
- lift d e (ⓑ{I} V1. T1) (ⓑ{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 (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
-qed.
-
-lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
-/2 width=4/ 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 width=1/
-| #j #d #e #Hj #i #Hi destruct /3 width=1/
-| #p #d #e #i #H destruct
-| #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 width=3/ 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_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
-#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_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5/ qed-.
-
-fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = ⓑ{I} V1.U1 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T2 = ⓑ{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
-| #p #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] ⓑ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T2 = ⓑ{I} V2. U2.
-/2 width=3/ qed-.
-
-fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T2 = ⓕ{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
-| #p #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] ⓕ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T2 = ⓕ{I} V2. U2.
-/2 width=3/ 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 width=1/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
-| #p #d #e #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed.
-
-(* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
- (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-/2 width=3/ qed-.
-
-(* Basic_1: was: lift_gen_lref_lt *)
-lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-(* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
- d ≤ i → i < d + e → False.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
-[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
-lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
-elim (lt_refl_false … H)
-qed-.
-
-(* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
-qed-.
-
-fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
-#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_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5/ qed-.
-
-fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = ⓑ{I} V2.U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T1 = ⓑ{I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #p #d #e #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
-]
-qed.
-
-(* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{I} V2. U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
- T1 = ⓑ{I} V1. U1.
-/2 width=3/ qed-.
-
-fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T1 = ⓕ{I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #p #d #e #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
-]
-qed.
-
-(* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓕ{I} V2. U2 →
- ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
- T1 = ⓕ{I} V1. U1.
-/2 width=3/ qed-.
-
-lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → False.
-#d #e #J #V elim V -V
-[ * #i #T #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * #I #W2 #U2 #IHW2 #_ #T #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
- ]
-]
-qed-.
-
-lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → False.
-#J #T elim T -T
-[ * #i #V #d #e #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * #I #W2 #U2 #_ #IHU2 #V #d #e #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
- ]
-]
-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-.
-
-lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒[T1] → 𝐒[T2].
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒[T2] → 𝐒[T1].
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-(* 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 ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
-qed.
-
-lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
-/2 width=1/ 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 width=1/
-| * /2 width=1/
-]
-qed.
-
-lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
-#T1 elim T1 -T1
-[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
-| * #I #V1 #T1 #IHV1 #IHT1 #d #e
- elim (IHV1 d e) -IHV1 #V2 #HV12
- [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
- | elim (IHT1 d e) -IHT1 /3 width=2/
- ]
-]
-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 -d1 -e2 -T1 -T2
-[ /3 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
- >(plus_minus_m_m e2 e1 ?) // /3 width=3/
-| /3 width=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) /2 width=1/ /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_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
-#T1 elim T1 -T1
-[ * [1,3: /3 width=2/ ] #i #d #e
- elim (lt_dec i d) #Hid
- [ /4 width=2/
- | lapply (false_lt_to_le … Hid) -Hid #Hid
- elim (lt_dec i (d + e)) #Hide
- [ @or_intror * #T1 #H
- elim (lift_inv_lref2_be … H Hid Hide)
- | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
- ]
- ]
-| * #I #V2 #T2 #IHV2 #IHT2 #d #e
- [ elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
- [ * #T1 #HT12 @or_introl /3 width=2/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2/
- ]
- | elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 d e) -IHT2
- [ * #T1 #HT12 /4 width=2/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2/
- ]
- ]
-]
-qed.
-
-(* Basic_1: removed theorems 7:
- lift_head lift_gen_head
- lift_weight_map lift_weight lift_weight_add lift_weight_add_O
- lift_tlt_dx
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Main properies ***********************************************************)
-
-(* Basic_1: was: lift_inj *)
-theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
-#d #e #T1 #U #H elim H -d -e -T1 -U
-[ #k #d #e #X #HX
- lapply (lift_inv_sort2 … HX) -HX //
-| #i #d #e #Hid #X #HX
- lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/
-| #p #d #e #X #HX
- lapply (lift_inv_gref2 … HX) -HX //
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-]
-qed-.
-
-(* Basic_1: was: lift_gen_lift *)
-theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T →
- d1 ≤ d2 →
- ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
- [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
- | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
- elim (le_inv_plus_l … H) -H #Hide2 #He2i
- lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
- >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
- /4 width=3/
- ]
-| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=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
- elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /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
- elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
- elim (IHU … HU2 ?) // /3 width=5/
-]
-qed.
-
-(* Note: apparently this was missing in Basic_1 *)
-theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
- ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
- e ≤ e1 → e1 ≤ e + e2 →
- ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
- [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/
- | >(lift_inv_lref2_ge … H ?) -H //
- lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
- elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
- @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12
- @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ]
- ]
-| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
- elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2
- elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
- elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- elim (IHV1 … HV2 ? ?) -V //
- elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /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 -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 //
-| #p #d #e #X #HX
- lapply (lift_inv_gref1 … 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 /3 width=1/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
-]
-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 -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 width=1/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
- | -Hd21 /2 width=1/
- ]
-| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
- >(lift_inv_gref1 … HT2) -HT2 //
-| #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
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1/
-| #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
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) // /2 width=1/
-]
-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 -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3/
-| #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 /3 width=3/ /4 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
- lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
- >plus_plus_comm_23 /4 width=3/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #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
- elim (IHV12 … HV20 ?) -IHV12 -HV20 //
- elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /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
- 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 -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3/
-| #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 width=1/ #Hid2e
- lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #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
- elim (IHV12 … HV20 ?) -IHV12 -HV20 //
- elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T
- <plus_minus /2 width=2/ /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20 ?) -IHV12 -HV20 //
- elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/lift_vector.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-(* Main properies ***********************************************************)
-
-theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s →
- ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s.
-#Ts #U1s #d #e #H elim H -Ts -U1s
-[ #U2s #H >(liftv_inv_nil1 … H) -H //
-| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
- elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
- >(lift_mono … HTU1 … HTU2) -T /3 width=1/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/substitution/lift.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-inductive liftv (d,e:nat) : relation (list term) ≝
-| liftv_nil : liftv d e ◊ ◊
-| liftv_cons: ∀T1s,T2s,T1,T2.
- ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s →
- liftv d e (T1 :: T1s) (T2 :: T2s)
-.
-
-interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
-#T1s #T2s #d #e * -T1s -T2s //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
-qed.
-
-lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
-/2 width=5/ qed-.
-
-fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
- ∀U1,U1s. T1s = U1 :: U1s →
- ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
- T2s = U2 :: U2s.
-#T1s #T2s #d #e * -T1s -T2s
-[ #U1 #U1s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5/
-]
-qed.
-
-lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 :: U1s ≡ T2s →
- ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
- T2s = U2 :: U2s.
-/2 width=3/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s.
-#d #e #T1s elim T1s -T1s
-[ /2 width=2/
-| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 d e) /3 width=2/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
-
-(* Basic_1: includes: csubst1_bind *)
-inductive ltps: nat → nat → relation lenv ≝
-| ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
-| ltps_pair: ∀L,I,V. ltps 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
-| ltps_tps2: ∀L1,L2,I,V1,V2,e.
- ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 →
- ltps 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2
-| ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
- ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 →
- ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
-.
-
-interpretation "parallel substritution (local environment)"
- 'PSubst L1 d e L2 = (ltps d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶ L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) /2 width=1/
-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. ⓑ{I} V1 [d, e] ▶ L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) /2 width=1/
-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 width=1/ * /2 width=1/
-qed.
-
-lemma ltps_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → L1 [0, |L2|] ▶ L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-// /3 width=2/ /3 width=3/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltps_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-normalize //
-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 -d -e -L1 -L2 //
-[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
- >(IHL12 ?) -IHL12 // >(tps_inv_refl_O2 … HV12) //
-]
-qed.
-
-lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶ L2 → L1 = L2.
-/2 width=4/ 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. ⓑ{I} V1 →
- ∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
- K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L2 = K2. ⓑ{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 /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶ L2 → 0 < e →
- ∃∃K2,V2. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L2 = K2. ⓑ{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. ⓑ{I} V1 →
- ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L2 = K2. ⓑ{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 /2 width=5/
-]
-qed.
-
-lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶ L2 → 0 < d →
- ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L2 = K2. ⓑ{I} V2.
-/2 width=3/ 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. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
- K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{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 /2 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{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. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{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 /2 width=5/
-]
-qed.
-
-lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=3/ 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
-
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_ldrop_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 -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
-]
-qed.
-
-lemma ltps_ldrop_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 -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
-| //
-| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
- elim (le_inv_plus_l … He12) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
-| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
-]
-qed.
-
-lemma ltps_ldrop_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 -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltps_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ▶ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L [0, d1 + e1 - e2] ▶ L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq … He2) -He2 #H destruct
- lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
- ]
-| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l … Hd1e2) #_ #He2
- <minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
- elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
-]
-qed.
-
-lemma ltps_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ▶ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 [d1 - e2, e1] ▶ L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK01 -HV01 <minus_le_minus_minus_comm //
- elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
-
-lemma ltps_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ▶ L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L [d1 - e2, e1] ▶ L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
-| /2 width=3/
-| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
- lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
- lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
-| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
- lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
- [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
- | -HK10 -HV10 <minus_le_minus_minus_comm //
- elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/substitution/ltps_ldrop.ma".
-
-(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
-
-lemma ltps_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
- ∀L1,d1,e1. L0 [d1, e1] ▶ L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 [d2, e2] ▶ U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ //
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
- lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
- lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-qed.
-
-(* Basic_1: was: subst1_subst1_back *)
-lemma ltps_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
- ∀L1,d1,e1. L0 [d1, e1] ▶ L1 →
- ∃∃T. L1 ⊢ T2 [d2, e2] ▶ T &
- L1 ⊢ U2 [d1, e1] ▶ T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltps_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
- elim (ltps_inv_tps11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltps_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
- elim (ltps_inv_tps21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK1) #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // normalize #HW01
- lapply (tps_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
- ]
- ]
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
- elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -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 -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_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
- @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
-| /3 width=4/
-]
-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 -L0 -T2 -U2 -d2 -e2
-[ /2 width=3/
-| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
- elim (lt_or_ge i2 d1) #Hi2d1
- [ elim (ltps_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
- elim (ltps_inv_tps12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
- | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
- [ elim (ltps_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
- elim (ltps_inv_tps22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
- elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
- lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // normalize #HW01
- lapply (tps_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
- | lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
- ]
- ]
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
- elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
- elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -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.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop.ma".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-inductive tps: nat → nat → lenv → relation term ≝
-| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
-| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K. ⓓ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. ⓑ{I} V2) T1 T2 →
- tps d e L (ⓑ{I} V1. T1) (ⓑ{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 (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "parallel substritution (term)"
- 'PSubst L T1 d e T2 = (tps d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tps_lsubs_conf: ∀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 -L1 -T1 -T2 -d -e
-[ //
-| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
-| /4 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ▶ T.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
- ∃∃T2,T. L ⊢ T1 [d, 1] ▶ T2 & ⇧[d, 1] T ≡ T2.
-#K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK /2 width=4/
- elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i ? ? ?) // /3 width=4/
-| * #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=8/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
- ]
-]
-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 -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 width=4/
-| /4 width=3/
-| /4 width=1/
-]
-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 -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
- lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
- lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
-| normalize /2 width=1/
-| /2 width=1/
-]
-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 -L -T1 -T2 -d -e
-[ /2 width=3/
-| #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 j d) in ⊢ (% → ?); // -Hdj /3 width=4/
- | -Hdi -Hdj #Hid
- generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
- >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 width=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 width=1/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (tps_lsubs_conf … HT1 (L. ⓑ{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 = ⓪{I} →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓ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 /2 width=1/
-| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /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 ⊢ ⓪{I} [d, e] ▶ T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV &
- ⇧[O, i + 1] V ≡ T2 &
- I = LRef i.
-/2 width=3/ 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. ⓓV &
- ⇧[O, i + 1] V ≡ T2.
-#L #T2 #i #d #e #H
-elim (tps_inv_atom1 … H) -H /2 width=1/
-* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
-qed-.
-
-lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶ T2 → T2 = §p.
-#L #T2 #p #d #e #H
-elim (tps_inv_atom1 … H) -H //
-* #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
- ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 &
- L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
- U2 = ⓑ{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 ⊢ ⓑ{I} V1. T1 [d, e] ▶ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 &
- L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
- U2 = ⓑ{I} V2. T2.
-/2 width=3/ qed-.
-
-fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
- ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
- U2 = ⓕ{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 ⊢ ⓕ{I} V1. T1 [d, e] ▶ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
- U2 = ⓕ{I} V2. T2.
-/2 width=3/ 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 -L -T1 -T2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
- elim (lt_refl_false … Hdd)
-| /3 width=1/
-| /3 width=1/
-]
-qed.
-
-lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ▶ T2 → T1 = T2.
-/2 width=6/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → #[T1] ≤ #[T2].
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
-/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop_ldrop.ma".
-include "Basic_2/substitution/tps.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → e = 1 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
-[ //
-| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
- >(le_to_le_to_eq … Hdi ?) /2 width=1/ -d #K #V #HLK
- lapply (ldrop_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 width=1/
-| #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. ⓛ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 -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 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
- lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
-| #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
- @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /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 /3 width=6/
-]
-qed.
-
-lemma tps_lift_be: ∀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 ≤ d → d ≤ dt + et →
- L ⊢ U1 [dt, et + e] ▶ U2.
-#K #T1 #T2 #dt #et #H elim 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 #Hdtd #_
- elim (lift_inv_lref1 … H) -H * #Hid #H destruct
- [ -Hdtd
- lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
- elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
- | -Hdti
- lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
- ]
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
- ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
-| #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 /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 -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
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
-| #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
- @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 /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 -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=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
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
-| #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
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_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
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
- ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ -Hdtd -Hidet
- lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
- | -Hdti -Hdedet
- lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
- elim (le_inv_plus_l … Hid) #Hdie #Hei
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
- ]
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
- /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHV12 … HLK … HWV1 ? ?) -V1 //
- elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /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 -L -U1 -U2 -dt -et
-[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
- | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=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
- elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
- elim (le_inv_plus_l … Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
- @ex2_1_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
-| #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
- elim (le_inv_plus_l … Hdetd) #_ #Hedt
- elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
- <plus_minus // /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
- elim (IHV12 … HLK … HWV1 ?) -V1 //
- elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
-]
-qed.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma tps_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 elim H -L -U1 -U2 -d -e
-[ //
-| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
- elim (lift_inv_lref2 … H) -H * #H
- [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
- elim (lt_refl_false … H)
- ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >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
- >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 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
-qed.
-
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 [dt, d - dt] ▶ T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
-elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem tps_conf_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.
-#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (tps_inv_lref1 … H) -H
- [ #HX destruct /4 width=4/
- | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
- ]
-| #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
- lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
- elim (IHV01 … HV02) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubs_conf … HT2 (L. ⓑ{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
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /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 -L1 -T0 -T1 -d1 -e1
-[ /2 width=3/
-| #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 /4 width=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
- elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
- elim (IHT01 … HT02 ?) -T0
- [ -H #T #HT1 #HT2
- lapply (tps_lsubs_conf … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubs_conf … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /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
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -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 //
- | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
- ]
-| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
- <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
-| #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
- lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- lapply (IHT10 … HT02 He) -T0 #HT12
- lapply (tps_lsubs_conf … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
-| #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 /3 width=1/
-]
-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 width=3/
-| #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 width=1/ -Hde2i1 #HWT2
- <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=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
- lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
- elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
- lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
- lapply (tps_lsubs_conf … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /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
- elim (IHV10 … HV02 ?) -V0 //
- elim (IHT10 … HT02 ?) -T0 // /3 width=6/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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.ma".
-
-(* DELIFT ON TERMS **********************************************************)
-
-definition delift: nat → nat → lenv → relation term ≝
- λd,e,L,T1,T2. ∃∃T. L ⊢ T1 [d, e] ▶* T & ⇧[d, e] T2 ≡ T.
-
-interpretation "delift (term)"
- 'TSubst L T1 d e T2 = (delift d e L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma delift_lsubs_conf: ∀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 * /3 width=3/
-qed.
-
-lemma delift_bind: ∀I,L,V1,V2,T1,T2,d,e.
- L ⊢ V1 [d, e] ≡ V2 → L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 →
- L ⊢ ⓑ{I} V1. T1 [d, e] ≡ ⓑ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
-qed.
-
-lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
- L ⊢ V1 [d, e] ≡ V2 → L ⊢ T1 [d, e] ≡ T2 →
- L ⊢ ⓕ{I} V1. T1 [d, e] ≡ ⓕ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma delift_fwd_sort1: ∀L,U2,d,e,k. L ⊢ ⋆k [d, e] ≡ U2 → U2 = ⋆k.
-#L #U2 #d #e #k * #U #HU
->(tpss_inv_sort1 … HU) -HU #HU2
->(lift_inv_sort2 … HU2) -HU2 //
-qed-.
-
-lemma delift_fwd_gref1: ∀L,U2,d,e,p. L ⊢ §p [d, e] ≡ U2 → U2 = §p.
-#L #U #d #e #p * #U #HU
->(tpss_inv_gref1 … HU) -HU #HU2
->(lift_inv_gref2 … HU2) -HU2 //
-qed-.
-
-lemma delift_fwd_bind1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓑ{I} V1. T1 [d, e] ≡ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
- L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 &
- U2 = ⓑ{I} V2. T2.
-#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
-lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-qed-.
-
-lemma delift_fwd_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓕ{I} V1. T1 [d, e] ≡ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
- L ⊢ T1 [d, e] ≡ T2 &
- U2 = ⓕ{I} V2. T2.
-#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
-elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
-elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
-qed-.
-
-(* Basic Inversion lemmas ***************************************************)
-
-lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≡ T2 → T1 = T2.
-#L #T1 #T2 #d * #T #HT1
->(tpss_inv_refl_O2 … HT1) -HT1 #HT2
->(lift_inv_refl_O2 … HT2) -HT2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_lift.ma".
-include "Basic_2/unfold/delift.ma".
-
-(* DELIFT ON TERMS **********************************************************)
-
-(* Advanced forward lemmas **************************************************)
-
-lemma delift_fwd_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 → i < d → U2 = #i.
-#L #U2 #i #d #e * #U #HU #HU2 #Hid
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_lt … HU2) //
-| * #K #V1 #V2 #Hdi
- lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
-
-lemma delift_fwd_lref1_be: ∀L,U2,d,e,i. L ⊢ #i [d, e] ≡ U2 →
- d ≤ i → i < d + e →
- ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
- K ⊢ V1 [0, d + e - i - 1] ≡ V2 &
- ⇧[0, d] V2 ≡ U2.
-#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
-| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
- elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
-]
-qed-.
-
-lemma delift_fwd_lref1_ge: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 →
- d + e ≤ i → U2 = #(i - e).
-#L #U2 #i #d #e * #U #HU #HU2 #Hdei
-elim (tpss_inv_lref1 … HU) -HU
-[ #H destruct >(lift_inv_lref2_ge … HU2) //
-| * #K #V1 #V2 #_ #Hide
- lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
- elim (lt_refl_false … Hi)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-inductive at: list2 nat nat → relation nat ≝
-| at_nil: ∀i. at ⟠ i i
-| at_lt : ∀des,d,e,i1,i2. i1 < d →
- at des i1 i2 → at ({d, e} :: des) i1 i2
-| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
- at des (i1 + e) i2 → at ({d, e} :: des) i1 i2
-.
-
-interpretation "application (generic relocation with pairs)"
- 'RAt i1 des i2 = (at des i1 i2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact at_inv_nil_aux: ∀des,i1,i2. @[i1] des ≡ i2 → des = ⟠ → i1 = i2.
-#des #i1 #i2 * -des -i1 -i2
-[ //
-| #des #d #e #i1 #i2 #_ #_ #H destruct
-| #des #d #e #i1 #i2 #_ #_ #H destruct
-]
-qed.
-
-lemma at_inv_nil: ∀i1,i2. @[i1] ⟠ ≡ i2 → i1 = i2.
-/2 width=3/ qed-.
-
-fact at_inv_cons_aux: ∀des,i1,i2. @[i1] des ≡ i2 →
- ∀d,e,des0. des = {d, e} :: des0 →
- i1 < d ∧ @[i1] des0 ≡ i2 ∨
- d ≤ i1 ∧ @[i1 + e] des0 ≡ i2.
-#des #i1 #i2 * -des -i1 -i2
-[ #i #d #e #des #H destruct
-| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
-| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
-]
-qed.
-
-lemma at_inv_cons: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- i1 < d ∧ @[i1] des ≡ i2 ∨
- d ≤ i1 ∧ @[i1 + e] des ≡ i2.
-/2 width=3/ qed-.
-
-lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- i1 < d → @[i1] des ≡ i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
-lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false … Hd)
-qed-.
-
-lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
- d ≤ i1 → @[i1 + e] des ≡ i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
-lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false … Hd)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-(* Main properties **********************************************************)
-
-theorem at_mono: ∀des,i,i1. @[i] des ≡ i1 → ∀i2. @[i] des ≡ i2 → i1 = i2.
-#des #i #i1 #H elim H -des -i -i1
-[ #i #x #H <(at_inv_nil … H) -x //
-| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H
- lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1/
-| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H
- lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-inductive minuss: nat → relation (list2 nat nat) ≝
-| minuss_nil: ∀i. minuss i ⟠ ⟠
-| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
- minuss i ({d, e} :: des1) ({d - i, e} :: des2)
-| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
- minuss i ({d, e} :: des1) des2
-.
-
-interpretation "minus (generic relocation with pairs)"
- 'RMinus des1 i des2 = (minuss i des1 des2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠.
-#des1 #des2 #i * -des1 -des2 -i
-[ //
-| #des1 #des2 #d #e #i #_ #_ #H destruct
-| #des1 #des2 #d #e #i #_ #_ #H destruct
-]
-qed.
-
-lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
-/2 width=4/ qed-.
-
-fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
- ∀d,e,des. des1 = {d, e} :: des →
- d ≤ i ∧ des ▭ e + i ≡ des2 ∨
- ∃∃des0. i < d & des ▭ i ≡ des0 &
- des2 = {d - i, e} :: des0.
-#des1 #des2 #i * -des1 -des2 -i
-[ #i #d #e #des #H destruct
-| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/
-| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/
-]
-qed.
-
-lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
- d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
- ∃∃des. i < d & des1 ▭ i ≡ des &
- des2 = {d - i, e} :: des.
-/2 width=3/ qed-.
-
-lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
- d ≤ i → des1 ▭ e + i ≡ des2.
-#des1 #des2 #d #e #i #H
-elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi
-lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
-elim (lt_refl_false … Hi)
-qed-.
-
-lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
- i < d →
- ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} :: des.
-#des1 #des2 #d #e #i #H
-elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid
-lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
-elim (lt_refl_false … Hi)
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/gr2.ma".
-
-(* GENERIC RELOCATION WITH PAIRS ********************************************)
-
-let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with
-[ nil2 ⇒ ⟠
-| cons2 d e des ⇒ {d + i, e} :: pluss des i
-].
-
-interpretation "plus (generic relocation with pairs)"
- 'plus x y = (pluss x y).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠.
-#i * // normalize
-#d #e #des #H destruct
-qed.
-
-lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} :: des2 →
- ∃∃des1. des1 + i = des2 & des = {d - i, e} :: des1.
-#i #d #e #des2 * normalize
-[ #H destruct
-| #d1 #e1 #des1 #H destruct /2 width=3/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop.ma".
-include "Basic_2/unfold/gr2_minus.ma".
-include "Basic_2/unfold/lifts.ma".
-
-(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
-
-inductive ldrops: list2 nat nat → relation lenv ≝
-| ldrops_nil : ∀L. ldrops ⟠ L L
-| ldrops_cons: ∀L1,L,L2,des,d,e.
- ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} :: des) L1 L2
-.
-
-interpretation "generic local environment slicing"
- 'RDropStar des T1 T2 = (ldrops des T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2.
-#L1 #L2 #des * -L1 -L2 -des //
-#L1 #L #L2 #d #e #des #_ #_ #H destruct
-qed.
-
-(* Basic_1: was: drop1_gen_pnil *)
-lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2.
-/2 width=3/ qed-.
-
-fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 →
- ∀d,e,tl. des = {d, e} :: tl →
- ∃∃L. ⇩*[tl] L1 ≡ L & ⇩[d, e] L ≡ L2.
-#L1 #L2 #des * -L1 -L2 -des
-[ #L #d #e #tl #H destruct
-| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
- /2 width=3/
-qed.
-
-(* Basic_1: was: drop1_gen_pcons *)
-lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} :: des] L1 ≡ L2 →
- ∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2.
-/2 width=3/ qed-.
-
-lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 →
- ∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 →
- ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
- ⇩*[des1] K1 ≡ K2 &
- ⇧*[des1] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#I #des #i #des2 #H elim H -des -i -des2
-[ #i #L1 #K2 #V2 #H
- >(ldrops_inv_nil … H) -L1 /2 width=7/
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (ldrops_inv_cons … H) -H #L #HL1 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct
- elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ]
- normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *)
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
- /4 width=7/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: drop1_skip_bind *)
-lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
- ∀I. ⇩*[des + 1] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
-#L1 #L2 #des #H elim H -L1 -L2 -des
-[ #L #V1 #V2 #HV12 #I
- >(lifts_inv_nil … HV12) -HV12 //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
- elim (lifts_inv_cons … H) -H /3 width=5/
-].
-qed.
-
-(* Basic_1: removed theorems 1: drop1_getl_trans
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrop_ldrop.ma".
-include "Basic_2/unfold/ldrops.ma".
-
-(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
-
-(* Properties concerning basic local environment slicing ********************)
-
-lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[des] L1 ≡ L → ∀L2,i. ⇩[0, i] L ≡ L2 →
- ∃∃L0,des0,i0. ⇩[0, i0] L1 ≡ L0 & ⇩*[des0] L0 ≡ L2 &
- @[i] des ≡ i0 & des ▭ i ≡ des0.
-#L1 #L #des #H elim H -L1 -L -des
-[ /2 width=7/
-| #L1 #L3 #L #des3 #d #e #_ #HL3 #IHL13 #L2 #i #HL2
- elim (lt_or_ge i d) #Hid
- [ elim (ldrop_trans_le … HL3 … HL2 ?) -L /2 width=2/ #L #HL3 #HL2
- elim (IHL13 … HL3) -L3 /3 width=7/
- | lapply (ldrop_trans_ge … HL3 … HL2 ?) -L // #HL32
- elim (IHL13 … HL32) -L3 /3 width=7/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/ldrops_ldrop.ma".
-
-(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop1_trans *)
-theorem ldrops_trans: ∀L,L2,des2. ⇩*[des2] L ≡ L2 → ∀L1,des1. ⇩*[des1] L1 ≡ L →
- ⇩*[des2 @ des1] L1 ≡ L2.
-#L #L2 #des2 #H elim H -L -L2 -des2 // /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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.ma".
-include "Basic_2/unfold/gr2_plus.ma".
-
-(* GENERIC TERM RELOCATION **************************************************)
-
-inductive lifts: list2 nat nat → relation term ≝
-| lifts_nil : ∀T. lifts ⟠ T T
-| lifts_cons: ∀T1,T,T2,des,d,e.
- ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} :: des) T1 T2
-.
-
-interpretation "generic relocation (term)"
- 'RLiftStar des T1 T2 = (lifts des T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2.
-#T1 #T2 #des * -T1 -T2 -des //
-#T1 #T #T2 #d #e #des #_ #_ #H destruct
-qed.
-
-lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
-/2 width=3/ qed-.
-
-fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
- ∀d,e,tl. des = {d, e} :: tl →
- ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2.
-#T1 #T2 #des * -T1 -T2 -des
-[ #T #d #e #tl #H destruct
-| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
- /2 width=3/
-qed.
-
-lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} :: des] T1 ≡ T2 →
- ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2.
-/2 width=3/ qed-.
-
-(* Basic_1: was: lift1_sort *)
-lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
-#T2 #k #des elim des -des
-[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #H
- elim (lifts_inv_cons … H) -H #X #H
- >(lift_inv_sort1 … H) -H /2 width=1/
-]
-qed-.
-
-(* Basic_1: was: lift1_lref *)
-lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 →
- ∃∃i2. @[i1] des ≡ i2 & T2 = #i2.
-#T2 #des elim des -des
-[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3/
-| #d #e #des #IH #i1 #H
- elim (lifts_inv_cons … H) -H #X #H1 #H2
- elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
- elim (IH … H2) -IH -H2 /3 width=3/
-]
-qed-.
-
-lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p.
-#T2 #p #des elim des -des
-[ #H <(lifts_inv_nil … H) -H //
-| #d #e #des #IH #H
- elim (lifts_inv_cons … H) -H #X #H
- >(lift_inv_gref1 … H) -H /2 width=1/
-]
-qed-.
-
-(* Basic_1: was: lift1_bind *)
-lemma lifts_inv_bind1: ∀I,T2,des,V1,U1. ⇧*[des] ⓑ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 &
- T2 = ⓑ{I} V2. U2.
-#I #T2 #des elim des -des
-[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5/
-| #d #e #des #IHdes #V1 #U1 #H
- elim (lifts_inv_cons … H) -H #X #H #HT2
- elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
- /3 width=5/
-]
-qed-.
-
-(* Basic_1: was: lift1_flat *)
-lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 &
- T2 = ⓕ{I} V2. U2.
-#I #T2 #des elim des -des
-[ #V1 #U1 #H
- <(lifts_inv_nil … H) -H /2 width=5/
-| #d #e #des #IHdes #V1 #U1 #H
- elim (lifts_inv_cons … H) -H #X #H #HT2
- elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
- /3 width=5/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒[T1] → 𝐒[T2].
-#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_dx/
-qed-.
-
-lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒[T2] → 𝐒[T1].
-#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_sn/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lifts_bind: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
- ∀T1. ⇧*[des + 1] T1 ≡ T2 →
- ⇧*[des] ⓑ{I} V1. T1 ≡ ⓑ{I} V2. T2.
-#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
-[ #V #T1 #H >(lifts_inv_nil … H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
- elim (lifts_inv_cons … H) -H /3 width=3/
-]
-qed.
-
-lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
- ∀T1. ⇧*[des] T1 ≡ T2 →
- ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
-#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
-[ #V #T1 #H >(lifts_inv_nil … H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
- elim (lifts_inv_cons … H) -H /3 width=3/
-]
-qed.
-
-lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
-#des elim des -des /2 width=2/
-#d #e #des #IH #T1
-elim (lift_total T1 d e) #T #HT1
-elim (IH T) -IH /3 width=4/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/unfold/gr2_minus.ma".
-include "Basic_2/unfold/lifts.ma".
-
-(* GENERIC TERM RELOCATION **************************************************)
-
-(* Properties concerning basic term relocation ******************************)
-
-(* Basic_1: was: lift1_xhg (right to left) *)
-lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 →
- ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2.
-#T1 #T #des #H elim H -T1 -T -des
-[ /2 width=3/
-| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2
- elim (IHT13 … HT2) -T #T #HT3 #HT2
- elim (lift_trans_le … HT13 … HT3 ?) -T3 // /3 width=5/
-]
-qed-.
-
-(* Basic_1: was: lift1_free (right to left) *)
-lemma lifts_lift_trans: ∀des,i,i0. @[i] des ≡ i0 →
- ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 →
- ∀T1,T0. ⇧*[des0] T1 ≡ T0 →
- ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 →
- ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2.
-#des elim des -des normalize
-[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2
- <(at_inv_nil … H1) -x #HT02
- lapply (minuss_inv_nil1 … H2) -H2 #H
- >(pluss_inv_nil2 … H) in HT10; -des0 #H
- >(lifts_inv_nil … H) -T1 /2 width=3/
-| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
- elim (at_inv_cons … H1) -H1 * #Hid #Hi0
- [ elim (minuss_inv_cons1_lt … H2 ?) -H2 [2: /2 width=1/ ] #des1 #Hdes1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
- elim (pluss_inv_cons2 … H) -H #des2 #H1 #H2 destruct
- elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
- elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02
- elim (lift_trans_le … HT1 … HT0 ?) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
- | >commutative_plus in Hi0; #Hi0
- lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hdes0
- elim (IHdes … Hi0 … Hdes0 … HT10 … HT02) -IHdes -Hi0 -Hdes0 -T0 #T0 #HT0 #HT02
- elim (lift_split … HT0 d (i+1) ? ? ?) -HT0 /2 width=1/ /3 width=5/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/unfold/lifts_lift.ma".
-include "Basic_2/unfold/lifts_vector.ma".
-
-(* GENERIC RELOCATION *******************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: lifts1_xhg (right to left) *)
-lemma liftsv_liftv_trans_le: ∀T1s,Ts,des. ⇧*[des] T1s ≡ Ts →
- ∀T2s:list term. ⇧[0, 1] Ts ≡ T2s →
- ∃∃T0s. ⇧[0, 1] T1s ≡ T0s & ⇧*[des + 1] T0s ≡ T2s.
-#T1s #Ts #des #H elim H -T1s -Ts
-[ #T1s #H
- >(liftv_inv_nil1 … H) -T1s /2 width=3/
-| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H
- elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct
- elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s
- elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/lifts_lift.ma".
-
-(* GENERIC RELOCATION *******************************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: lift1_lift1 (left to right) *)
-theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 →
- ⇧*[des1 @ des2] T1 ≡ T2.
-#T1 #T #des1 #H elim H -T1 -T -des1 // /3 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_vector.ma".
-include "Basic_2/unfold/lifts.ma".
-
-(* GENERIC TERM VECTOR RELOCATION *******************************************)
-
-inductive liftsv (des:list2 nat nat) : relation (list term) ≝
-| liftsv_nil : liftsv des ◊ ◊
-| liftsv_cons: ∀T1s,T2s,T1,T2.
- ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s →
- liftsv des (T1 :: T1s) (T2 :: T2s)
-.
-
-interpretation "generic relocation (vector)"
- 'RLiftStar des T1s T2s = (liftsv des T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 →
- ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 &
- T2 = Ⓐ V2s. U2.
-#V1s elim V1s -V1s normalize
-[ #T1 #T2 #des #HT12
- @(ex3_2_intro) [3,4: // |1,2: skip | // ] (**) (* explicit constructor *)
-| #V1 #V1s #IHV1s #T1 #X #des #H
- elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
- elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
- @(ex3_2_intro) [4: // |3: /2 width=2/ |1,2: skip | // ] (**) (* explicit constructor *)
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 →
- ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
-#V1s #V2s #des #H elim H -V1s -V2s // /3 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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:predicate lenv. 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 width=3/ qed.
-
-lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
-/2 width=1/ qed.
-
-lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
-#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2
->(ltps_fwd_length … HL2) /3 width=5/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
-#L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
-/2 width=3 by ltps_fwd_length/
-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
->(ltps_inv_atom1 … HL2) -HL2 //
-qed-.
-
-fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
-#d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
-#L2 #L #_ #HL2 #IHL2 #H destruct
-lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
-qed.
-
-lemma ltpss_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. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
- K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{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 /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. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
- L1 = K1. ⓑ{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. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{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 /2 width=5/
-]
-qed.
-
-lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
- K2 ⊢ V1 [d - 1, e] ▶ V2 &
- L1 = K1. ⓑ{I} V1.
-/2 width=1/ qed.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_ldrop.ma".
-include "Basic_2/unfold/ltpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
-
-lemma ltpss_ldrop_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 @(ltpss_ind … H) -L1 // /3 width=6/
-qed.
-
-lemma ltpss_ldrop_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 @(ltpss_ind … H) -L0 // /3 width=6/
-qed.
-
-lemma ltpss_ldrop_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 @(ltpss_ind … H) -L1
-[ /2 width=3/
-| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
- elim (IHL … HL02 Hd1e2 He2de1) -L0 #L0 #HL20 #HL0
- elim (ltps_ldrop_conf_be … HL1 … HL0 Hd1e2 He2de1) -L /3 width=3/
-]
-qed.
-
-lemma ltpss_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. L [0, d1 + e1 - e2] ▶* L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
-[ /2 width=3/
-| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
- elim (ltps_ldrop_trans_be … HL0 … HL02 Hd1e2 He2de1) -L0 #L0 #HL02 #HL0
- elim (IHL … HL0 Hd1e2 He2de1) -L /3 width=3/
-]
-qed.
-
-lemma ltpss_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ▶* L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L2 [d1 - e2, e1] ▶* L & ⇩[0, e2] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
-[ /2 width=3/
-| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #He2d1
- elim (IHL … HL02 He2d1) -L0 #L0 #HL20 #HL0
- elim (ltps_ldrop_conf_le … HL1 … HL0 He2d1) -L /3 width=3/
-]
-qed.
-
-lemma ltpss_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. L [d1 - e2, e1] ▶* L2 & ⇩[0, e2] L1 ≡ L.
-#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
-[ /2 width=3/
-| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #He2d1
- elim (ltps_ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL02 #HL0
- elim (IHL … HL0 He2d1) -L /3 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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_tpss.ma".
-
-(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
-
-(* Main properties **********************************************************)
-
-theorem ltpss_trans_eq: ∀L1,L,L2,d,e.
- L1 [d, e] ▶* L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
-/2 width=3/ qed.
-
-lemma ltpss_tpss2: ∀L1,L2,I,V1,V2,e.
- L1 [0, e] ▶* L2 → L2 ⊢ V1 [0, e] ▶* V2 →
- L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #H @(tpss_ind … H) -V2
-[ /2 width=1/
-| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2 width=1/
-]
-qed.
-
-lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶* V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
- L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶* V2 →
- L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #H @(tpss_ind … H) -V2
-[ /2 width=1/
-| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2 width=1/
-]
-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. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) // /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 width=1/
-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 width=1/ (**) (* /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 width=1/
-| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2
- lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 -HTU2 /2 width=1/
-]
-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 width=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 width=1/ (**) (* /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 width=3/
-| -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 width=3/
-]
-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 width=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 width=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 width=3/
-]
-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 width=1/ qed.
-
-(* Advanced properties ******************************************************)
-
-lemma ltpss_tps2: ∀L1,L2,I,e.
- L1 [0, e] ▶* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ▶ V2 →
- L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #e #H @(ltpss_ind … H) -L2
-[ /3 width=1/
-| #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. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
->(plus_minus_m_m e 1) // /2 width=1/
-qed.
-
-lemma ltpss_tps1: ∀L1,L2,I,d,e. L1 [d, e] ▶* L2 →
- ∀V1,V2. L2 ⊢ V1 [d, e] ▶ V2 →
- L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2
-[ /3 width=1/
-| #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. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
-#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
->(plus_minus_m_m d 1) // /2 width=1/
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma ltpss_fwd_tpss21: ∀e,K1,I,V1,L2. 0 < e → K1. ⓑ{I} V1 [0, e] ▶* L2 →
- ∃∃K2,V2. K1 [0, e - 1] ▶* K2 & K1 ⊢ V1 [0, e - 1] ▶* V2 &
- L2 = K2. ⓑ{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
- 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. ⓑ{I} V1 [d, e] ▶* L2 →
- ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
- K1 ⊢ V1 [d - 1, e] ▶* V2 &
- L2 = K2. ⓑ{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
- 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-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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:predicate term. 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 width=3/ qed.
-
-lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶* T2 →
- ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ▶* T2.
-/3 width=3/ qed.
-
-lemma tpss_refl: ∀d,e,L,T. L ⊢ T [d, e] ▶* T.
-/2 width=1/ qed.
-
-lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 [d, e] ▶* V2 →
- ∀I,T1,T2. L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 →
- L ⊢ ⓑ{I} V1. T1 [d, e] ▶* ⓑ{I} V2. T2.
-#L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #I #T1 #T2 #HT12 elim 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_lsubs_conf … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #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 ⊢ ⓕ{I} V1. T1 [d, e] ▶* ⓕ{I} V2. T2.
-#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #HT12 elim HT12 -T2
- [ /3 width=1/
- | #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) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
-]
-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) -T2
-[ //
-| #T #T2 #_ #HT12 #IHT
- lapply (tps_weak_top … HT12) -HT12 /2 width=3/
-]
-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) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_sort1 … HT2) -HT2 //
-]
-qed-.
-
-(* Note: this can be derived from tpss_inv_atom1 *)
-lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶* T2 → T2 = §p.
-#L #T2 #p #d #e #H @(tpss_ind … H) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT destruct
- >(tps_inv_gref1 … HT2) -HT2 //
-]
-qed-.
-
-lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 [d, e] ▶* U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 &
- L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 &
- U2 = ⓑ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
-]
-qed-.
-
-lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 [d, e] ▶* U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 & L ⊢ T1 [d, e] ▶* T2 &
- U2 = ⓕ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
-[ /2 width=5/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- 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) -T2
-[ //
-| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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. ⓓ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) -U1
-[ /3 width=4/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU … HU0) -IHU #H
- lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
- lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
-]
-qed.
-
-(* Advanced inverion lemmas *************************************************)
-
-lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} [d, e] ▶* T2 →
- T2 = ⓪{I} ∨
- ∃∃K,V1,V2,i. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓ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) -T2
-[ /2 width=1/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
- | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (ldrop_fwd_ldrop2 … HLK) #H
- elim (tps_inv_lift1_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
- @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
- ]
-]
-qed-.
-
-lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ▶* T2 →
- T2 = #i ∨
- ∃∃K,V1,V2. d ≤ i & i < d + e &
- ⇩[O, i] L ≡ K. ⓓV1 &
- K ⊢ V1 [0, d + e - i - 1] ▶* V2 &
- ⇧[O, i + 1] V2 ≡ T2.
-#L #T2 #i #d #e #H
-elim (tpss_inv_atom1 … H) -H /2 width=1/
-* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
-qed-.
-
-lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ▶* T2 →
- ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
-#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
-#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶* T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
- ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- L ⊢ U1 [dt, et] ▶* U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … 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_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-qed.
-
-lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶* T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
- ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
- ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 [dt, et + e] ▶* U2.
-#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … 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_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
-]
-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) -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 width=3/
-]
-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) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
-
-lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶* U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 [dt, et - e] ▶* T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=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) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=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) -U2 //
-#U #U2 #_ #HU2 #IHU destruct
-<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
-qed.
-
-lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶* U2 →
- ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
- dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. K ⊢ T1 [dt, d - dt] ▶* T2 & ⇧[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
-[ /2 width=3/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 width=3/
-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 width=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 width=3/
-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 width=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 width=3/ (**) (* /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
- lapply (ldrop_fwd_lw … HLK1) normalize #H1
- elim (ltps_ldrop_trans_be … HL10 … HLK1 ? ?) -HL10 -HLK1 // /2 width=2/ #X #H #HLK0
- elim (ltps_inv_tps22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
- lapply (tps_fwd_tw … HV01) #H2
- lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 /2 width=1/ -H2 #H
- lapply (IH … HV01 … HK01 ? ?) -IH -HV01 -HK01
- [1,3: // |2,4: skip | normalize /2 width=1/ | /3 width=6/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct
- lapply (tps_lsubs_conf … HT12 (L. ⓑ{I} V1) ?) -HT12 /2 width=1/ #HT12
- lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=2/ ] #HV12
- lapply (IH … HT12 (L0. ⓑ{I} V1) ? ? ?) -IH -HT12 [1,3,5: /2 width=2/ |2,4: skip | normalize // ] -HL0 #HT12
- lapply (tpss_lsubs_conf … HT12 (L0. ⓑ{I} V2) ?) -HT12 /2 width=1/
-| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct
- lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=3/ ]
- lapply (IH … HT12 … HL0 ? ?) -IH -HT12 [1,3,5: normalize // |2,4: skip ] -HL0 /2 width=1/
-]
-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 width=3/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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) -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 width=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 width=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 width=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 width=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) -T2
-[ /2 width=3/
-| #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 /2 width=1/ ]
- /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
-]
-qed.
-
-lemma tpss_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 (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
-lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
-qed.
-
-(* Main properties **********************************************************)
-
-theorem tpss_conf_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 width=3/ qed.
-
-theorem tpss_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.
-/3 width=3/ qed.
-
-theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
- L ⊢ T1 [d, e] ▶* T → L ⊢ T [d, e] ▶* T2 →
- L ⊢ T1 [d, e] ▶* T2.
-/2 width=3/ qed.
-
-theorem tpss_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.
-/3 width=3/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "arithmetics/nat.ma".
-include "Ground_2/star.ma".
-
-(* ARITHMETICAL PROPERTIES **************************************************)
-
-(* Equations ****************************************************************)
-
-lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
-/2 by plus_minus/ qed.
-
-lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
-/2 by plus_minus/ qed.
-
-lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
-#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
-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 width=1/
-qed.
-
-lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
-/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
-
-lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
- a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
-#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/
-qed.
-
-(* inversion & forward lemmas ***********************************************)
-
-axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
-
-axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
-
-lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
-#m #n elim (lt_or_ge m n) /2 width=1/
-#H elim H -m /2 width=1/
-#m #Hm * #H /2 width=1/ /3 width=1/
-qed-.
-
-lemma lt_refl_false: ∀n. n < n → False.
-#n #H elim (lt_to_not_eq … H) -H /2 width=1/
-qed-.
-
-lemma lt_zero_false: ∀n. n < 0 → False.
-#n #H elim (lt_to_not_le … H) -H /2 width=1/
-qed-.
-
-lemma false_lt_to_le: ∀x,y. (x < y → False) → y ≤ x.
-#x #y #H elim (decidable_lt x y) /2 width=1/
-#Hxy elim (H Hxy)
-qed-.
-
-(*
-lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
-/3 width=2/
-
-lemma le_or_ge: ∀m,n. m ≤ n ∨ n ≤ m.
-#m #n elim (lt_or_ge m n) /2 width=1/ /3 width=2/
-qed-.
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* 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 all A (R:predicate A) (l:list A) on l ≝
- match l with
- [ nil ⇒ True
- | cons hd tl ⇒ R hd ∧ all A R tl
- ].
-
-inductive list2 (A1,A2:Type[0]) : Type[0] :=
- | nil2 : list2 A1 A2
- | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
-
-interpretation "nil (list of pairs)" 'Nil2 = (nil2 ? ?).
-
-interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
-
-let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with
-[ nil2 ⇒ l2
-| cons2 a1 a2 tl ⇒ {a1, a2} :: append2 A1 A2 tl l2
-].
-
-interpretation "append (list of pairs)"
- 'Append l1 l2 = (append2 ? ? l1 l2).
-
-let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with
-[ nil2 ⇒ 0
-| cons2 _ _ l ⇒ length2 A1 A2 l + 1
-].
-
-interpretation "length (list of pairs)"
- 'card l = (length2 ? ? l).
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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( ◊ )"
- non associative with precedence 90
- for @{'Nil}.
-
-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 }.
-
-notation "hvbox( ⟠ )"
- non associative with precedence 90
- for @{'Nil2}.
-
-notation "hvbox( { hd1 , break hd2 } :: break tl )"
- non associative with precedence 47
- for @{'Cons $hd1 $hd2 $tl}.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-include "Ground_2/notation.ma".
-
-(* PROPERTIES OF RELATIONS **************************************************)
-
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → False).
-
-definition confluent2: ∀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 transitive2: ∀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. confluent2 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 -a1
-[ #a1 #Ha01 #a2 #Ha02
- elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
- elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
- elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_strip2: ∀A,R1,R2. confluent2 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 -a2
-[ #a2 #Ha02 #a1 #Ha01
- elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
- elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
- elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_confluent2: ∀A,R1,R2.
- confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
-#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
-[ #a1 #Ha01 #a2 #Ha02
- elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
-| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
- elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
- elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_strap1: ∀A,R1,R2. transitive2 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 -a0
-[ #a0 #Ha10 #a2 #Ha02
- elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
- elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
- elim (IHa … Ha0) -a /4 width=3/
-]
-qed.
-
-lemma TC_strap2: ∀A,R1,R2. transitive2 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 -a2
-[ #a2 #Ha02 #a1 #Ha10
- elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
-| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
- elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
- elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
-]
-qed.
-
-lemma TC_transitive2: ∀A,R1,R2.
- transitive2 A R1 R2 → transitive2 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 width=3/
-| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
- elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
- elim (IHa … Ha0) -a /4 width=3/
-]
-qed.
-
-definition NF: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
-
-inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → False) → SN A R S a2) → SN A R S a1
-.
-
-lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
-#A #R #S #a1 #Ha1
-@SN_intro #a2 #HRa12 #HSa12
-elim (HSa12 ?) -HSa12 /2 width=1/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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".
-
-(* TRICOTOMY FUNCTION *******************************************************)
-
-let rec tri (A:Type[0]) m n a b c on m : A ≝
- match m with
- [ O ⇒ match n with [ O ⇒ b | S n ⇒ a ]
- | S m ⇒ match n with [ O ⇒ c | S n ⇒ tri A m n a b c ]
- ].
-
-(* Basic properties *********************************************************)
-
-lemma tri_lt: ∀A,a,b,c,n,m. m < n → tri A m n a b c = a.
-#A #a #b #c #n elim n -n
-[ #m #H elim (lt_zero_false … H)
-| #n #IH #m elim m -m // /3 width=1/
-]
-qed.
-
-lemma tri_eq: ∀A,a,b,c,m. tri A m m a b c = b.
-#A #a #b #c #m elim m -m normalize //
-qed.
-
-lemma tri_gt: ∀A,a,b,c,m,n. n < m → tri A m n a b c = c.
-#A #a #b #c #m elim m -m
-[ #n #H elim (lt_zero_false … H)
-| #m #IH #n elim n -n // /3 width=1/
-]
-qed.
-
+++ /dev/null
-<?xml version="1.0" encoding="utf-8"?>
-<helm_registry>
- <section name="matita">
- <key name="rt_base_dir">$(MATITA_RT_BASE_DIR)</key>
-<!--
- <key name="system">false</key>
- <key name="map_unicode_to_tex">false</key>
- <key name="do_heavy_checks">true</key>
- <key name="include_path">lib</key>
--->
- </section>
- <section name="xoa">
- <key name="output_dir">contribs/lambda_delta/Ground_2/</key>
- <key name="objects">xoa</key>
- <key name="notations">xoa_notation</key>
- <key name="include">basics/pts.ma</key>
- <key name="ex">1 2</key>
- <key name="ex">2 1</key>
- <key name="ex">2 2</key>
- <key name="ex">2 3</key>
- <key name="ex">3 1</key>
- <key name="ex">3 2</key>
- <key name="ex">3 3</key>
- <key name="ex">4 2</key>
- <key name="ex">4 3</key>
- <key name="ex">4 4</key>
- <key name="ex">5 2</key>
- <key name="ex">5 3</key>
- <key name="ex">5 4</key>
- <key name="ex">6 4</key>
- <key name="ex">6 6</key>
- <key name="ex">7 6</key>
- <key name="or">3</key>
- <key name="or">4</key>
- <key name="and">3</key>
- </section>
-</helm_registry>
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 (1, 2) *)
-
-inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝
- | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ?
-.
-
-interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
-
-(* 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 (2, 3) *)
-
-inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝
- | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? 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, 2) *)
-
-inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝
- | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
-
-(* 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).
-
-(* multiple conjunction connective (3) *)
-
-inductive and3 (P0,P1,P2:Prop) : Prop ≝
- | and3_intro: P0 → P1 → P2 → and3 ? ? ?
-.
-
-interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 (1, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }.
-
-(* 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 (2, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
- 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) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
- 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) }.
-
-(* 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, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 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}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$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.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }.
-
-(* 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 }.
-
-(* multiple conjunction connective (3) *)
-
-notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)"
- non associative with precedence 35
- for @{ 'And $P0 $P1 $P2 }.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 width=3/
-qed.
@printf ' %-10s' ''
@printf '\x1B[0m\n'
-# sum ########################################################################
-
-tbls: $(PACKAGES:%=etc/ld_%_sum.tbl)
-
-etc/ld_%_sum.tbl: MAS = $(shell find $* -name "*.ma")
-
-%.tbl: V1 = $(shell ls $(MAS) | wc -l)
-%.tbl: C1 = $(shell grep "inductive \|record " $(MAS) | wc -l)
-%.tbl: C2 = $(shell grep "definition \|let rec " $(MAS) | wc -l)
-%.tbl: C3 = $(shell grep "inductive \|record \|definition \|let rec " $(MAS) | wc -l)
-%.tbl: P1 = $(shell grep "theorem " $(MAS) | wc -l)
-%.tbl: P2 = $(shell grep "lemma " $(MAS) | wc -l)
-%.tbl: P3 = $(shell grep "lemma \|theorem " $(MAS) | wc -l)
-
-etc/ld_%_sum.tbl: $(MAS)
- @mkdir -p etc
- @printf 'Summary for: $*\n'
- @printf 'name "$(basename $(@F))"\n\n' > $@
- @printf 'table {\n' >> $@
- @printf ' class "grey" [ "category"\n' >> $@
- @printf ' [ "objects" * ]\n' >> $@
- @printf ' ]\n' >> $@
- @printf ' class "cyan" [ "volume"\n' >> $@
- @printf ' [ "files" "$(V1)" * ]\n' >> $@
- @printf ' [ 4 ]\n' >> $@
- @printf ' ]\n' >> $@
- @printf ' class "green" [ "propositions"\n' >> $@
- @printf ' [ "theorems" "$(P1)" * ]\n' >> $@
- @printf ' [ "lemmas" "$(P2)" * ]\n' >> $@
- @printf ' [ "total" "$(P3)" * ]\n' >> $@
- @printf ' ]\n' >> $@
- @printf ' class "yellow" [ "concepts"\n' >> $@
- @printf ' [ "declared" "$(C1)" * ]\n' >> $@
- @printf ' [ "defined" "$(C2)" * ]\n' >> $@
- @printf ' [ "total" "$(C3)" * ]\n' >> $@
- @printf ' ]\n' >> $@
- @printf '}\n\n' >> $@
- @printf 'class "component" { 0 }\n\n' >> $@
- @printf 'class "plane" { 1 } { 3 } { 5 }\n\n' >> $@
- @printf 'class "number" { 2 } { 4 } { 6 }\n\n' >> $@
+# summary ####################################################################
+
+define SUMMARY_TEMPLATE
+ TBL_$(1) := $(1)/ld_$(1)_sum.tbl
+ MAS_$(1) := $$(shell find $(1) -name "*.ma")
+ TBLS += $$(TBL_$(1))
+
+ $$(TBL_$(1)): V1 := $$(shell ls $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): V2 := $$(shell cat $$(MAS_$(1)) | wc -c)
+ $$(TBL_$(1)): C1 := $$(shell grep "inductive \|record " $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): C2 := $$(shell grep "definition \|let rec " $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): C3 := $$(shell grep "inductive \|record \|definition \|let rec " $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): P1 := $$(shell grep "theorem " $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): P2 := $$(shell grep "lemma " $$(MAS_$(1)) | wc -l)
+ $$(TBL_$(1)): P3 := $$(shell grep "lemma \|theorem " $$(MAS_$(1)) | wc -l)
+
+ $$(TBL_$(1)): $$(MAS_$(1))
+ @printf ' SUMMARY $(1)\n'
+ @printf 'name "$$(basename $$(@F))"\n\n' > $$@
+ @printf 'table {\n' >> $$@
+ @printf ' class "grey" [ "category"\n' >> $$@
+ @printf ' [ "objects" * ]\n' >> $$@
+ @printf ' ]\n' >> $$@
+ @printf ' class "cyan" [ "sizes"\n' >> $$@
+ @printf ' [ "files" "$$(V1)" ]\n' >> $$@
+ @printf ' [ "bytes" "$$(V2)" ]\n' >> $$@
+ @printf ' [ * ]\n' >> $$@
+ @printf ' ]\n' >> $$@
+ @printf ' class "green" [ "propositions"\n' >> $$@
+ @printf ' [ "theorems" "$$(P1)" ]\n' >> $$@
+ @printf ' [ "lemmas" "$$(P2)" ]\n' >> $$@
+ @printf ' [ "total" "$$(P3)" ]\n' >> $$@
+ @printf ' ]\n' >> $$@
+ @printf ' class "yellow" [ "concepts"\n' >> $$@
+ @printf ' [ "declared" "$$(C1)" ]\n' >> $$@
+ @printf ' [ "defined" "$$(C2)" ]\n' >> $$@
+ @printf ' [ "total" "$$(C3)" ]\n' >> $$@
+ @printf ' ]\n' >> $$@
+ @printf '}\n\n' >> $$@
+ @printf 'class "component" { 0 }\n\n' >> $$@
+ @printf 'class "plane" { 1 } { 3 } { 5 }\n\n' >> $$@
+ @printf 'class "number" { 2 } { 4 } { 6 }\n\n' >> $$@
+endef
+
+$(foreach PKG, $(PACKAGES), $(eval $(call SUMMARY_TEMPLATE,$(PKG))))
+
+tbls: $(TBLS)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tri.ma".
+include "Basic_2/substitution/lift.ma".
+include "Apps_2/functional/notation.ma".
+
+(* RELOCATION ***************************************************************)
+
+let rec flift d e U on U ≝ match U with
+[ TAtom I ⇒ match I with
+ [ Sort _ ⇒ U
+ | LRef i ⇒ #(tri … i d i (i + e) (i + e))
+ | GRef _ ⇒ U
+ ]
+| TPair I V T ⇒ match I with
+ [ Bind2 I ⇒ ⓑ{I} (flift d e V). (flift (d+1) e T)
+ | Flat2 I ⇒ ⓕ{I} (flift d e V). (flift d e T)
+ ]
+].
+
+interpretation "functional relocation" 'Lift d e T = (flift d e T).
+
+(* Main properties **********************************************************)
+
+theorem flift_lift: ∀T,d,e. ⇧[d, e] T ≡ ↑[d, e] T.
+#T elim T -T
+[ * #i #d #e //
+ elim (lt_or_eq_or_gt i d) #Hid normalize
+ [ >(tri_lt ?????? Hid) /2 width=1/
+ | /2 width=1/
+ | >(tri_gt ?????? Hid) /3 width=2/
+ ]
+| * /2/
+]
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem flift_inv_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ↑[d, e] T1 = T2.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
+[ #i #d #e #Hid >(tri_lt ?????? Hid) //
+| #i #d #e #Hid
+ elim (le_to_or_lt_eq … Hid) -Hid #Hid
+ [ >(tri_gt ?????? Hid) //
+ | destruct //
+ ]
+]
+qed-.
+
+(* Derived properties *******************************************************)
+
+lemma flift_join: ∀e1,e2,T. ⇧[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T.
+#e1 #e2 #T
+lapply (flift_lift T 0 (e1+e2)) #H
+elim (lift_split … H e1 e1 ? ? ?) -H // #U #H
+>(flift_inv_lift … H) -H //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "functional" COMPONENT ********************************)
+
+notation "hvbox( ↑ [ d , break e ] break T )"
+ non associative with precedence 55
+ for @{ 'Lift $d $e $T }.
+
+notation "hvbox( [ d ← break V ] break T )"
+ non associative with precedence 55
+ for @{ 'Subst $V $d $T }.
+
+notation "hvbox( T1 ⇨ break T2 )"
+ non associative with precedence 45
+ for @{ 'SRed $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/grammar/genv.ma".
+
+(* REDUCTION AND TYPE MACHINE ***********************************************)
+
+(* machine local environment *)
+inductive xenv: Type[0] ≝
+| XAtom: xenv (* empty *)
+| XQuad: xenv → bind2 → nat → xenv → term → xenv (* entry *)
+.
+
+interpretation "atom (ext. local environment)"
+ 'Star = XAtom.
+
+interpretation "environment construction (quad)"
+ 'DxItem4 L I u K V = (XQuad L I u K V).
+
+(* machine stack *)
+definition stack: Type[0] ≝ list2 xenv term.
+
+(* machine status *)
+record rtm: Type[0] ≝
+{ rg: genv; (* global environment *)
+ ru: nat; (* current de Bruijn's level *)
+ re: xenv; (* extended local environment *)
+ rs: stack; (* application stack *)
+ rt: term (* code *)
+}.
+
+(* initial state *)
+definition rtm_i: genv → term → rtm ≝
+ λG,T. mk_rtm G 0 (⋆) (⟠) T.
+
+(* update code *)
+definition rtm_t: rtm → term → rtm ≝
+ λM,T. match M with
+ [ mk_rtm G u E _ _ ⇒ mk_rtm G u E (⟠) T
+ ].
+
+(* update closure *)
+definition rtm_u: rtm → xenv → term → rtm ≝
+ λM,E,T. match M with
+ [ mk_rtm G u _ _ _ ⇒ mk_rtm G u E (⟠) T
+ ].
+
+(* get global environment *)
+definition rtm_g: rtm → genv ≝
+ λM. match M with
+ [ mk_rtm G _ _ _ _ ⇒ G
+ ].
+
+(* get local reference level *)
+definition rtm_l: rtm → nat ≝
+ λM. match M with
+ [ mk_rtm _ u E _ _ ⇒ match E with
+ [ XAtom ⇒ u
+ | XQuad _ _ u _ _ ⇒ u
+ ]
+ ].
+
+(* get stack *)
+definition rtm_s: rtm → stack ≝
+ λM. match M with
+ [ mk_rtm _ _ _ S _ ⇒ S
+ ].
+
+(* get code *)
+definition rtm_c: rtm → term ≝
+ λM. match M with
+ [ mk_rtm _ _ _ _ T ⇒ T
+ ].
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "Apps_2/functional/rtm.ma".
+
+(* REDUCTION AND TYPE MACHINE ***********************************************)
+
+(* transitions *)
+inductive rtm_step: relation rtm ≝
+| rtm_ldrop : ∀G,u,E,I,t,F,V,S,i.
+ rtm_step (mk_rtm G u (E. ④{I} {t, F, V}) S (#(i + 1)))
+ (mk_rtm G u E S (#i))
+| rtm_ldelta: ∀G,u,E,t,F,V,S.
+ rtm_step (mk_rtm G u (E. ④{Abbr} {t, F, V}) S (#0))
+ (mk_rtm G u F S V)
+| rtm_ltype : ∀G,u,E,t,F,V,S.
+ rtm_step (mk_rtm G u (E. ④{Abst} {t, F, V}) S (#0))
+ (mk_rtm G u F S V)
+| rtm_gdrop : ∀G,I,V,u,E,S,p. p < |G| →
+ rtm_step (mk_rtm (G. ⓑ{I} V) u E S (§p))
+ (mk_rtm G u E S (§p))
+| rtm_gdelta: ∀G,V,u,E,S,p. p = |G| →
+ rtm_step (mk_rtm (G. ⓓV) u E S (§p))
+ (mk_rtm G u E S V)
+| rtm_gtype : ∀G,V,u,E,S,p. p = |G| →
+ rtm_step (mk_rtm (G. ⓛV) u E S (§p))
+ (mk_rtm G u E S V)
+| rtm_tau : ∀G,u,E,S,W,T.
+ rtm_step (mk_rtm G u E S (ⓣW. T))
+ (mk_rtm G u E S T)
+| rtm_appl : ∀G,u,E,S,V,T.
+ rtm_step (mk_rtm G u E S (ⓐV. T))
+ (mk_rtm G u E ({E, V} :: S) T)
+| rtm_beta : ∀G,u,E,F,V,S,W,T.
+ rtm_step (mk_rtm G u E ({F, V} :: S) (ⓛW. T))
+ (mk_rtm G u (E. ④{Abbr} {u, F, V}) S T)
+| rtm_push : ∀G,u,E,W,T.
+ rtm_step (mk_rtm G u E ⟠ (ⓛW. T))
+ (mk_rtm G (u + 1) (E. ④{Abst} {u, E, W}) ⟠ T)
+| rtm_theta : ∀G,u,E,S,V,T.
+ rtm_step (mk_rtm G u E S (ⓓV. T))
+ (mk_rtm G u (E. ④{Abbr} {u, E, V}) S T)
+.
+
+interpretation "sequential reduction (RTM)"
+ 'SRed O1 O2 = (rtm_step O1 O2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/delift_lift.ma".
+include "Apps_2/functional/lift.ma".
+
+(* CORE SUBSTITUTION ********************************************************)
+
+let rec fsubst W d U on U ≝ match U with
+[ TAtom I ⇒ match I with
+ [ Sort _ ⇒ U
+ | LRef i ⇒ tri … i d (#i) (↑[0, i] W) (#(i-1))
+ | GRef _ ⇒ U
+ ]
+| TPair I V T ⇒ match I with
+ [ Bind2 I ⇒ ⓑ{I} (fsubst W d V). (fsubst W (d+1) T)
+ | Flat2 I ⇒ ⓕ{I} (fsubst W d V). (fsubst W d T)
+ ]
+].
+
+interpretation "functional core substitution" 'Subst V d T = (fsubst V d T).
+
+(* Main properties **********************************************************)
+
+theorem fsubst_delift: ∀K,V,T,L,d.
+ ⇩[0, d] L ≡ K. ⓓV → L ⊢ T [d, 1] ≡ [d ← V] T.
+#K #V #T elim T -T
+[ * #i #L #d #HLK normalize in ⊢ (? ? ? ? ? %); /2 width=3/
+ elim (lt_or_eq_or_gt i d) #Hid
+ [ -HLK >(tri_lt ?????? Hid) /3 width=3/
+ | destruct >tri_eq /4 width=4 by tpss_strap, tps_subst, le_n, ex2_1_intro/ (**) (* too slow without trace *)
+ | -HLK >(tri_gt ?????? Hid) /3 width=3/
+ ]
+| * /3 width=1/ /4 width=1/
+]
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem fsubst_inv_delift: ∀K,V,T1,L,T2,d. ⇩[0, d] L ≡ K. ⓓV →
+ L ⊢ T1 [d, 1] ≡ T2 → [d ← V] T1 = T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #T2 #d #HLK #H
+ [ -HLK >(delift_fwd_sort1 … H) -H //
+ | elim (lt_or_eq_or_gt i d) #Hid normalize
+ [ -HLK >(delift_fwd_lref1_lt … H) -H // /2 width=1/
+ | destruct
+ elim (delift_fwd_lref1_be … H ? ?) -H // #K0 #V0 #V2 #HLK0
+ lapply (ldrop_mono … HLK0 … HLK) -HLK0 -HLK #H >minus_plus <minus_n_n #HV2 #HVT2 destruct
+ >(delift_inv_refl_O2 … HV2) -V >(flift_inv_lift … HVT2) -V2 //
+ | -HLK >(delift_fwd_lref1_ge … H) -H // /2 width=1/
+ ]
+ | -HLK >(delift_fwd_gref1 … H) -H //
+ ]
+| * #I #V1 #T1 #IHV1 #IHT1 #L #X #d #HLK #H
+ [ elim (delift_fwd_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 // /2 width=1/
+ | elim (delift_fwd_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ <(IHV1 … HV12) -IHV1 -HV12 // <(IHT1 … HT12) -IHT1 -HT12 //
+ ]
+]
+qed-.
--- /dev/null
+# 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/csuba csubc_csuba
+csubc/drop1 drop1_csubc_trans
+csubc/drop drop_csubc_trans
+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/props drop1_cons_tail
+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_cons_tail
+
+# check #############################################################
+
+lift1/fwd lifts1_flat
+lift1/fwd lifts1_nil
+lift1/fwd lifts1_cons
+lift1/props lift1_free
+lift/props thead_x_lift_y_y
+lift/props lifts_tapp
+
+# waiting ####################################################################
+
+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/pr3 nf2_pr3_unfold
+nf2/pr3 nf2_pr3_confluence
+
+nf2/props nfs2_tapp
+nf2/props nf2_appls_lref
+
+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 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_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
+
+# check #############################################################
+
+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
+pr2/fwd pr2_gen_void
+pr2/props pr2_ctail
+pr2/subst1 pr2_gen_cabbr
+
+pr3/fwd pr3_gen_abst
+pr3/fwd pr3_gen_lref
+pr3/fwd pr3_gen_void
+pr3/fwd pr3_gen_appl
+pr3/fwd pr3_gen_bind
+pr3/iso pr3_iso_appls_abbr
+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/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_flat
+pr3/props pr3_eta
+pr3/subst1 pr3_subst1
+pr3/subst1 pr3_gen_cabbr
+sn3/props sn3_cpr3_trans
+
+sn3/props sn3_shift
+sn3/props sn3_change
+sn3/props sn3_gen_def
+sn3/props sn3_cdelta
+sn3/props sn3_appl_lref
+sn3/props sn3_appl_abbr
+sn3/props sn3_appl_cast
+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_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
+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 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
+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
+
+# check ######################################################################
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrops.ma".
+
+(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
+
+definition CP1 ≝ λRR:lenv→relation term. λRS:relation term.
+ ∀L,k. NF … (RR L) RS (⋆k).
+
+definition CP2 ≝ λRR:lenv→relation term. λRS:relation term.
+ ∀L,K,W,i. ⇩[0,i] L ≡ K. ⓛW → NF … (RR L) RS (#i).
+
+definition CP3 ≝ λRR:lenv→relation term. λRP:lenv→predicate term.
+ ∀L,V,k. RP L (ⓐ⋆k.V) → RP L V.
+
+definition CP4 ≝ λRR:lenv→relation term. λRS:relation term.
+ ∀L0,L,T,T0,d,e. NF … (RR L) RS T →
+ ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → NF … (RR L0) RS T0.
+
+definition CP4s ≝ λRR:lenv→relation term. λRS:relation term.
+ ∀L0,L,des. ⇩*[des] L0 ≡ L →
+ ∀T,T0. ⇧*[des] T ≡ T0 →
+ NF … (RR L) RS T → NF … (RR L0) RS T0.
+
+(* requirements for abstract computation properties *)
+record acp (RR:lenv->relation term) (RS:relation term) (RP:lenv→predicate term) : Prop ≝
+{ cp1: CP1 RR RS;
+ cp2: CP2 RR RS;
+ cp3: CP3 RR RP;
+ cp4: CP4 RR RS
+}.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_lift1 *)
+lemma acp_lifts: ∀RR,RS. CP4 RR RS → CP4s RR RS.
+#RR #RS #HRR #L1 #L2 #des #H elim H -L1 -L2 -des
+[ #L #T1 #T2 #H #HT1
+ <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+ elim (lifts_inv_cons … H) -H /3 width=9/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/lifts_lifts.ma".
+include "Basic_2/unfold/ldrops_ldrops.ma".
+include "Basic_2/static/aaa_lifts.ma".
+include "Basic_2/static/aaa_aaa.ma".
+include "Basic_2/computation/lsubc_ldrops.ma".
+
+(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
+
+(* Main propertis ***********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP.
+ acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,T,A. L1 ⊢ T ÷ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
+ ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 [RP] ⊑ L0 →
+ ⦃L2, T0⦄ [RP] ϵ 〚A〛.
+#RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A
+[ #L #k #L0 #des #HL0 #X #H #L2 #HL20
+ >(lifts_inv_sort1 … H) -H
+ lapply (aacr_acr … H1RP H2RP ⓪) #HAtom
+ @(s2 … HAtom … ◊) // /2 width=2/
+| #I #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20
+ lapply (aacr_acr … H1RP H2RP B) #HB
+ elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b
+ elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1
+ >(at_mono … Hi1 … Hi0) -i1
+ elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct
+ elim (lsubc_ldrop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H
+ elim (lsubc_inv_pair2 … H) -H *
+ [ #K2 #HK20 #H destruct
+ generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2
+ [ elim (lift_total V0 0 (i0 +1)) #V #HV0
+ elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
+ @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *)
+ | @(s2 … HB … ◊) // /2 width=3/
+ ]
+ | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
+ #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b
+ lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
+ >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B
+ elim (lift_total V2 0 (i0 +1)) #V #HV2
+ @(s4 … HB … ◊ … HV2 HLK2)
+ @(s7 … HB … HKV2B) //
+ ]
+| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (aacr_acr … H1RP H2RP A) #HA
+ lapply (aacr_acr … H1RP H2RP B) #HB
+ lapply (s1 … HB) -HB #HB
+ @(s5 … HA … ◊ ◊) // /3 width=5/
+| #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
+ elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+ @(aacr_abst … H1RP H2RP)
+ [ lapply (aacr_acr … H1RP H2RP B) #HB
+ @(s1 … HB) /2 width=5/
+ | -IHB
+ #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
+ elim (lifts_total des3 W0) #W2 #HW02
+ elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
+ lapply (aaa_lifts … L2 W2 … (des @ des3) … HLWB) -HLWB /2 width=3/ #HLW2B
+ @(IHA (L2. ⓛW2) … (des + 1 @ des3 + 1)) -IHA
+ /2 width=3/ /3 width=5/
+ ]
+| #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ /3 width=10/
+| #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (aacr_acr … H1RP H2RP A) #HA
+ lapply (s1 … HA) #H
+ @(s6 … HA … ◊) /2 width=5/ /3 width=5/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L,T,A. L ⊢ T ÷ A → ⦃L, T⦄ [RP] ϵ 〚A〛.
+/2 width=8/ qed.
+
+lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L,T,A. L ⊢ T ÷ A → RP L T.
+#RR #RS #RP #H1RP #H2RP #L #T #A #HT
+lapply (aacr_acr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/aarity.ma".
+include "Basic_2/unfold/gr2_gr2.ma".
+include "Basic_2/unfold/lifts_lift_vector.ma".
+include "Basic_2/unfold/ldrops_ldrop.ma".
+include "Basic_2/computation/acp.ma".
+
+(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:lenv→predicate term.
+ ∀L,T. C L T → RP L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term.
+ ∀L,Vs. all … (RP L) Vs →
+ ∀T. 𝐒[T] → NF … (RR L) RS T → C L (ⒶVs.T).
+
+(* Note: this is Tait's ii *)
+definition S3 ≝ λRP,C:lenv→predicate term.
+ ∀L,Vs,V,T,W. C L (ⒶVs. ⓓV. T) → RP L W → C L (ⒶVs. ⓐV. ⓛW. T).
+
+definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i.
+ C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 →
+ ⇩[0, i] L ≡ K. ⓓV1 → C L (Ⓐ Vs. #i).
+
+definition S5 ≝ λRP,C:lenv→predicate term.
+ ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓV. T).
+
+definition S6 ≝ λRP,C:lenv→predicate term.
+ ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓣW. T).
+
+definition S7 ≝ λC:lenv→predicate term. ∀L2,L1,T1,d,e.
+ C L1 T1 → ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2.
+
+definition S7s ≝ λC:lenv→predicate term.
+ ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
+
+(* properties of the abstract candidate of reducibility *)
+record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝
+{ s1: S1 RP C;
+ s2: S2 RR RS RP C;
+ s3: S3 RP C;
+ s4: S4 RP C;
+ s5: S5 RP C;
+ s6: S6 RP C;
+ s7: S7 C
+}.
+
+(* the abstract candidate of reducibility associated to an atomic arity *)
+let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
+λT. match A with
+[ AAtom ⇒ RP L T
+| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
+ aacr RP A L0 (ⓐV0. T0)
+].
+
+interpretation
+ "candidate of reducibility of an atomic arity (abstract)"
+ 'InEInt RP L T A = (aacr RP A L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: sc3_lift1 *)
+lemma acr_lifts: ∀C. S7 C → S7s C.
+#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
+[ #L #T1 #T2 #H #HT1
+ <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+ elim (lifts_inv_cons … H) -H /3 width=9/
+]
+qed.
+
+lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+ ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 →
+ RP L V → RP L0 V0.
+#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV
+@acr_lifts /width=6/
+@(s7 … HRP)
+qed.
+
+(* Basic_1: was only: sns3_lifts1 *)
+lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+ ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[des] L0 ≡ L →
+ all … (RP L) Vs → all … (RP L0) V0s.
+#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s
+@conj /2 width=1/ /2 width=6 by rp_lifts/
+qed.
+
+(* Basic_1: was:
+ sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
+*)
+lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀A. acr RR RS RP (aacr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
+#B #A #IHB #IHA @mk_acr normalize
+[ #L #T #H
+ lapply (H ? (⋆0) ? ⟠ ? ? ?) -H
+ [1,3: // |2,4: skip
+ | @(s2 … IHB … ◊) // /2 width=2/
+ | #H @(cp3 … H1RP … 0) @(s1 … IHA) //
+ ]
+| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+ lapply (s1 … IHB … HB) #HV0
+ @(s2 … IHA … (V0 :: V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/
+| #L #Vs #U #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
+ elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
+ @(s3 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /4 width=5/
+| #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
+ elim (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
+ >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
+ elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
+ elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
+ elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
+ >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
+ @(s4 … IHA … (V0 :: V0s) … HW12 HL02) /3 width=4/
+| #L #V1s #V2s #HV12s #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+ elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
+ elim (lift_total V10 0 1) #V20 #HV120
+ elim (liftv_total 0 1 V10s) #V20s #HV120s
+ @(s5 … IHA … (V10 :: V10s) (V20 :: V20s)) /2 width=1/ /2 width=6 by rp_lifts/
+ @(HA … (des + 1)) /2 width=1/
+ [ @(s7 … IHB … HB … HV120) /2 width=1/
+ | @lifts_applv //
+ elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
+ >(liftv_mono … HV12s … HV10s) -V1s //
+ ]
+| #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
+ @(s6 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /3 width=4/
+| /3 width=7/
+]
+qed.
+
+lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L,W,T,A,B. RP L W → (
+ ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 →
+ ⦃L0, V0⦄ [RP] ϵ 〚B〛 → ⦃L0. ⓓV0, T0⦄ [RP] ϵ 〚A〛
+ ) →
+ ⦃L, ⓛW. T⦄ [RP] ϵ 〚②B. A〛.
+#RR #RS #RP #H1RP #H2RP #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
+lapply (aacr_acr … H1RP H2RP A) #HCA
+lapply (aacr_acr … H1RP H2RP B) #HCB
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (s1 … HCB) -HCB #HCB
+@(s3 … HCA … ◊) /2 width=6 by rp_lifts/
+@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Basic_1: includes: pr3_pr2 *)
+definition cprs: lenv → relation term ≝
+ λL. TC … (cpr L).
+
+interpretation "context-sensitive parallel computation (term)"
+ 'PRedStar L T1 T2 = (cprs L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
+ ∀T2. L ⊢ T1 ➡* T2 → R T2.
+#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+axiom cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) →
+ ∀T1. L ⊢ T1 ➡* T2 → R T1.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: pr3_refl *)
+lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
+/2 width=1/ qed.
+
+lemma cprs_strap1: ∀L,T1,T,T2.
+ L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2.
+/2 width=3/ qed.
+
+(* Basic_1: was: pr3_step *)
+lemma cprs_strap2: ∀L,T1,T,T2.
+ L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+/2 width=3/ qed.
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+lemma cprs_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡* T2 →
+ ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡* T2.
+/3 width=3/
+qed.
+
+lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 →
+ L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
+#T #T2 #_ #HT2 #IHT2
+@(cprs_strap1 … IHT2) -IHT2 /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_1: was: pr3_gen_sort *)
+lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k.
+#L #U2 #k #H @(cprs_ind … H) -U2 //
+#U2 #U #_ #HU2 #IHU2 destruct
+>(cpr_inv_sort1 … HU2) -HU2 //
+qed-.
+
+(* Basic_1: was: pr3_gen_cast *)
+lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓣW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
+ ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓣW2.T2.
+#L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U2 #U #_ #HU2 * /3 width=3/ *
+#W #T #HW1 #HT1 #H destruct
+elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
+#W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
+qed-.
+
+(* Basic_1: removed theorems 2: clear_pr3_trans pr3_cflat *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr_lift.ma".
+include "Basic_2/reducibility/cpr_cpr.ma".
+include "Basic_2/reducibility/lcpr_cpr.ma".
+include "Basic_2/computation/cprs_lcpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1
+[ /3 width=5/
+| #T1 #T #HT1 #_ #IHT1
+ @(cprs_strap2 … IHT1) -IHT1 /2 width=1/
+]
+qed.
+
+lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+/3 width=1/ qed.
+
+lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12
+lapply (lcpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr3_gen_appl *)
+lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
+ U2 = ⓐV2. T2
+ | ∃∃V2,W,T. L ⊢ V1 ➡* V2 &
+ L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ➡* U2
+ | ∃∃V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
+ L ⊢ T1 ➡* ⓓV. T & L ⊢ ⓓV. ⓐV2. T ➡* U2.
+#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_appl1 … HU2) -HU2 *
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+ | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/
+ | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
+ @or3_intro2 @(ex4_4_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
+ ]
+| /4 width=8/
+| /4 width=10/
+]
+qed-.
+
+(* Main propertis ***********************************************************)
+
+(* Basic_1: was: pr3_confluence *)
+theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 →
+ ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0.
+/3 width=3/ qed.
+
+(* Basic_1: was: pr3_t *)
+theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+/2 width=3/ qed.
+
+lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 →
+ L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+#I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/
+#V #V2 #_ #HV2 #IHV1
+@(cprs_trans … IHV1) -IHV1 /2 width=1/
+qed.
+
+(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
+lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT2
+@(cprs_trans … IHT2) /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/ltpr_tps.ma".
+include "Basic_2/reducibility/cpr_ltpss.ma".
+include "Basic_2/reducibility/lcpr.ma".
+include "Basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties concerning context-sensitive parallel reduction on lenv's *****)
+
+lemma ltpr_tpss_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2,d,e. L2 ⊢ T1 [d, e] ▶* T2 →
+ ∃∃T. L1 ⊢ T1 [d, e] ▶* T & L1 ⊢ T ➡* T2.
+#L1 #L2 #HL12 #T1 #T2 #d #e #H @(tpss_ind … H) -T2
+[ /2 width=3/
+| #T #T2 #_ #HT2 * #T0 #HT10 #HT0
+ elim (ltpr_tps_trans … HT2 … HL12) -L2 #T3 #HT3 #HT32
+ @(ex2_1_intro … HT10) -T1 (**) (* explicit constructors *)
+ @(cprs_strap1 … T3 …) /2 width=1/ -HT32
+ @(cprs_strap1 … HT0) -HT0 /3 width=3/
+]
+qed.
+
+(* Basic_1: was just: pr3_pr0_pr2_t *)
+lemma ltpr_cpr_trans: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #HL12 #T1 #T2 * #T #HT1
+<(ltpr_fwd_length … HL12) #HT2
+elim (ltpr_tpss_trans … HL12 … HT2) -L2 /3 width=3/
+qed.
+
+(* Basic_1: was just: pr3_pr2_pr2_t *)
+lemma lcpr_cpr_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 * /3 width=7/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/cprs_cprs.ma".
+include "Basic_2/computation/lcprs_lcprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Properties exploiting context-senstive computation on local environments *)
+
+(* Basic_1: was just: pr3_pr3_pr3_t *)
+lemma lcprs_cprs_trans: ∀L1,L2. L1 ⊢ ➡* L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #HL12 @(lcprs_ind … HL12) -L2 // /3 width=3/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr3_gen_abbr *)
+lemma cprs_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡* U2 →
+ (∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
+ U2 = ⓓV2. T2
+ ) ∨
+ ∃∃U. ⇧[0, 1] U2 ≡ U & L. ⓓV1 ⊢ T1 ➡* U.
+#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U0 #U2 #_ #HU02 * *
+[ #V0 #T0 #HV10 #HT10 #H destruct
+ elim (cpr_inv_abbr1 … HU02) -HU02 *
+ [ #V #V2 #T2 #HV0 #HV2 #HT02 #H destruct
+ lapply (cpr_intro … HV0 … HV2) -HV2 #HV02
+ lapply (ltpr_cpr_trans (L.ⓓV0) … HT02) /2 width=1/ -V #HT02
+ lapply (lcprs_cprs_trans (L. ⓓV1) … HT02) -HT02 /2 width=1/ /4 width=5/
+ | -V0 #T2 #HT20 #HTU2
+ elim (lift_total U2 0 1) #U0 #HU20
+ lapply (cpr_lift (L.ⓓV1) … HT20 … HU20 HTU2) -T2 /2 width=1/ /4 width=5/
+ ]
+| #U1 #HU01 #HTU1
+ elim (lift_total U2 0 1) #U #HU2
+ lapply (cpr_lift (L.ⓓV1) … HU01 … HU2 HU02) -U0 /2 width=1/ /4 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr_lift.ma".
+include "Basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr3_lift *)
+lemma cprs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
+ ∀T2. K ⊢ T1 ➡* T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 ➡* U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #HT12 @(cprs_ind … HT12) -T2
+[ -HLK #T2 #HT12
+ <(lift_mono … HTU1 … HT12) -T1 //
+| -HTU1 #T #T2 #_ #HT2 #IHT2 #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (cpr_lift … HLK … HTU … HTU2 … HT2) -T2 -HLK /3 width=3/
+]
+qed.
+
+(* Basic_1: was: pr3_gen_lift *)
+lemma cprs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡* U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡* T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 #HU12 @(cprs_ind … HU12) -U2 /2 width=3/
+-HTU1 #U #U2 #_ #HU2 * #T #HTU #HT1
+elim (cpr_inv_lift … HLK … HTU … HU2) -U -HLK /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tstc.ma".
+(*
+include "Basic_2/reducibility/cpr_lift.ma".
+include "Basic_2/reducibility/lcpr_cpr.ma".
+*)
+include "Basic_2/computation/cprs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Forward lemmas involving same top term constructor ***********************)
+(*
+lemma cpr_fwd_beta: ∀L,V,W,T,U. L ⊢ ⓐV. ⓛW. T ➡ U →
+ ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⓓV. T ➡* U.
+#L #V #W #T #U #H
+elim (cpr_inv_appl1 … H) -H *
+[ #V0 #X #_ #_ #H destruct /2 width=1/
+| #V0 #W0 #T1 #T2 #HV0 #HT12 #H1 #H2 destruct
+ lapply (lcpr_cpr_trans (L. ⓓV) … HT12) -HT12 /2 width=1/ /3 width=1/
+| #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma cpr_fwd_theta: ∀L,V1,V,T,U. L ⊢ ⓐV1. ⓓV. T ➡ U →
+ ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1. ⓓV. T ≃ U ∨
+ L ⊢ ⓓV. ⓐV2. T ➡* U.
+#L #V1 #V #T #U #H #V2 #HV12
+elim (cpr_inv_appl1 … H) -H *
+[ -HV12 #V0 #X #_ #_ #H destruct /2 width=1/
+| -HV12 #V0 #W #T1 #T2 #_ #_ #H destruct
+| #V0 #V3 #W1 #W2 #T1 #T2 #HV10 #HW12 #HT12 #HV03 #H1 #H2 destruct
+ lapply (cpr_lift (L.ⓓW1) … HV12 … HV03 … HV10) -V0 -HV12 /2 width=1/ #HV23
+ lapply (lcpr_cpr_trans (L. ⓓW1) … HT12) -HT12 /2 width=1/ #HT12
+ /4 width=1/
+]
+qed-.
+*)
+lemma cprs_fwd_tau: ∀L,W,T,U. L ⊢ ⓣW. T ➡* U →
+ ⓣW. T ≃ U ∨ L ⊢ T ➡* U.
+#L #W #T #U #H
+elim (cprs_inv_cast1 … H) -H /2 width=1/ *
+#W0 #T0 #_ #_ #H destruct /2 width=1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tstc_vector.ma".
+include "Basic_2/substitution/lift_vector.ma".
+include "Basic_2/computation/cprs_tstc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Vector form of forward lemmas involving same top term constructor ********)
+(*
+lemma cpr_fwd_beta_vector: ∀L,V,W,T,U,Vs. L ⊢ ⒶVs. ⓐV. ⓛW. T ➡ U →
+ ⒶVs. ⓐV. ⓛW. T ≃ U ∨ L ⊢ ⒶVs. ⓓV. T ➡* U.
+#L #V #W #T #U * /2 width=1 by cpr_fwd_beta/
+#V0 #Vs #H
+elim (cpr_inv_appl1_simple … H ?) -H
+[ #V1 #T1 #_ #_ #H destruct /2 width=1/
+| elim Vs -Vs //
+]
+qed-.
+
+lemma cpr_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡ U →
+ ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
+#L #V1s #V2s * -V1s -V2s /3 width=1/
+#V1s #V2s #V1a #V2a #HV12a * -V1s -V2s /2 width=1 by cpr_fwd_theta/ -HV12a
+#V1s #V2s #V1b #V2b #_ #_ #V #U #T #H
+elim (cpr_inv_appl1_simple … H ?) -H //
+#V0 #T0 #_ #_ #H destruct /2 width=1/
+qed-.
+*)
+
+(* Basic_1: was: pr3_iso_appls_cast *)
+lemma cprs_fwd_tau_vector: ∀L,Vs,W,T,U. L ⊢ ⒶVs. ⓣW. T ➡* U →
+ ⒶVs. ⓣW. T ≃ U ∨ L ⊢ ⒶVs. T ➡* U.
+#L #Vs elim Vs -Vs /2 width=1 by cprs_fwd_tau/
+#V #Vs #IHVs #W #T #U #H
+elim (cprs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1/
+| #V0 #W0 #T0 #HV0 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+ | @or_intror
+ @(cprs_trans … HU) -HU
+ @(cprs_strap1 … (ⓐV0.ⓛW0.T0)) /2 width=1/ -HV0 -HT0 /3 width=1/
+ ]
+| #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (tstc_inv_bind_appls_simple … HT0 ?) //
+ | @or_intror
+ @(cprs_trans … HU) -HU
+ @(cprs_strap1 … (ⓐV0.ⓓV2.T0)) /2 width=1/ -HV0 -HT0 /3 width=3/
+]
+qed-.
+
+axiom cprs_fwd_theta_vector: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T,U. L ⊢ ⒶV1s. ⓓV. T ➡* U →
+ ⒶV1s. ⓓV. T ≃ U ∨ L ⊢ ⓓV. ⒶV2s. T ➡* U.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr.ma".
+include "Basic_2/reducibility/cnf.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …).
+
+interpretation
+ "context-sensitive strong normalization (term)"
+ 'SN L T = (csn L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csn_ind: ∀L. ∀R:predicate term.
+ (∀T1. L ⊢ ⬇* T1 →
+ (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → R T2) →
+ R T1
+ ) →
+ ∀T. L ⊢ ⬇* T → R T.
+#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: sn3_pr2_intro *)
+lemma csn_intro: ∀L,T1.
+ (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇* T2) → L ⊢ ⬇* T1.
+#L #T1 #H
+@(SN_intro … H)
+qed.
+
+(* Basic_1: was: sn3_nf2 *)
+lemma csn_cnf: ∀L,T. L ⊢ 𝐍[T] → L ⊢ ⬇* T.
+/2 width=1/ qed.
+
+lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬇* T2.
+#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+@csn_intro #T #HLT2 #HT2
+elim (term_eq_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1/
+| -HT1 -HT2 /3 width=4/
+qed.
+
+(* Basic_1: was: sn3_cast *)
+lemma csn_cast: ∀L,W. L ⊢ ⬇* W → ∀T. L ⊢ ⬇* T → L ⊢ ⬇* ⓣW. T.
+#L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
+@csn_intro #X #H1 #H2
+elim (cpr_inv_cast1 … H1) -H1
+[ * #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2
+ [ /3 width=3/
+ | -HLW0 * #H destruct /3 width=1/
+ ]
+| /3 width=3/
+]
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬇* T.
+#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csn_intro #T2 #HLT2 #HT2
+@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_flat *)
+lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
+/2 width=5/ qed-.
+
+(* Basic_1: removed theorems 3: sn3_gen_cflat sn3_cflat sn3_bind *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/acp_aaa.ma".
+include "Basic_2/computation/csn_lcpr_vector.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Properties concerning atomic arity assignment ****************************)
+
+lemma csn_aaa: ∀L,T,A. L ⊢ T ÷ A → L ⊢ ⬇* T.
+#L #T #A #H
+@(acp_aaa … csn_acp csn_acr … H)
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr_cpr.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Advanced forvard lemmas **************************************************)
+
+fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬇* V.
+#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csn_intro #V2 #HLV2 #HV2
+@(IH (②{I} V2. T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_head *)
+lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬇* ②{I} V. T → L ⊢ ⬇* V.
+/2 width=5/ qed.
+
+fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬇* U →
+ ∀I,V,T. U = ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
+#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csn_intro #T2 #HLT2 #HT2
+@(IH (ⓑ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_bind *)
+lemma csn_fwd_bind_dx: ∀I,L,V,T. L ⊢ ⬇* ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
+/2 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/cprs.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Properties concerning context-sensitive computation on terms *************)
+
+definition csns: lenv → predicate term ≝ λL. SN … (cprs L) (eq …).
+
+interpretation
+ "context-sensitive strong normalization (term)"
+ 'SNStar L T = (csns L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma csns_ind: ∀L. ∀R:predicate term.
+ (∀T1. L ⊢ ⬇** T1 →
+ (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
+ ) →
+ ∀T. L ⊢ ⬇** T → R T.
+#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: sn3_intro *)
+lemma csns_intro: ∀L,T1.
+ (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
+#L #T1 #H
+@(SN_intro … H)
+qed.
+
+fact csns_intro_aux: ∀L,T1.
+ (∀T,T2. L ⊢ T ➡* T2 → T1 = T → (T1 = T2 → False) → L ⊢ ⬇** T2) → L ⊢ ⬇** T1.
+/4 width=3/ qed-.
+
+(* Basic_1: was: sn3_pr3_trans (old version) *)
+lemma csns_cprs_trans: ∀L,T1. L ⊢ ⬇** T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇** T2.
+#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+@csns_intro #T #HLT2 #HT2
+elim (term_eq_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1/
+| -HT1 -HT2 /3 width=4/
+qed.
+
+(* Basic_1: was: sn3_pr2_intro (old version) *)
+lemma csns_intro_cpr: ∀L,T1.
+ (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇** T2) →
+ L ⊢ ⬇** T1.
+#L #T1 #H
+@csns_intro_aux #T #T2 #H @(cprs_ind_dx … H) -T
+[ -H #H destruct #H
+ elim (H ?) //
+| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
+ elim (term_eq_dec T0 T) #HT0
+ [ -HLT1 -HLT2 -H /3 width=1/
+ | -IHT -HT12 /4 width=3/
+ ]
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem csn_csns: ∀L,T. L ⊢ ⬇* T → L ⊢ ⬇** T.
+#L #T #H @(csn_ind … H) -T /4 width=1/
+qed.
+
+theorem csns_csn: ∀L,T. L ⊢ ⬇** T → L ⊢ ⬇* T.
+#L #T #H @(csns_ind … H) -T /4 width=1/
+qed.
+
+(* Basic_1: was: sn3_pr3_trans *)
+lemma csn_cprs_trans: ∀L,T1. L ⊢ ⬇* T1 → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ ⬇* T2.
+/4 width=3/ qed.
+
+(* Basic_1: was: nf2_sn3 *)
+lemma csn_cwn: ∀L,T1. L ⊢ ⬇* T1 →
+ ∃∃T2. L ⊢ T1 ➡* T2 & L ⊢ 𝐍[T2].
+#L #T1 #H elim H -T1 #T1 #_ #IHT1
+elim (cnf_dec L T1)
+[ -IHT1 /2 width=3/
+| * #T2 #HLT12 #HT12
+ elim (IHT1 T2 ? ?) -IHT1 // /2 width=1/ -HT12 /3 width=3/
+]
+qed.
+
+(* Main eliminators *********************************************************)
+
+lemma csn_ind_cprs: ∀L. ∀R:predicate term.
+ (∀T1. L ⊢ ⬇* T1 →
+ (∀T2. L ⊢ T1 ➡* T2 → (T1 = T2 → False) → R T2) → R T1
+ ) →
+ ∀T. L ⊢ ⬇* T → R T.
+#L #R #H0 #T1 #H @(csns_ind … (csn_csns … H)) -T1 #T1 #HT1 #IHT1
+@H0 -H0 /2 width=1/ -HT1 /3 width=1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tstc_tstc.ma".
+include "Basic_2/computation/cprs_cprs.ma".
+include "Basic_2/computation/csn_lift.ma".
+include "Basic_2/computation/csn_cpr.ma".
+include "Basic_2/computation/csn_cprs.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma csn_lcpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
+#L1 #L2 #HL12 #T #H @(csn_ind_cprs … H) -T #T #_ #IHT
+@csn_intro #T0 #HLT0 #HT0
+@IHT /2 width=2/ -IHT -HT0 /2 width=3/
+qed.
+
+lemma csn_abbr: ∀L,V. L ⊢ ⬇* V → ∀T. L. ⓓV ⊢ ⬇* T → L ⊢ ⬇* ⓓV. T.
+#L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_cprs … HT) -T #T #HT #IHT
+@csn_intro #X #H1 #H2
+elim (cpr_inv_abbr1 … H1) -H1 *
+[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
+ lapply (cpr_intro … HLV0 HLV01) -HLV01 #HLV1
+ lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
+ elim (eq_false_inv_tpair_sn … H2) -H2
+ [ #HV1 @IHV // /2 width=1/ -HV1
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
+ | -IHV -HLV1 * #H destruct /3 width=1/
+ ]
+| -IHV -IHT -H2 #T0 #HT0 #HLT0
+ lapply (csn_inv_lift … HT … HT0) -HT /2 width=3/
+]
+qed.
+
+fact csn_appl_beta_aux: ∀L,W. L ⊢ ⬇* W → ∀U. L ⊢ ⬇* U →
+ ∀V,T. U = ⓓV. T → L ⊢ ⬇* ⓐV. ⓛW. T.
+#L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V #T #H destruct
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) #HT -HVT
+@csn_intro #X #H #H2
+elim (cpr_inv_appl1 … H) -H *
+[ #V0 #Y #HLV0 #H #H0 destruct
+ elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_beta … H2) -H2
+ [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+ | -IHW -HLW0 -HV -HT * #H #HVT0 destruct
+ @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/
+ ]
+| -IHW -IHVT -H2 #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct
+ lapply (lcpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+| -IHW -IHVT -HV -HT -H2 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: sn3_beta *)
+lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* (ⓓV. T) → (**)
+ L ⊢ ⬇* ⓐV. ⓛW. T.
+/2 width=3/ qed.
+
+fact csn_appl_theta_aux: ∀L,U. L ⊢ ⬇* U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+ ∀V,T. U = ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
+#L #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
+@csn_intro #X #HL #H
+elim (cpr_inv_appl1 … HL) -HL *
+[ -HV #V0 #Y #HLV10 #HL #H0 destruct
+ elim (cpr_inv_abbr1 … HL) -HL *
+ [ #V3 #V4 #T3 #HV3 #HLV34 #HLT3 #H0 destruct
+ lapply (cpr_intro … HV3 HLV34) -HLV34 #HLV34
+ elim (lift_total V0 0 1) #V5 #HV05
+ elim (term_eq_dec (ⓓV.ⓐV2.T) (ⓓV4.ⓐV5.T3))
+ [ -IHVT #H0 destruct
+ elim (eq_false_inv_tpair_sn … H) -H
+ [ -HLV10 -HLV34 -HV3 -HLT3 -HVT
+ >(lift_inj … HV12 … HV05) -V5
+ #H elim (H ?) //
+ | * #_ #H elim (H ?) //
+ ]
+ | -H -HVT #H
+ lapply (cpr_lift (L. ⓓV) … HV12 … HV05 HLV10) -HLV10 -HV12 /2 width=1/ #HV25
+ lapply (ltpr_cpr_trans (L. ⓓV) … HLT3) /2 width=1/ -HLT3 #HLT3
+ @(IHVT … H … HV05) -IHVT // -H -HV05 /3 width=1/
+ ]
+ | -H -IHVT #T0 #HT0 #HLT0
+ @(csn_cpr_trans … (ⓐV1.T0)) /2 width=1/ -V0 -Y
+ @(csn_inv_lift … 0 1 HVT) /2 width=1/
+ ]
+| -HV -HV12 -HVT -IHVT -H #V0 #W0 #T0 #T1 #_ #_ #H destruct
+| -IHVT -H #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HLW01 #HLT01 #HV03 #H1 #H2 destruct
+ lapply (cpr_lift (L. ⓓW0) … HV12 … HV03 HLV10) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
+ lapply (lcpr_cpr_trans (L. ⓓW0) … HLT01) -HLT01 /2 width=1/ #HLT01
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓW0)) /2 width=1/ -W1
+ @(csn_cprs_trans … HVT) -HVT /2 width=1/
+]
+qed.
+
+(* Basic_1: was: sn3_appl_bind *)
+lemma csn_appl_theta: ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
+ ∀L,V,T. L ⊢ ⬇* ⓓV. ⓐV2. T → L ⊢ ⬇* ⓐV1. ⓓV. T.
+/2 width=5/ qed.
+
+(* Basic_1: was only: sn3_appl_appl *)
+lemma csn_appl_simple_tstc: ∀L,V. L ⊢ ⬇* V → ∀T1.
+ L ⊢ ⬇* T1 →
+ (∀T2. L ⊢ T1 ➡* T2 → (T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
+#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csn_intro #X #HL #H
+elim (cpr_inv_appl1_simple … HL ?) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (eq_false_inv_tpair_sn … H) -H
+[ -IHT1 #HV0
+ @(csn_cpr_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
+ @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
+ #T2 #HLT12 #HT12
+ @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+ @H2T1 -H2T1 // -HLT12 /2 width=1/
+| -IHV -H1T1 -HLV0 * #H #H1T10 destruct
+ elim (tstc_dec T1 T0) #H2T10
+ [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
+ #T2 #HLT02 #HT02
+ @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
+ | -IHT1 -H3T1 -H1T10
+ @H2T1 -H2T1 /2 width=1/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/acp_cr.ma".
+include "Basic_2/computation/cprs_tstc_vector.ma".
+include "Basic_2/computation/csn_lcpr.ma".
+include "Basic_2/computation/csn_vector.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
+
+(* Advanced properties ******************************************************)
+(*
+(* Basic_1: was only: sn3_appl_appls *)
+lemma csn_appl_appls_simple_tstc: ∀L,Vs,V,T1. L ⊢ ⬇* V → L ⊢ ⬇* T1 →
+ (∀T2. L ⊢ ⒶVs.T1 ➡* T2 → (ⒶVs.T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒[T1] → L ⊢ ⬇* ⓐV. ⒶVs. T1.
+#L *
+[ #V #T1 #HV
+ @csn_appl_simple_tstc //
+| #V0 #Vs #V #T1 #HV #H1T1 #H2T1 #H3T1
+ @csn_appl_simple_tstc // -HV
+ [ @H2T1
+]
+qed.
+*)
+lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+ ∀V,T. L ⊢ ⬇* ⓓV. ⒶV2s. T → L ⊢ ⬇* V →
+ L ⊢ ⬇* ⒶV1s. ⓓV. T.
+#L #V1s #V2s * -V1s -V2s /2 width=1/
+#V1s #V2s #V1 #V2 #HV12 #H
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1s -V2s /2 width=3/
+#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H #HV
+lapply (csn_appl_theta … HW12 … H) -H -HW12 #H
+lapply (csn_fwd_pair_sn … H) #HW1
+lapply (csn_fwd_flat_dx … H) #H1
+@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2
+elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
+[ -H #H elim (H2 ?) -H2 //
+| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
+]
+qed.
+
+lemma csn_applv_tau: ∀L,W. L ⊢ ⬇* W →
+ ∀Vs,T. L ⊢ ⬇* ⒶVs. T →
+ L ⊢ ⬇* ⒶVs. ⓣW. T.
+#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
+#V #Vs #IHV #T #H1T
+lapply (csn_fwd_pair_sn … H1T) #HV
+lapply (csn_fwd_flat_dx … H1T) #H2T
+@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
+[ #X #H #H0
+ elim (cprs_fwd_tau_vector … H) -H #H
+ [ -H1T elim (H0 ?) -H0 //
+ | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
+ ]
+| -H1T elim Vs -Vs //
+]
+qed.
+(*
+theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
+@mk_acr //
+[
+|
+|
+| #L #V1 #V2 #HV12 #V #T #H #HVT
+ @(csn_applv_theta … HV12) -HV12 //
+ @(csn_abbr) //
+| /2 width=1/
+| @csn_lift
+]
+qed.
+*)
+axiom csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cnf_lift.ma".
+include "Basic_2/computation/acp.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: sn3_lift *)
+lemma csn_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
+ ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → L2 ⊢ ⬇* T2.
+#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+@csn_intro #T #HLT2 #HT2
+elim (cpr_inv_lift … HL21 … HT12 … HLT2) -HLT2 #T0 #HT0 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_mono … HT0 … HT12) in HT2; -T0 /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_lift *)
+lemma csn_inv_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
+ ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → L2 ⊢ ⬇* T2.
+#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+@csn_intro #T #HLT2 #HT2
+elim (lift_total T d e) #T0 #HT0
+lapply (cpr_lift … HL12 … HT21 … HT0 HLT2) -HLT2 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_inj … HT0 … HT21) in HT2; -T0 /2 width=1/
+qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was: sn3_abbr *)
+lemma csn_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → K ⊢ ⬇* V → L ⊢ ⬇* #i.
+#L #K #V #i #HLK #HV
+@csn_intro #X #H #Hi
+elim (cpr_inv_lref1 … H) -H
+[ #H destruct elim (Hi ?) //
+| -Hi * #K0 #V0 #V1 #HLK0 #HV01 #HV1 #_
+ lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
+ @(csn_lift … HLK HV1) -HLK -HV1
+ @(csn_cpr_trans … HV) -HV
+ @(cpr_intro … HV01) -HV01 //
+]
+qed.
+
+lemma csn_abst: ∀L,W. L ⊢ ⬇* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬇* T → L ⊢ ⬇* ⓛW. T.
+#L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT
+@csn_intro #X #H1 #H2
+elim (cpr_inv_abst1 … H1 I V) -H1
+#W0 #T0 #HLW0 #HLT0 #H destruct
+elim (eq_false_inv_tpair_sn … H2) -H2
+[ /3 width=5/
+| -HLW0 * #H destruct /3 width=1/
+]
+qed.
+
+lemma csn_appl_simple: ∀L,V. L ⊢ ⬇* V → ∀T1.
+ (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
+#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
+@csn_intro #X #H1 #H2
+elim (cpr_inv_appl1_simple … H1 ?) // -H1
+#V0 #T0 #HLV0 #HLT10 #H destruct
+elim (eq_false_inv_tpair_dx … H2) -H2
+[ -IHV -HT1 #HT10
+ @(csn_cpr_trans … (ⓐV.T0)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // /2 width=1/
+| -HLT10 * #H #HV0 destruct
+ @IHV -IHV // -HT1 /2 width=1/ -HV0
+ #T2 #HLT02 #HT02
+ @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // -HLT02 /2 width=1/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem csn_acp: acp cpr (eq …) (csn …).
+@mk_acp
+[ /2 width=1/
+| /2 width=3/
+| /2 width=5/
+| @cnf_lift
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
+
+inductive csnv (L:lenv): predicate (list term) ≝
+| csnv_nil: csnv L ◊
+| csn_cons: ∀Ts,T. L ⊢ ⬇* T → csnv L Ts → csnv L (T :: Ts)
+.
+
+interpretation
+ "context-sensitive strong normalization (term vector)"
+ 'SN L Ts = (csnv L Ts).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact csnv_inv_cons_aux: ∀L,Ts. L ⊢ ⬇* Ts → ∀U,Us. Ts = U :: Us →
+ L ⊢ ⬇* U ∧ L ⊢ ⬇* Us.
+#L #Ts * -Ts
+[ #U #Us #H destruct
+| #Ts #T #HT #HTs #U #Us #H destruct /2 width=1/
+]
+qed.
+
+lemma csnv_inv_cons: ∀L,T,Ts. L ⊢ ⬇* T :: Ts → L ⊢ ⬇* T ∧ L ⊢ ⬇* Ts.
+/2 width=3/ qed-.
+
+include "Basic_2/computation/csn_cpr.ma".
+
+lemma csn_fwd_applv: ∀L,T,Vs. L ⊢ ⬇* Ⓐ Vs. T → L ⊢ ⬇* Vs ∧ L ⊢ ⬇* T.
+#L #T #Vs elim Vs -Vs /2 width=1/
+#V #Vs #IHVs #HVs
+lapply (csn_fwd_pair_sn … HVs) #HV
+lapply (csn_fwd_flat_dx … HVs) -HVs #HVs
+elim (IHVs HVs) -IHVs -HVs /3 width=1/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/lcpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
+
+definition lcprs: relation lenv ≝ TC … lcpr.
+
+interpretation
+ "context-sensitive parallel computation (environment)"
+ 'CPRedStar L1 L2 = (lcprs L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma lcprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
+ (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
+ ∀L2. L1 ⊢ ➡* L2 → R L2.
+#L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lcprs_refl: ∀L. L ⊢ ➡* L.
+/2 width=1/ qed.
+
+lemma lcprs_strap1: ∀L1,L,L2.
+ L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
+/2 width=3/ qed.
+
+lemma lcprs_strap2: ∀L1,L,L2.
+ L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
+/2 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/lcpr_cpr.ma".
+include "Basic_2/computation/cprs.ma".
+include "Basic_2/computation/lcprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
+
+(* Advanced properties ******************************************************)
+
+lemma lcprs_pair_dx: ∀I,L1,L2. L1 ⊢ ➡ L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
+ L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
+#I #L1 #L2 #HL12 #V1 #V2 #H @(cprs_ind … H) -V2
+/3 width=1/ /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/lcprs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *************)
+
+(* Main properties **********************************************************)
+
+theorem lcprs_trans: ∀L1,L. L1 ⊢ ➡* L → ∀L2. L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
+/2 width=3/ qed.
+
+lemma lcprs_pair: ∀L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
+ ∀I. L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
+#L1 #L2 #H @(lcprs_ind … H) -L2 /2 width=1/
+#L #L2 #_ #HL2 #IHL1 #V1 #V2 #HV12 #I
+@(lcprs_trans … (L.ⓑ{I}V1)) /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/aaa.ma".
+include "Basic_2/computation/acp_cr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+inductive lsubc (RP:lenv→predicate term): relation lenv ≝
+| lsubc_atom: lsubc RP (⋆) (⋆)
+| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ÷ A →
+ lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+ "local environment refinement (abstract candidates of reducibility)"
+ 'CrSubEq L1 RP L2 = (lsubc RP L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+#RP #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
+ (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#RP #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
+/2 width=4/ qed-.
+
+fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L1 = K1. ⓓV & I = Abst.
+#RP #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+]
+qed.
+
+(* Basic_1: was: csubc_gen_head_l *)
+lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. ⓑ{I} W →
+ (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ÷ A &
+ K1 [RP] ⊑ K2 &
+ L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: csubc_refl *)
+lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
+#RP #L elim L -L // /2 width=1/
+qed.
+
+(* Basic_1: removed theorems 2: csubc_clear_conf csubc_getl_conf *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/aaa_lift.ma".
+include "Basic_2/computation/acp_cr.ma".
+include "Basic_2/computation/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Basic_1: was: csubc_drop_conf_O *)
+(* Note: the constant (0) can not be generalized *)
+lemma lsubc_ldrop_O1_trans: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 [RP] ⊑ K2.
+#RP #L1 #L2 #H elim H -L1 -L2
+[ #X #e #H
+ >(ldrop_inv_atom1 … H) -H /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #X #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H destruct
+ [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=3/
+ | elim (IHL12 … H) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #X #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H destruct
+ [ elim (IHL12 L2 0 ?) -IHL12 // #X #H <(ldrop_inv_refl … H) -H /3 width=7/
+ | elim (IHL12 … H) -L2 /3 width=3/
+ ]
+qed-.
+
+(* Basic_1: was: csubc_drop_conf_rev *)
+lemma ldrop_lsubc_trans: ∀RR,RS,RP.
+ acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
+ ∃∃L2. L1 [RP] ⊑ L2 & ⇩[d, e] L2 ≡ K2.
+#RR #RS #RP #Hacp #Hacr #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H
+ >(lsubc_inv_atom1 … H) -H /2 width=3/
+| #L1 #I #V1 #X #H
+ elim (lsubc_inv_pair1 … H) -H *
+ [ #K1 #HLK1 #H destruct /3 width=3/
+ | #K1 #W1 #A #HV1 #HW1 #HLK1 #H1 #H2 destruct /3 width=3/
+ ]
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+| #L1 #K1 #I #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H
+ elim (lsubc_inv_pair1 … H) -H *
+ [ #K2 #HK12 #H destruct
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+ | #K2 #W2 #A #HV2 #HW2 #HK12 #H1 #H2 destruct
+ elim (IHLK1 … HK12) #K #HL1K #HK2
+ lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA
+ lapply (s7 … HA … HV2 … HLK1 HV21) -HV2
+ elim (lift_total W2 d e) /4 width=9/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/computation/lsubc_ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+(* Properties concerning generic local environment slicing ******************)
+
+(* Basic_1: was: csubc_drop1_conf_rev *)
+lemma ldrops_lsubc_trans: ∀RR,RS,RP.
+ acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,K1,des. ⇩*[des] L1 ≡ K1 → ∀K2. K1 [RP] ⊑ K2 →
+ ∃∃L2. L1 [RP] ⊑ L2 & ⇩*[des] L2 ≡ K2.
+#RR #RS #RP #Hacp #Hacr #L1 #K1 #des #H elim H -L1 -K1 -des
+[ /2 width=3/
+| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12
+ elim (ldrop_lsubc_trans … Hacp Hacr … HLK1 … HK12) -Hacp -Hacr -K1 #K #HLK #HK2
+ elim (IHL … HLK) -IHL -HLK /3 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/lsuba.ma".
+include "Basic_2/computation/acp_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
+
+(* properties concerning lenv refinement for atomic arity assignment ********)
+
+lemma lsubc_lsuba: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+ ∀L1,L2. L1 ÷⊑ L2 → L1 [RP] ⊑ L2.
+#RR #RS #RP #H1RP #H2RP #L1 #L2 #H elim H -L1 -L2
+// /2 width=1/ /3 width=4/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
+
+definition cpc: lenv → relation term ≝
+ λL,T1,T2. L ⊢ T1 ➡ T2 ∨ L ⊢ T2 ➡ T1.
+
+interpretation
+ "context-sensitive parallel conversion (term)"
+ 'PConv L T1 T2 = (cpc L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpc_refl: ∀L,T. L ⊢ T ⬌ T.
+/2 width=1/ qed.
+
+lemma cpc_sym: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → L ⊢ T2 ⬌ T1.
+#L #T1 #T2 * /2 width=1/
+qed.
+
+lemma cpc_cpr: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
+#L #T1 #T2 * /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/conversion/cpc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
+
+(* Main properties **********************************************************)
+
+theorem cpc_conf: ∀L,T0,T1,T2. L ⊢ T0 ⬌ T1 → L ⊢ T0 ⬌ T2 →
+ ∃∃T. L ⊢ T1 ⬌ T & L ⊢ T2 ⬌ T.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/conversion/cpc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+definition cpcs: lenv → relation term ≝
+ λL. TC … (cpc L).
+
+interpretation "context-sensitive parallel equivalence (term)"
+ 'PConvStar L T1 T2 = (cpcs L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
+ ∀T2. L ⊢ T1 ⬌* T2 → R T2.
+#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: pc3_refl *)
+lemma cpcs_refl: ∀L,T. L ⊢ T ⬌* T.
+/2 width=1/ qed.
+
+lemma cpcs_strap1: ∀L,T1,T,T2.
+ L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
+/2 width=3/ qed.
+
+lemma cpcs_strap2: ∀L,T1,T,T2.
+ L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+/2 width=3/ qed.
+
+(* Basic_1: removed theorems 1: clear_pc3_trans *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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
+ * Support for abstract candidates of reducibility closed: 2012 January 27
+ * Confluence of context-sensitive parallel reduction closed: 2011 September 21
+ * Confluence of context-free parallel reduction closed: 2011 September 6
+ * Specification started: 2011 April 17
+ * - Patience on me to gain peace and perfection! -
+ * [ suggested invocation to start formal specifications with ]
+ *)
+
+include "Ground_2/star.ma".
+include "Basic_2/notation.ma".
+
+(* ATOMIC ARITY *************************************************************)
+
+inductive aarity: Type[0] ≝
+ | AAtom: aarity (* atomic aarity construction *)
+ | APair: aarity → aarity → aarity (* binary aarity construction *)
+.
+
+interpretation "aarity construction (atomic)"
+ 'Item0 = AAtom.
+
+interpretation "aarity construction (binary)"
+ 'SnItem2 A1 A2 = (APair A1 A2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma discr_apair_xy_x: ∀A,B. ②B. A = B → False.
+#A #B elim B -B
+[ #H destruct
+| #Y #X #IHY #_ #H destruct
+ -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
+ /2 width=1/
+]
+qed-.
+
+lemma discr_tpair_xy_y: ∀B,A. ②B. A = A → False.
+#B #A elim A -A
+[ #H destruct
+| #Y #X #_ #IHX #H destruct
+ -H (**) (* destruct: the destucted equality is not erased *)
+ /2 width=1/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma aarity_eq_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
+#A1 elim A1 -A1
+[ #A2 elim A2 -A2 /2 width=1/
+ #B2 #A2 #_ #_ @or_intror #H destruct
+| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
+ [ -IHB1 -IHA1 @or_intror #H destruct
+ | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
+ [ #H destruct elim (IHA1 A2) -IHA1
+ [ #H destruct /2 width=1/
+ | #HA12 @or_intror #H destruct /2 width=1/
+ ]
+ | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
+ ]
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 (ⓑ{I} V. T)
+].
+
+interpretation "shift (closure)" 'Append L T = (shift L T).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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→predicate term.
+ (∀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. ⓑ{I} V, T] < #[K, ②{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 width=2/ | skip ]
+qed.
+
+(* Basic_1: removed theorems 6:
+ flt_thead_sx flt_thead_dx flt_arith0 flt_arith1 flt_arith2 flt_trans
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+include "Basic_2/grammar/term.ma".
+
+(* GLOBAL ENVIRONMENTS ******************************************************)
+
+(* global environments *)
+definition genv ≝ list2 bind2 term.
+
+interpretation "sort (global environment)"
+ 'Star = (nil2 bind2 term).
+
+interpretation "environment construction (binary)"
+ 'DxItem2 L I T = (cons2 bind2 term I T L).
+
+interpretation "environment binding construction (binary)"
+ 'DxBind2 L I T = (cons2 bind2 term I T L).
+
+interpretation "abbreviation (global environment)"
+ 'DxAbbr L T = (cons2 bind2 term Abbr T L).
+
+interpretation "abstraction (global environment)"
+ 'DxAbst L T = (cons2 bind2 term Abst T L).
+
+(* Basic properties *********************************************************)
+
+axiom genv_eq_dec: ∀T1,T2:genv. Decidable (T1 = T2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "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 *)
+ | GRef: nat → item0 (* reference by position: 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] ≝
+ | Bind2: bind2 → item2 (* binding item *)
+ | Flat2: flat2 → item2 (* non-binding item *)
+.
+
+coercion Bind2.
+
+coercion Flat2.
+
+(* Basic properties *********************************************************)
+
+axiom item0_eq_dec: ∀I1,I2:item0. Decidable (I1 = I2).
+
+(* Basic_1: was: bind_dec *)
+axiom bind2_eq_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
+
+(* Basic_1: was: flat_dec *)
+axiom flat2_eq_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
+
+(* Basic_1: was: kind_dec *)
+axiom item2_eq_dec: ∀I1,I2:item2. Decidable (I1 = I2).
+
+(* Basic_1: removed theorems 21:
+ 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
+ not_abbr_abst bind_dec_not
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 construction (binary)"
+ 'DxItem2 L I T = (LPair L I T).
+
+interpretation "environment binding construction (binary)"
+ 'DxBind2 L I T = (LPair L I T).
+
+interpretation "abbreviation (local environment)"
+ 'DxAbbr L T = (LPair L Abbr T).
+
+interpretation "abstraction (local environment)"
+ 'DxAbst L T = (LPair L Abst T).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 REFINEMENT FOR SUBSTITUTION ****************************)
+
+inductive lsubs: nat → nat → relation lenv ≝
+| lsubs_sort: ∀d,e. lsubs d e (⋆) (⋆)
+| lsubs_OO: ∀L1,L2. lsubs 0 0 L1 L2
+| lsubs_abbr: ∀L1,L2,V,e. lsubs 0 e L1 L2 →
+ lsubs 0 (e + 1) (L1. ⓓV) (L2.ⓓV)
+| lsubs_abst: ∀L1,L2,I,V1,V2,e. lsubs 0 e L1 L2 →
+ lsubs 0 (e + 1) (L1. ⓛV1) (L2.ⓑ{I} V2)
+| lsubs_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
+ lsubs d e L1 L2 → lsubs (d + 1) e (L1. ⓑ{I1} V1) (L2. ⓑ{I2} V2)
+.
+
+interpretation
+ "local environment refinement (substitution)"
+ 'SubEq L1 d e L2 = (lsubs d e L1 L2).
+
+definition lsubs_conf: ∀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_lsubs_conf: ∀S,R. lsubs_conf S R → lsubs_conf S (λL. (TC … (R L))).
+#S #R #HR #L1 #s1 #s2 #H elim H -s2
+[ /3 width=5/
+| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
+ lapply (HR … Hs2 … HL12) -HR -Hs2 -HL12 /3 width=3/
+]
+qed.
+
+lemma lsubs_bind_eq: ∀L1,L2,e. L1 [0, e] ≼ L2 → ∀I,V.
+ L1. ⓑ{I} V [0, e + 1] ≼ L2.ⓑ{I} V.
+#L1 #L2 #e #HL12 #I #V elim I -I /2 width=1/
+qed.
+
+lemma lsubs_refl: ∀d,e,L. L [d, e] ≼ L.
+#d elim d -d
+[ #e elim e -e // #e #IHe #L elim L -L // /2 width=1/
+| #d #IHd #e #L elim L -L // /2 width=1/
+]
+qed.
+
+lemma lsubs_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≼ L2 → 0 < d →
+ ∀I1,I2,V1,V2. L1. ⓑ{I1} V1 [d, e] ≼ L2. ⓑ{I2} V2.
+
+#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) // /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic forward lemmas ***************************************************)
+
+fact lsubs_fwd_length_full1_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
+ d = 0 → e = |L1| → |L1| ≤ |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubs_fwd_length_full1: ∀L1,L2. L1 [0, |L1|] ≼ L2 → |L1| ≤ |L2|.
+/2 width=5/ qed-.
+
+fact lsubs_fwd_length_full2_aux: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
+ d = 0 → e = |L2| → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize
+[ //
+| /2 width=1/
+| /3 width=1/
+| /3 width=1/
+| #L1 #L2 #_ #_ #_ #_ #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma lsubs_fwd_length_full2: ∀L1,L2. L1 [0, |L2|] ≼ L2 → |L2| ≤ |L1|.
+/2 width=5/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "term construction (atomic)"
+ 'Item0 I = (TAtom I).
+
+interpretation "term construction (binary)"
+ 'SnItem2 I T1 T2 = (TPair I T1 T2).
+
+interpretation "term binding construction (binary)"
+ 'SnBind2 I T1 T2 = (TPair (Bind2 I) T1 T2).
+
+interpretation "term flat construction (binary)"
+ 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
+
+interpretation "sort (term)"
+ 'Star k = (TAtom (Sort k)).
+
+interpretation "local reference (term)"
+ 'LRef i = (TAtom (LRef i)).
+
+interpretation "global reference (term)"
+ 'GRef p = (TAtom (GRef p)).
+
+interpretation "abbreviation (term)"
+ 'SnAbbr T1 T2 = (TPair (Bind2 Abbr) T1 T2).
+
+interpretation "abstraction (term)"
+ 'SnAbst T1 T2 = (TPair (Bind2 Abst) T1 T2).
+
+interpretation "application (term)"
+ 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+
+interpretation "native type annotation (term)"
+ 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: term_dec *)
+axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → False.
+#I #T #V elim V -V
+[ #J #H destruct
+| #J #W #U #IHW #_ #H destruct
+ -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destucted equality is not erased *)
+ /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: thead_x_y_y *)
+lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → False.
+#I #V #T elim T -T
+[ #J #H destruct
+| #J #W #U #_ #IHU #H destruct
+ -H (**) (* destruct: the destucted equality is not erased *)
+ /2 width=1/
+]
+qed-.
+
+lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
+ (②{I} V1. T1 = ②{I} V2. T2 → False) →
+ (V1 = V2 → False) ∨ (V1 = V2 ∧ (T1 = T2 → False)).
+#I #V1 #T1 #V2 #T2 #H
+elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1/
+qed-.
+
+lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
+ (②{I} V1. T1 = ②{I} V2. T2 → False) →
+ (T1 = T2 → False) ∨ (T1 = T2 ∧ (V1 = V2 → False)).
+#I #V1 #T1 #V2 #T2 #H
+elim (term_eq_dec T1 T2) /3 width=1/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1/
+qed-.
+
+lemma eq_false_inv_beta: ∀V1,V2,W1,W2,T1,T2.
+ (ⓐV1. ⓛW1. T1 = ⓐV2. ⓛW2 .T2 →False) →
+ (W1 = W2 → False) ∨
+ (W1 = W2 ∧ (ⓓV1. T1 = ⓓV2. T2 → False)).
+#V1 #V2 #W1 #W2 #T1 #T2 #H
+elim (eq_false_inv_tpair_sn … H) -H
+[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/
+ #H destruct @or_intror @conj // #H destruct /2 width=1/
+| * #H1 #H2 destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/
+ * #H #HT12 destruct
+ @or_intror @conj // #H destruct /2 width=1/
+]
+qed.
+
+(* Basic_1: removed theorems 3:
+ not_void_abst not_abbr_void not_abst_void
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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: predicate term ≝
+ | simple_atom: ∀I. simple (⓪{I})
+ | simple_flat: ∀I,V,T. simple (ⓕ{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 = ⓑ{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[ⓑ{I} V. T] → False.
+/2 width=6/ qed-.
+
+lemma simple_inv_pair: ∀I,V,T. 𝐒[②{I}V.T] → ∃J. I = Flat2 J.
+* /2 width=2/ #I #V #T #H
+elim (simple_inv_bind … H)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+include "Basic_2/grammar/term.ma".
+
+(* TERMS ********************************************************************)
+
+let rec applv Vs T on Vs ≝
+ match Vs with
+ [ nil ⇒ T
+ | cons hd tl ⇒ ⓐhd. (applv tl T)
+ ].
+
+interpretation "application o vevtor (term)"
+ 'SnApplV Vs T = (applv Vs T).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************************************************)
+
+(* Basic_1: was: tweight_lt *)
+lemma tw_pos: ∀T. 1 ≤ #[T].
+#T elim T -T //
+qed.
+
+(* Basic eliminators ********************************************************)
+
+axiom tw_wf_ind: ∀R:predicate term.
+ (∀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
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* SAME HEAD TERM FORMS *****************************************************)
+
+inductive tshf: relation term ≝
+ | tshf_atom: ∀I. tshf (⓪{I}) (⓪{I})
+ | tshf_abst: ∀V1,V2,T1,T2. tshf (ⓛV1. T1) (ⓛV2. T2)
+ | tshf_appl: ∀V1,V2,T1,T2. tshf T1 T2 → 𝐒[T1] → 𝐒[T2] →
+ tshf (ⓐV1. T1) (ⓐV2. T2)
+.
+
+interpretation "same head form (term)" 'napart T1 T2 = (tshf T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tshf_sym: ∀T1,T2. T1 ≈ T2 → T2 ≈ T1.
+#T1 #T2 #H elim H -T1 -T2 /2 width=1/
+qed.
+
+lemma tshf_refl2: ∀T1,T2. T1 ≈ T2 → T2 ≈ T2.
+#T1 #T2 #H elim H -T1 -T2 // /2 width=1/
+qed.
+
+lemma tshf_refl1: ∀T1,T2. T1 ≈ T2 → T1 ≈ T1.
+/3 width=2/ qed.
+
+lemma simple_tshf_repl_dx: ∀T1,T2. T1 ≈ T2 → 𝐒[T1] → 𝐒[T2].
+#T1 #T2 #H elim H -T1 -T2 //
+#V1 #V2 #T1 #T2 #H
+elim (simple_inv_bind … H)
+qed. (**) (* remove from index *)
+
+lemma simple_tshf_repl_sn: ∀T1,T2. T1 ≈ T2 → 𝐒[T2] → 𝐒[T1].
+/3 width=3/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tshf_inv_bind1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓑ{I}W1.U1 →
+ ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
+#T1 #T2 * -T1 -T2
+[ #J #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/
+| #V1 #V2 #T1 #T2 #H_ #_ #_ #I #W1 #U1 #H destruct
+]
+qed.
+
+lemma tshf_inv_bind1: ∀I,W1,U1,T2. ⓑ{I}W1.U1 ≈ T2 →
+ ∃∃W2,U2. I = Abst & T2 = ⓛW2. U2.
+/2 width=5/ qed-.
+
+fact tshf_inv_flat1_aux: ∀T1,T2. T1 ≈ T2 → ∀I,W1,U1. T1 = ⓕ{I}W1.U1 →
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
+ I = Appl & T2 = ⓐW2. U2.
+#T1 #T2 * -T1 -T2
+[ #J #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #HT12 #HT1 #HT2 #I #W1 #U1 #H destruct /2 width=5/
+]
+qed.
+
+lemma tshf_inv_flat1: ∀I,W1,U1,T2. ⓕ{I}W1.U1 ≈ T2 →
+ ∃∃W2,U2. U1 ≈ U2 & 𝐒[U1] & 𝐒[U2] &
+ I = Appl & T2 = ⓐW2. U2.
+/2 width=4/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* SAME TOP TERM CONSTRUCTOR ************************************************)
+
+inductive tstc: relation term ≝
+ | tstc_atom: ∀I. tstc (⓪{I}) (⓪{I})
+ | tstc_pair: ∀I,V1,V2,T1,T2. tstc (②{I} V1. T1) (②{I} V2. T2)
+.
+
+interpretation "same top constructor (term)" 'Iso T1 T2 = (tstc T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tstc_inv_atom1_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T1 = ⓪{I} → T2 = ⓪{I}.
+#T1 #T2 * -T1 -T2 //
+#J #V1 #V2 #T1 #T2 #I #H destruct
+qed.
+
+(* Basic_1: was: iso_gen_sort iso_gen_lref *)
+lemma tstc_inv_atom1: ∀I,T2. ⓪{I} ≃ T2 → T2 = ⓪{I}.
+/2 width=3/ qed-.
+
+fact tstc_inv_pair1_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W1,U1. T1 = ②{I}W1.U1 →
+ ∃∃W2,U2. T2 = ②{I}W2. U2.
+#T1 #T2 * -T1 -T2
+[ #J #I #W1 #U1 #H destruct
+| #J #V1 #V2 #T1 #T2 #I #W1 #U1 #H destruct /2 width=3/
+]
+qed.
+
+(* Basic_1: was: iso_gen_head *)
+lemma tstc_inv_pair1: ∀I,W1,U1,T2. ②{I}W1.U1 ≃ T2 →
+ ∃∃W2,U2. T2 = ②{I}W2. U2.
+/2 width=5/ qed-.
+
+fact tstc_inv_atom2_aux: ∀T1,T2. T1 ≃ T2 → ∀I. T2 = ⓪{I} → T1 = ⓪{I}.
+#T1 #T2 * -T1 -T2 //
+#J #V1 #V2 #T1 #T2 #I #H destruct
+qed.
+
+lemma tstc_inv_atom2: ∀I,T1. T1 ≃ ⓪{I} → T1 = ⓪{I}.
+/2 width=3/ qed-.
+
+fact tstc_inv_pair2_aux: ∀T1,T2. T1 ≃ T2 → ∀I,W2,U2. T2 = ②{I}W2.U2 →
+ ∃∃W1,U1. T1 = ②{I}W1. U1.
+#T1 #T2 * -T1 -T2
+[ #J #I #W2 #U2 #H destruct
+| #J #V1 #V2 #T1 #T2 #I #W2 #U2 #H destruct /2 width=3/
+]
+qed.
+
+lemma tstc_inv_pair2: ∀I,T1,W2,U2. T1 ≃ ②{I}W2.U2 →
+ ∃∃W1,U1. T1 = ②{I}W1. U1.
+/2 width=5/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+lemma tstc_refl: ∀T. T ≃ T.
+#T elim T -T //
+qed.
+
+lemma tstc_sym: ∀T1,T2. T1 ≃ T2 → T2 ≃ T1.
+#T1 #T2 #H elim H -T1 -T2 //
+qed.
+
+lemma tstc_dec: ∀T1,T2. Decidable (T1 ≃ T2).
+* #I1 [2: #V1 #T1 ] * #I2 [2,4: #V2 #T2 ]
+[ elim (item2_eq_dec I1 I2) #HI12
+ [ destruct /2 width=1/
+ | @or_intror #H
+ elim (tstc_inv_pair1 … H) -H #V #T #H destruct /2 width=1/
+ ]
+| @or_intror #H
+ lapply (tstc_inv_atom1 … H) -H #H destruct
+| @or_intror #H
+ lapply (tstc_inv_atom2 … H) -H #H destruct
+| elim (item0_eq_dec I1 I2) #HI12
+ [ destruct /2 width=1/
+ | @or_intror #H
+ lapply (tstc_inv_atom2 … H) -H #H destruct /2 width=1/
+ ]
+]
+qed.
+
+lemma simple_tstc_repl_dx: ∀T1,T2. T1 ≃ T2 → 𝐒[T1] → 𝐒[T2].
+#T1 #T2 * -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #H
+elim (simple_inv_pair … H) -H #J #H destruct //
+qed. (**) (* remove from index *)
+
+lemma simple_tstc_repl_sn: ∀T1,T2. T1 ≃ T2 → 𝐒[T2] → 𝐒[T1].
+/3 width=3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tstc.ma".
+
+(* SAME TOP TERM CONSTRUCTOR ************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: iso_trans *)
+theorem tstc_trans: ∀T1,T. T1 ≃ T → ∀T2. T ≃ T2 → T1 ≃ T2.
+#T1 #T * -T1 -T //
+#I #V1 #V #T1 #T #X #H
+elim (tstc_inv_pair1 … H) -H #V2 #T2 #H destruct //
+qed.
+
+theorem tstc_canc_sn: ∀T,T1. T ≃ T1 → ∀T2. T ≃ T2 → T1 ≃ T2.
+/3 width=3/ qed.
+
+theorem tstc_canc_dx: ∀T1,T. T1 ≃ T → ∀T2. T2 ≃ T → T1 ≃ T2.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/grammar/tstc.ma".
+
+(* SAME TOP TERM CONSTRUCTOR ************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
+lemma tstc_inv_bind_appls_simple: ∀I,Vs,V2,T1,T2. ⒶVs.T1 ≃ ⓑ{I} V2. T2 →
+ 𝐒[T1] → False.
+#I #Vs #V2 #T1 #T2 #H
+elim (tstc_inv_pair2 … H) -H #V0 #T0
+elim Vs -Vs normalize
+[ #H destruct #H
+ @(simple_inv_bind … H)
+| #V #Vs #_ #H destruct
+]
+qed.
+
--- /dev/null
+NAMING CONVENTIONS FOR METAVARIABLES
+
+A,B : arity
+C,D : candidate of reducibility
+E,F : RTM environment
+G : global environment
+H : reserved: transient premise
+IH : reserved: inductive premise
+I,J : item
+K,L : local environment
+M,N : reserved: future use
+O :
+P,Q : reserved: future use
+R : generic predicate (relation)
+S : RTM stack
+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
+p,q : global reference position
+t,u : local reference position level (de Bruijn's)
+x,y,z : reserved: transient objet denoted by a small letter
+
+NAMING CONVENTIONS FOR CONSTRUCTORS
+
+0: atomic
+2: binary
+
+A: application to vector
+
+a: application
+b: binder
+d: abbreviation
+f: flat
+l: abstraction
+t: native type annotation
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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( ⓪ )"
+ non associative with precedence 90
+ for @{ 'Item0 }.
+
+notation "hvbox( ⓪ { I } )"
+ non associative with precedence 90
+ for @{ 'Item0 $I }.
+
+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 i )"
+ non associative with precedence 90
+ for @{ 'LRef $i }.
+
+notation "hvbox( § term 90 p )"
+ non associative with precedence 90
+ for @{ 'GRef $p }.
+
+notation "hvbox( ② term 90 T1 . break term 90 T )"
+ non associative with precedence 90
+ for @{ 'SnItem2 $T1 $T }.
+
+notation "hvbox( ② { I } break term 90 T1 . break term 90 T )"
+ non associative with precedence 90
+ for @{ 'SnItem2 $I $T1 $T }.
+
+notation "hvbox( ⓑ { I } break term 90 T1 . break term 90 T )"
+ non associative with precedence 90
+ for @{ 'SnBind2 $I $T1 $T }.
+
+notation "hvbox( ⓕ { I } break term 90 T1 . break term 90 T )"
+ non associative with precedence 90
+ for @{ 'SnFlat2 $I $T1 $T }.
+
+notation "hvbox( ⓓ term 90 T1 . break term 90 T2 )"
+ non associative with precedence 90
+ for @{ 'SnAbbr $T1 $T2 }.
+
+notation "hvbox( ⓛ term 90 T1 . break term 90 T2 )"
+ non associative with precedence 90
+ for @{ 'SnAbst $T1 $T2 }.
+
+notation "hvbox( ⓐ term 90 T1 . break term 90 T2 )"
+ non associative with precedence 90
+ for @{ 'SnAppl $T1 $T2 }.
+
+notation "hvbox( ⓣ term 90 T1 . break term 90 T2 )"
+ non associative with precedence 90
+ for @{ 'SnCast $T1 $T2 }.
+
+notation "hvbox( Ⓐ term 90 T1 . break term 90 T )"
+ non associative with precedence 90
+ for @{ 'SnApplV $T1 $T }.
+
+notation > "hvbox( T . break ②{ I } break term 47 T1 )"
+ non associative with precedence 46
+ for @{ 'DxBind2 $T $I $T1 }.
+
+notation "hvbox( T . break ⓑ { I } break term 90 T1 )"
+ non associative with precedence 89
+ for @{ 'DxBind2 $T $I $T1 }.
+
+notation "hvbox( T1 . break ⓓ T2 )"
+ left associative with precedence 48
+ for @{ 'DxAbbr $T1 $T2 }.
+
+notation "hvbox( T1 . break ⓛ T2 )"
+ left associative with precedence 49
+ for @{ 'DxAbst $T1 $T2 }.
+
+notation "hvbox( T . break ④ { I } break { T1 , break T2 , break T3 } )"
+ non associative with precedence 47
+ for @{ 'DxItem4 $T $I $T1 $T2 $T3 }.
+
+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( L ⊢ break term 90 T1 ≈ break T2 )"
+ non associative with precedence 45
+ for @{ 'Hom $L $T1 $T2 }.
+
+notation "hvbox( T1 ≃ break T2 )"
+ non associative with precedence 45
+ for @{ 'Iso $T1 $T2 }.
+
+notation "hvbox( T1 break [ d , break e ] ≼ break T2 )"
+ non associative with precedence 45
+ for @{ 'SubEq $T1 $d $e $T2 }.
+
+(* Substitution *************************************************************)
+
+notation "hvbox( L ⊢ break [ d , break e ] break 𝐑 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Reducible $L $d $e $T }.
+
+notation "hvbox( L ⊢ break [ d , break e ] break 𝐈 [ T ] )"
+ non associative with precedence 45
+ for @{ 'NotReducible $L $d $e $T }.
+
+notation "hvbox( L ⊢ break [ d , break e ] break 𝐍 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Normal $L $d $e $T }.
+
+notation "hvbox( ⇧ [ d , break e ] break T1 ≡ break T2 )"
+ non associative with precedence 45
+ for @{ 'RLift $d $e $T1 $T2 }.
+
+notation "hvbox( ⇩ [ e ] break L1 ≡ break L2 )"
+ non associative with precedence 45
+ for @{ 'RDrop $e $L1 $L2 }.
+
+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 f ≡ break T2 )"
+ non associative with precedence 45
+ for @{ 'RAt $T1 $f $T2 }.
+
+notation "hvbox( T1 ▭ break T2 ≡ break T )"
+ non associative with precedence 45
+ for @{ 'RMinus $T1 $T2 $T }.
+
+notation "hvbox( ⇧ * [ e ] break T1 ≡ break T2 )"
+ non associative with precedence 45
+ for @{ 'RLiftStar $e $T1 $T2 }.
+
+notation "hvbox( ⇩ * [ e ] break L1 ≡ break L2 )"
+ non associative with precedence 45
+ for @{ 'RDropStar $e $L1 $L2 }.
+
+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 }.
+
+notation "hvbox( T1 break [ d , break e ] ≡ break T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $T1 $d $e $T2 }.
+
+notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≡ break T2 )"
+ non associative with precedence 45
+ for @{ 'TSubst $L $T1 $d $e $T2 }.
+
+(* Static Typing ************************************************************)
+
+notation "hvbox( L ⊢ break term 90 T ÷ break A )"
+ non associative with precedence 45
+ for @{ 'AtomicArity $L $T $A }.
+
+notation "hvbox( T1 ÷ ⊑ break T2 )"
+ non associative with precedence 45
+ for @{ 'CrSubEqA $T1 $T2 }.
+
+(* Reducibility *************************************************************)
+
+notation "hvbox( 𝐑 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Reducible $T }.
+
+notation "hvbox( L ⊢ break 𝐑 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Reducible $L $T }.
+
+notation "hvbox( 𝐈 [ T ] )"
+ non associative with precedence 45
+ for @{ 'NotReducible $T }.
+
+notation "hvbox( L ⊢ break 𝐈 [ T ] )"
+ non associative with precedence 45
+ for @{ 'NotReducible $L $T }.
+
+notation "hvbox( 𝐍 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Normal $T }.
+
+notation "hvbox( L ⊢ break 𝐍 [ T ] )"
+ non associative with precedence 45
+ for @{ 'Normal $L $T }.
+
+notation "hvbox( 𝐖𝐇𝐑 [ T ] )"
+ non associative with precedence 45
+ for @{ 'WHdReducible $T }.
+
+notation "hvbox( L ⊢ break 𝐖𝐇𝐑 [ T ] )"
+ non associative with precedence 45
+ for @{ 'WHdReducible $L $T }.
+
+notation "hvbox( 𝐖𝐇𝐈 [ T ] )"
+ non associative with precedence 45
+ for @{ 'NotWHdReducible $T }.
+
+notation "hvbox( L ⊢ break 𝐖𝐇𝐈 [ T ] )"
+ non associative with precedence 45
+ for @{ 'NotWHdReducible $L $T }.
+
+notation "hvbox( 𝐖𝐇𝐍 [ T ] )"
+ non associative with precedence 45
+ for @{ 'WHdNormal $T }.
+
+notation "hvbox( L ⊢ break 𝐖𝐇𝐍 [ T ] )"
+ non associative with precedence 45
+ for @{ 'WHdNormal $L $T }.
+
+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 }.
+
+notation "hvbox( ⬇ * T )"
+ non associative with precedence 45
+ for @{ 'SN $T }.
+
+notation "hvbox( L ⊢ ⬇ * T )"
+ non associative with precedence 45
+ for @{ 'SN $L $T }.
+
+notation "hvbox( L ⊢ ⬇ * * T )"
+ non associative with precedence 45
+ for @{ 'SNStar $L $T }.
+
+notation "hvbox( ⦃ L, break T ⦄ break [ R ] ϵ break 〚 A 〛 )"
+ non associative with precedence 45
+ for @{ 'InEInt $R $L $T $A }.
+
+notation "hvbox( T1 break [ R ] ⊑ break T2 )"
+ non associative with precedence 45
+ for @{ 'CrSubEq $T1 $R $T2 }.
+
+(* Conversion ***************************************************************)
+
+notation "hvbox( L ⊢ break term 90 T1 ⬌ break T2 )"
+ non associative with precedence 45
+ for @{ 'PConv $L $T1 $T2 }.
+
+(* Equivalence **************************************************************)
+
+notation "hvbox( L ⊢ break term 90 T1 ⬌* break T2 )"
+ non associative with precedence 45
+ for @{ 'PConvStar $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+definition cnf: lenv → predicate term ≝ λL. NF … (cpr L) (eq …).
+
+interpretation
+ "context-sensitive normality (term)"
+ 'Normal L T = (cnf L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_sort *)
+lemma cnf_sort: ∀L,k. L ⊢ 𝐍[⋆k].
+#L #k #X #H
+>(cpr_inv_sort1 … H) //
+qed.
+
+axiom cnf_dec: ∀L,T1. L ⊢ 𝐍[T1] ∨
+ ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → False).
+
+(* Basic_1: removed theorems 1: nf2_abst_shift *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr_lift.ma".
+include "Basic_2/reducibility/cnf.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was only: nf2_csort_lref *)
+lemma cnf_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍[#i].
+#L #i #HLK #X #H
+elim (cpr_inv_lref1 … H) // *
+#K0 #V0 #V1 #HLK0 #_ #_ #_
+lapply (ldrop_mono … HLK … HLK0) -L #H destruct
+qed.
+
+(* Basic_1: was: nf2_lref_abst *)
+lemma cnf_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍[#i].
+#L #K #V #i #HLK #X #H
+elim (cpr_inv_lref1 … H) // *
+#K0 #V0 #V1 #HLK0 #_ #_ #_
+lapply (ldrop_mono … HLK … HLK0) -L #H destruct
+qed.
+
+(* Basic_1: was: nf2_abst *)
+lemma cnf_abst: ∀I,L,V,W,T. L ⊢ 𝐍[W] → L. ⓑ{I} V ⊢ 𝐍[T] → L ⊢ 𝐍[ⓛW.T].
+#I #L #V #W #T #HW #HT #X #H
+elim (cpr_inv_abst1 … H I V) -H #W0 #T0 #HW0 #HT0 #H destruct
+>(HW … HW0) -W >(HT … HT0) -T //
+qed.
+
+(* Basic_1: was only: nf2_appl_lref *)
+lemma cnf_appl_simple: ∀L,V,T. L ⊢ 𝐍[V] → L ⊢ 𝐍[T] → 𝐒[T] → L ⊢ 𝐍[ⓐV.T].
+#L #V #T #HV #HT #HS #X #H
+elim (cpr_inv_appl1_simple … H ?) -H // #V0 #T0 #HV0 #HT0 #H destruct
+>(HV … HV0) -V >(HT … HT0) -T //
+qed.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: nf2_lift *)
+lemma cnf_lift: ∀L0,L,T,T0,d,e.
+ L ⊢ 𝐍[T] → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍[T0].
+#L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
+elim (cpr_inv_lift … HL0 … HT0 … H) -L0 #T1 #HT10 #HT1
+>(HLT … HT1) in HT0; -L #HT0
+>(lift_mono … HT10 … HT0) -T1 -X //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/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 *********************************************************)
+
+lemma cpr_intro: ∀L,T1,T,T2,d,e. T1 ➡ T → L ⊢ T [d, e] ▶* T2 → L ⊢ T1 ➡ T2.
+/4 width=3/ qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma cpr_pr: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
+/2 width=3/ 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 width=1/ 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 ⊢ ⓕ{I} V1. T1 ➡ ⓕ{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 ⊢ ⓣV. T1 ➡ T2.
+#L #V #T1 #T2 * /3 width=3/
+qed.
+
+(* Note: it does not hold replacing |L1| with |L2| *)
+(* Basic_1: was only: pr2_change *)
+lemma cpr_lsubs_conf: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀L2. L1 [0, |L1|] ≼ L2 → L2 ⊢ T1 ➡ T2.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+lapply (tpss_lsubs_conf … HT2 … HL12) -HT2 -HL12 /3 width=4/
+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_abbr *)
+lemma cpr_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡ U2 →
+ (∃∃V,V2,T2. V1 ➡ V & L ⊢ V [O, |L|] ▶* V2 &
+ L. ⓓV ⊢ T1 ➡ T2 &
+ U2 = ⓓV2. T2
+ ) ∨
+ ∃∃T. ⇧[0,1] T ≡ T1 & L ⊢ T ➡ U2.
+#L #V1 #T1 #Y * #X #H1 #H2
+elim (tpr_inv_abbr1 … H1) -H1 *
+[ #V #T0 #T #HV1 #HT10 #HT0 #H destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tps_lsubs_conf … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
+ lapply (tps_weak_all … HT0) -HT0 #HT0
+ lapply (tpss_lsubs_conf … HT2 (L. ⓓV) ?) -HT2 /2 width=1/ #HT2
+ lapply (tpss_weak_all … HT2) -HT2 #HT2
+ lapply (tpss_strap … HT0 HT2) -T /4 width=7/
+| /4 width=5/
+]
+qed-.
+
+(* Basic_1: was: pr2_gen_cast *)
+lemma cpr_inv_cast1: ∀L,V1,T1,U2. L ⊢ ⓣV1. T1 ➡ U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U2 = ⓣV2. T2
+ ) ∨ L ⊢ T1 ➡ U2.
+#L #V1 #T1 #U2 * #X #H #HU2
+elim (tpr_inv_cast1 … H) -H /3 width=3/
+* #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
+qed-.
+
+(* Basic_1: removed theorems 4:
+ 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
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr_tpr.ma".
+include "Basic_2/reducibility/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 ⊢ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
+@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /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. ⓑ{I} V2 ⊢ T1 ➡ T2 →
+ L ⊢ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #HV12 * #T #HT1 normalize #HT2
+elim (tpss_split_up … HT2 1 ? ?) -HT2 // #T0 <minus_n_O #HT0 normalize <minus_plus_m_m #HT02
+lapply (tpss_lsubs_conf … HT0 (⋆. ⓑ{I} V2) ?) -HT0 /2 width=1/ #HT0
+lapply (tpss_tps … HT0) -HT0 #HT0
+@ex2_1_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
+qed.
+
+(* Basic_1: was only: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
+* /2 width=1/ /3 width=1/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @ T1 ➡ L @ T2.
+#L elim L -L
+[ #T1 #T2 #HT12 @(cpr_inv_atom … HT12)
+| normalize /3 width=1/
+].
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: pr2_confluence *)
+theorem cpr_conf: ∀L,U0,T1,T2. L ⊢ U0 ➡ T1 → L ⊢ U0 ➡ T2 →
+ ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
+#L #U0 #T1 #T2 * #U1 #HU01 #HUT1 * #U2 #HU02 #HUT2
+elim (tpr_conf … HU01 HU02) -U0 #U #HU1 #HU2
+elim (tpr_tpss_ltpr ? L … HU1 … HUT1) -U1 // #U1 #HTU1 #HU1
+elim (tpr_tpss_ltpr ? L … HU2 … HUT2) -U2 // #U2 #HTU2 #HU2
+elim (tpss_conf_eq … HU1 … HU2) -U /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_lift.ma".
+include "Basic_2/reducibility/tpr_lift.ma".
+include "Basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
+ ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 [0, |L| - i - 1] ▶* W1 →
+ ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
+#L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
+lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+@ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 [0, |L| - i - 1] ▶* T1 &
+ ⇧[0, i + 1] T1 ≡ T2 &
+ i < |L|.
+#L #T2 #i * #X #H
+>(tpr_inv_atom1 … H) -H #H
+elim (tpss_inv_lref1 … H) -H /2 width=1/
+* /3 width=6/
+qed-.
+
+(* Basic_1: was: pr2_gen_abst *)
+lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
+#L #V1 #T1 #Y * #X #H1 #H2 #I #W
+elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+lapply (tpss_lsubs_conf … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
+qed-.
+
+(* Basic_1: was pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+ U0 = ⓛW. T1 &
+ U2 = ⓓV2. T2
+ | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓW1. T1 &
+ U2 = ⓓW2. ⓐV. T2.
+#L #V1 #U0 #Y * #X #H1 #H2
+elim (tpr_inv_appl1 … H1) -H1 *
+[ #V #U #HV1 #HU0 #H destruct
+ elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
+| #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
+| #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
+ elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
+ elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
+ elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
+ lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒[T1] →
+ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+ U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr2_lift *)
+lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
+ K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
+elim (lift_total T d e) #U #HTU
+lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
+elim (lt_or_ge (|K|) d) #HKd
+[ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
+| lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
+]
+qed.
+
+(* Basic_1: was: pr2_gen_lift *)
+lemma cpr_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
+#L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
+elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
+elim (lt_or_ge (|L|) d) #HLd
+[ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
+| elim (lt_or_ge (|L|) (d + e)) #HLde
+ [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=2/ ]
+ | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr_tpss.ma".
+include "Basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning parallel unfold on terms ***************************)
+
+(* Note: we could invoke tpss_weak_all instead of ltpr_fwd_length *)
+(* Basic_1: was only: pr2_subst1 *)
+lemma cpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L2 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
+#L1 #L2 #HL12 #T1 #T2 * #T #HT1 #HT2 #d #e #U1 #HTU1
+elim (tpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U #HU1 #HTU
+elim (tpss_conf_eq … HT2 … HTU) -T /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_eq: ∀L1,T1,T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 ➡ T & T2 ➡ T.
+#L1 #T1 #T2 * #T #HT1 #HT2 #L2 #HL12
+>(ltpr_fwd_length … HL12) in HT2; #HT2
+elim (tpr_tpss_ltpr … HL12 … HT2) -L1 /3 width=3/
+qed.
+
+lemma cpr_ltpr_conf_tpss: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
+ ∃∃U2. L2 ⊢ U1 ➡ U2 & L2 ⊢ T2 ➡ U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1
+elim (cpr_ltpr_conf_eq … HT12 … HL12) -HT12 #T #HT1 #HT2
+elim (cpr_tpss_ltpr … HL12 … HT1 … HTU1) -L1 -HT1 #U2 #HU12 #HTU2
+lapply (tpss_weak_all … HTU2) -HTU2 #HTU2 /3 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_tpss.ma".
+include "Basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Properties concerning partial unfold on local environments ***************)
+
+lemma ltpss_cpr_trans: ∀L1,L2,d,e. L1 [d, e] ▶* L2 →
+ ∀T1,T2. L2 ⊢ T1 ➡ T2 → L1 ⊢ T1 ➡ T2.
+#L1 #L2 #d #e #HL12 #T1 #T2 *
+lapply (ltpss_weak_all … HL12)
+<(ltpss_fwd_length … HL12) -HL12 /3 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+include "Basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************)
+
+definition lcpr: relation lenv ≝
+ λL1,L2. ∃∃L. L1 ➡ L & L [0, |L|] ▶* L2
+.
+
+interpretation
+ "context-sensitive parallel reduction (environment)"
+ 'CPRed L1 L2 = (lcpr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lcpr_refl: ∀L. L ⊢ ➡ L.
+/2 width=3/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lcpr_inv_atom1: ∀L2. ⋆ ⊢ ➡ L2 → L2 = ⋆.
+#L2 * #L #HL >(ltpr_inv_atom1 … HL) -HL #HL2 >(ltpss_inv_atom1 … HL2) -HL2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/cpr.ma".
+include "Basic_2/reducibility/lcpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS *************)
+
+(* Advanced properties ****************************************************)
+
+lemma lcpr_pair: ∀L1,L2. L1 ⊢ ➡ L2 → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
+ ∀I. L1. ⓑ{I} V1 ⊢ ➡ L2. ⓑ{I} V2.
+#L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
+<(ltpss_fwd_length … HL2) /4 width=5/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+inductive ltpr: relation lenv ≝
+| ltpr_stom: ltpr (⋆) (⋆)
+| ltpr_pair: ∀K1,K2,I,V1,V2.
+ ltpr K1 K2 → V1 ➡ V2 → ltpr (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) (*ⓑ*)
+.
+
+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 width=1/
+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 width=3/ qed-.
+
+fact ltpr_inv_pair1_aux: ∀L1,L2. L1 ➡ L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
+#L1 #L2 * -L1 -L2
+[ #K1 #I #V1 #H destruct
+| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
+]
+qed.
+
+(* Basic_1: was: wcpr0_gen_head *)
+lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
+ ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3/ 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 width=3/ qed-.
+
+fact ltpr_inv_pair2_aux: ∀L1,L2. L1 ➡ L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+#L1 #L2 * -L1 -L2
+[ #K2 #I #V2 #H destruct
+| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
+]
+qed.
+
+lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
+#L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.
+
+(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr_lift.ma".
+include "Basic_2/reducibility/ltpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Basic_1: was: wcpr0_drop *)
+lemma ltpr_ldrop_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 -L1 -K1 -d -e
+[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
+| #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
+ elim (IHLK1 … HL12) -L1 /3 width=3/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (tpr_inv_lift … HV12 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5/
+]
+qed.
+
+(* Basic_1: was: wcpr0_drop_back *)
+lemma ldrop_ltpr_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 -L1 -K1 -d -e
+[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
+| #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) -K1 /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
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (tpr_lift … HW12 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+]
+qed.
+
+fact ltpr_ldrop_trans_O1_aux: ∀L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. L1 ➡ L2 →
+ d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
+#L2 #K2 #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #X #H >(ltpr_inv_atom2 … H) -H /2 width=3/
+| #K2 #I #V2 #X #H
+ elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
+| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
+ elim (ltpr_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
+| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpr_ldrop_trans_O1: ∀L1,L2. L1 ➡ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
+/2 width=5/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+(* Properties concerning parallel substitution on terms *********************)
+
+lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 [d, e] ▶ T2 → ∀L1. L1 ➡ L2 →
+ ∃∃T. L1 ⊢ T1 [d, e] ▶ T & T ➡ T2.
+#L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
+[ /2 width=3/
+| #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
+ elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
+ elim (lift_total V1 0 (i+1)) #W1 #HVW1
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
+| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
+| #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L2 /3 width=5/
+]
+qed.
+
+lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶ T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 [d, e] ▶ T & T2 ➡ T.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ /2 width=3/
+| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
+ elim (lift_total V2 0 (i+1)) #W2 #HVW2
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L1 /3 width=5/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/trf.ma".
+
+(* CONTEXT-FREE IRREDUCIBLE TERMS *******************************************)
+
+definition tif: predicate term ≝ λT. 𝐑[T] → False.
+
+interpretation "context-free irreducibility (term)"
+ 'NotReducible T = (tif T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tif_inv_abbr: ∀V,T. 𝐈[ⓓV.T] → False.
+/2 width=1/ qed-.
+
+lemma tif_inv_abst: ∀V,T. 𝐈[ⓛV.T] → 𝐈[V] ∧ 𝐈[T].
+/4 width=1/ qed-.
+
+lemma tif_inv_appl: ∀V,T. 𝐈[ⓐV.T] → ∧∧ 𝐈[V] & 𝐈[T] & 𝐒[T].
+#V #T #HVT @and3_intro /3 width=1/
+generalize in match HVT; -HVT elim T -T //
+* // * #U #T #_ #_ #H elim (H ?) -H /2 width=1/
+qed-.
+
+lemma tif_inv_cast: ∀V,T. 𝐈[ⓣV.T] → False.
+/2 width=1/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tif_atom: ∀I. 𝐈[⓪{I}].
+/2 width=4/ qed.
+
+lemma tif_abst: ∀V,T. 𝐈[V] → 𝐈[T] → 𝐈[ⓛV.T].
+#V #T #HV #HT #H
+elim (trf_inv_abst … H) -H /2 width=1/
+qed.
+
+lemma tif_appl: ∀V,T. 𝐈[V] → 𝐈[T] → 𝐒[T] → 𝐈[ⓐV.T].
+#V #T #HV #HT #S #H
+elim (trf_inv_appl … H) -H /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr.ma".
+
+(* CONTEXT-FREE NORMAL TERMS ************************************************)
+
+definition tnf: predicate term ≝ NF … tpr (eq …).
+
+interpretation
+ "context-free normality (term)"
+ 'Normal T = (tnf T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tnf_inv_abst: ∀V,T. 𝐍[ⓛV.T] → 𝐍[V] ∧ 𝐍[T].
+#V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
+]
+qed-.
+
+lemma tnf_inv_appl: ∀V,T. 𝐍[ⓐV.T] → ∧∧ 𝐍[V] & 𝐍[T] & 𝐒[T].
+#V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
+| generalize in match HVT1; -HVT1 elim T1 -T1 * // * #W1 #U1 #_ #_ #H
+ [ elim (lift_total V1 0 1) #V2 #HV12
+ lapply (H (ⓓW1.ⓐV2.U1) ?) -H /2 width=3/ -HV12 #H destruct
+ | lapply (H (ⓓV1.U1) ?) -H /2 width=1/ #H destruct
+]
+qed-.
+
+lemma tnf_inv_abbr: ∀V,T. 𝐍[ⓓV.T] → False.
+#V #T #H elim (is_lift_dec T 0 1)
+[ * #U #HTU
+ lapply (H U ?) -H /2 width=3/ #H destruct
+ elim (lift_inv_pair_xy_y … HTU)
+| #HT
+ elim (tps_full (⋆) V T (⋆. ⓓV) 0 ?) // #T2 #T1 #HT2 #HT12
+ lapply (H (ⓓV.T2) ?) -H /2 width=3/ -HT2 #H destruct /3 width=2/
+]
+qed.
+
+lemma tnf_inv_cast: ∀V,T. 𝐍[ⓣV.T] → False.
+#V #T #H lapply (H T ?) -H /2 width=1/ #H
+@(discr_tpair_xy_y … H)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tif.ma".
+include "Basic_2/reducibility/tnf.ma".
+
+(* CONTEXT-FREE NORMAL TERMS ************************************************)
+
+(* Main properties properties ***********************************************)
+
+lemma tpr_tif_eq: ∀T1,T2. T1 ➡ T2 → 𝐈[T1] → T1 = T2.
+#T1 #T2 #H elim H -T1 -T2
+[ //
+| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+ [ elim (tif_inv_appl … H) -H #HV1 #HT1 #_
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ | elim (tif_inv_cast … H)
+ ]
+| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (tif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #H
+ [ -HT2 -IHV1 -IHT1 elim (tif_inv_abbr … H)
+ | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
+ elim (tif_inv_abst … H) -H #HV1 #HT1
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ ]
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (tif_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #V1 #T1 #T2 #T #_ #_ #_ #H
+ elim (tif_inv_abbr … H)
+| #V1 #T1 #T #_ #_ #H
+ elim (tif_inv_cast … H)
+]
+qed.
+
+theorem tif_tnf: ∀T1. 𝐈[T1] → 𝐍[T1].
+/2 width=1/ qed.
+
+(* Note: this property is unusual *)
+lemma tnf_trf_false: ∀T1. 𝐑[T1] → 𝐍[T1] → False.
+#T1 #H elim H -T1
+[ #V #T #_ #IHV #H elim (tnf_inv_abst … H) -H /2 width=1/
+| #V #T #_ #IHT #H elim (tnf_inv_abst … H) -H /2 width=1/
+| #V #T #_ #IHV #H elim (tnf_inv_appl … H) -H /2 width=1/
+| #V #T #_ #IHV #H elim (tnf_inv_appl … H) -H /2 width=1/
+| #V #T #H elim (tnf_inv_abbr … H)
+| #V #T #H elim (tnf_inv_cast … H)
+| #V #W #T #H elim (tnf_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+qed.
+
+theorem tnf_tif: ∀T1. 𝐍[T1] → 𝐈[T1].
+/2 width=3/ qed.
+
+lemma tnf_abst: ∀V,T. 𝐍[V] → 𝐍[T] → 𝐍[ⓛV.T].
+/4 width=1/ qed.
+
+lemma tnf_appl: ∀V,T. 𝐍[V] → 𝐍[T] → 𝐒[T] → 𝐍[ⓐV.T].
+/4 width=1/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 (⓪{I}) (⓪{I})
+| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
+ tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| tpr_beta : ∀V1,V2,W,T1,T2.
+ tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛW. T1) (ⓓV2. T2)
+| tpr_delta: ∀I,V1,V2,T1,T2,T.
+ tpr V1 V2 → tpr T1 T2 → ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T →
+ tpr (ⓑ{I} V1. T1) (ⓑ{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 (ⓐV1. ⓓW1. T1) (ⓓW2. ⓐV. T2)
+| tpr_zeta : ∀V,T,T1,T2. ⇧[0,1] T1 ≡ T → tpr T1 T2 → tpr (ⓓV. T) T2
+| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓣ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 →
+ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
+/2 width=3/ qed.
+
+(* Basic_1: was by definition: pr0_refl *)
+lemma tpr_refl: ∀T. T ➡ T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{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. ⓪{I} ➡ U2 → U2 = ⓪{I}.
+/2 width=3/ qed-.
+
+fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
+ (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
+ ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T &
+ U2 = ⓑ{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 /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 /3 width=3/
+| #V #T1 #T2 #_ #I0 #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_bind1: ∀V1,T1,U2,I. ⓑ{I} V1. T1 ➡ U2 →
+ (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
+ ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ T &
+ U2 = ⓑ{I} V2. T
+ ) ∨
+ ∃∃T. ⇧[0,1] T ≡ T1 & T ➡ U2 & I = Abbr.
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_abbr *)
+lemma tpr_inv_abbr1: ∀V1,T1,U2. ⓓV1. T1 ➡ U2 →
+ (∃∃V2,T2,T. V1 ➡ V2 & T1 ➡ T2 &
+ ⋆. ⓓV2 ⊢ T2 [0, 1] ▶ T &
+ U2 = ⓓ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 = ⓕ{I} V1. U0 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛW. T1 &
+ U2 = ⓓV2. T2 & I = Appl
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓW1. T1 &
+ U2 = ⓓW2. ⓐ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 /3 width=5/
+| #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /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 /3 width=12/
+| #V #T #T1 #T2 #_ #_ #J #V0 #T0 #H destruct
+| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
+]
+qed.
+
+lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓕ{I} V2. T2
+ | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛW. T1 &
+ U2 = ⓓV2. T2 & I = Appl
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓW1. T1 &
+ U2 = ⓓW2. ⓐV. T2 &
+ I = Appl
+ | (U0 ➡ U2 ∧ I = Cast).
+/2 width=3/ qed-.
+
+(* Basic_1: was pr0_gen_appl *)
+lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
+ ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
+ U2 = ⓐV2. T2
+ | ∃∃V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U0 = ⓛW. T1 &
+ U2 = ⓓV2. T2
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
+ ⇧[0,1] V2 ≡ V &
+ U0 = ⓓW1. T1 &
+ U2 = ⓓW2. ⓐV. T2.
+#V1 #U0 #U2 #H
+elim (tpr_inv_flat1 … H) -H * /3 width=12/ #_ #H destruct
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒[T1] →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
+ U = ⓐV2. T2.
+#V1 #T1 #U #H #HT1
+elim (tpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+(* Basic_1: was: pr0_gen_cast *)
+lemma tpr_inv_cast1: ∀V1,T1,U2. ⓣV1. T1 ➡ U2 →
+ (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓣ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 = ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓣV. T.
+#T1 #T2 * -T1 -T2
+[ #I #i #H destruct /2 width=1/
+| #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 width=4/
+]
+qed.
+
+lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T,T0. ⇧[O,1] T0 ≡ T & T0 ➡ #i &
+ T1 = ⓓV. T
+ | ∃∃V,T. T ➡ #i & T1 = ⓣV. T.
+/2 width=3/ qed-.
+
+(* Basic_1: removed theorems 3:
+ pr0_subst0_back pr0_subst0_fwd pr0_subst0
+ Basic_1: removed local theorems: 1: pr0_delta_tau
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/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 -T1 -T2
+[ * #i #d #e #U1 #HU1 #U2 #HU2
+ lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
+ [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
+ | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
+ | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
+ ]
+| #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
+ elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
+| #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
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
+| #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
+ elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
+ elim (lift_total T2 (d + 1) e) #U2 #HTU2
+ @tpr_delta
+ [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/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
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
+ elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
+ elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
+ elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
+| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20
+ elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct
+ elim (lift_trans_ge … 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 /3 width=4/
+]
+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 -T1 -T2
+[ * #i #d #e #U1 #HU1
+ [ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct /2 width=3/
+ | lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … HU1) -HU1 #H destruct /2 width=3/
+ ]
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /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
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1
+ elim (IHT12 … HT01) -T1 /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
+ elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
+ elim (IHT12 … HUT1) -T1 #U2 #HUT2 #HU12
+ elim (tps_inv_lift1_le … HT20 … HUT2 ?) -T2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
+ @ex2_1_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /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
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
+ elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
+ elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
+ elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
+ elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
+ @ex2_1_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /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
+ elim (lift_div_le … HT1 … HT0 ?) -T // #T #HT0 #HT1
+ elim (IHT12 … HT1) -T1 /3 width=5/
+| #V #T1 #T2 #_ #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
+ elim (IHT12 … HT01) -T1 /3 width=3/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tpr_inv_abst1_aux: ∀U1,U2. U1 ➡ U2 → ∀V1,T1. U1 = ⓛV1. T1 →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ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
+ <(tps_inv_refl_SO2 … 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. ⓛV1. T1 ➡ U2 →
+ ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛV2. T2.
+/2 width=3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/tpr_tpss.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Confluence lemmas ********************************************************)
+
+fact tpr_conf_atom_atom: ∀I. ∃∃X. ⓪{I} ➡ X & ⓪{I} ➡ X.
+/2 width=3/ 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. ⓕ{I} V1. T1 ➡ T0 & ⓕ{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 -IH // /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 → ⓛW0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓ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
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
+elim (IH … HT02 … HU01) -HT02 -HU01 -IH /2 width=1/ /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 → ⓓW0. U0 ➡ T1 →
+ ∃∃X. ⓐV1. T1 ➡ X & ⓓW2. ⓐ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 /2 width=1/ #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
+ elim (IH … HW02 … HWW2) -HW02 -HWW2 /2 width=1/ #W #HW02 #HWW2
+ elim (IH … HU02 … HUU02) -HU02 -HUU02 -IH /2 width=1/ #U #HU2 #HUUU2
+ elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -UU2 #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 /2 width=1/ -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. ⓣV1. T1 ➡ X & X2 ➡ X.
+#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH // /3 width=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. ⓓV1. T1 ➡X & ⓓV2. T2 ➡ X.
+#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 -HV02 /2 width=1/
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ /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 →
+ ⋆. ⓑ{I1} V1 ⊢ T1 [O, 1] ▶ TT1 →
+ ⋆. ⓑ{I1} V2 ⊢ T2 [O, 1] ▶ TT2 →
+ ∃∃X. ⓑ{I1} V1. TT1 ➡ X & ⓑ{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) -T1 #U1 #TTU1 #HTU1
+elim (tpr_tps_bind … HV2 HT2 … HTT2) -T2 #U2 #TTU2 #HTU2
+elim (tps_conf_eq … HTU1 … HTU2) -T #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 → ⋆. ⓓV1 ⊢ T1 [O,1] ▶ TT1 →
+ T2 ➡ X2 → ⇧[O, 1] T2 ≡ T0 →
+ ∃∃X. ⓓ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
+lapply (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HTX2 … HTT2) -HTX2 -HTT2 -IH // /3 width=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. ⓓW1. ⓐVV1. T1 ➡ X & ⓓW2. ⓐ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 /2 width=1/ #V #HV1 #HV2
+elim (IH … HW01 … HW02) -HW01 -HW02 /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02) -HT01 -HT02 -IH /2 width=1/ #T #HT1 #HT2
+elim (lift_total V 0 1) #VV #HVV
+lapply (tpr_lift … HV1 … HVV1 … HVV) -V1 #HVV1
+lapply (tpr_lift … HV2 … HVV2 … HVV) -V2 -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
+lapply (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HT01 … HTX2) -HT01 -HTX2 -IH // /2 width=3/
+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 width=3/
+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
+ lapply (tpr_inv_atom1 … H1) -H1
+(* case 1: atom, atom *)
+ #H1 destruct //
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
+ elim (tpr_inv_flat1 … H1) -H1 *
+(* case 2: flat, flat *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ /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
+ /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
+ /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
+(* case 5: flat, tau *)
+ | #HT02 #H destruct
+ /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
+ ]
+| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 6: beta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
+(* case 7: beta, beta *)
+ | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct
+ /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
+ elim (tpr_inv_bind1 … H1) -H1 *
+(* case 9: delta, delta *)
+ [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
+ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 10: delta, zata *)
+ | #T2 #HT20 #HTX2 #H destruct
+ /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
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 11: theta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct
+ @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
+ /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+ ]
+| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct
+ elim (tpr_inv_abbr1 … H1) -H1 *
+(* case 14: zeta, delta (repeated) *)
+ [ #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct
+ @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
+ elim (tpr_inv_cast1 … H1) -H1
+(* case 16: tau, flat (repeated) *)
+ [ * #V2 #T2 #HV02 #HT02 #H destruct
+ @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
+(* case 17: tau, tau *)
+ | #HT02
+ /3 width=5 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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/ltpr_ldrop.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Unfold properties ********************************************************)
+
+(* 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 -T1 -T2
+[ #I #L1 #d #e #X #H
+ elim (tps_inv_atom1 … H) -H
+ [ #H destruct /2 width=3/
+ | * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 #HU12
+ @ex2_1_intro [2: @HU12 | skip | /3 width=4/ ] (**) (* /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
+ elim (IHV12 … HVW1 … HL12) -V1
+ elim (IHT12 … HTU1 … HL12) -T1 -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
+ elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHT12 … HTT1 (L2. ⓛWW) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ lapply (tpss_lsubs_conf … HTT2 (L2. ⓓ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
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHT12 … HTT1 (L2. ⓑ{I} VV2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (tpss_strip_neq … HTT2 … HTU2 ?) -T2 /2 width=1/ #T2 #HTT2 #HUT2
+ lapply (tps_lsubs_conf … HTT2 (L2. ⓑ{I} V2) ?) -HTT2 /2 width=1/ #HTT2
+ elim (ltpss_tps_conf … HTT2 (L2. ⓑ{I} VV2) (d + 1) e ?) -HTT2 /2 width=1/ #W2 #HTTW2 #HTW2
+ lapply (tpss_lsubs_conf … HTTW2 (⋆. ⓑ{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
+ lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2
+ lapply (tpss_lsubs_conf … HTW2 (L2. ⓑ{I} VV2) ?) -HTW2 /2 width=1/ #HTW2
+ lapply (tpss_trans_eq … HUT2 … HTW2) -T2 /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
+ elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct
+ elim (IHV12 … HVV1 … HL12) -V1 #VV2 #HVV12 #HVV2
+ elim (IHW12 … HWW1 … HL12) -W1 #WW2 #HWW12 #HWW2
+ elim (IHT12 … HTT1 (L2. ⓓWW2) ?) -T1 /2 width=1/ -HL12 #TT2 #HTT12 #HTT2
+ elim (lift_total VV2 0 1) #VV #H2VV
+ lapply (tpss_lift_ge … HVV2 (L2. ⓓWW2) … HV2 … H2VV) -V2 /2 width=1/ #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 #HTT1 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct
+ elim (tps_inv_lift1_ge … HTT12 L1 … HTT1 ?) -TT1 /2 width=1/ #T2 #HT12 #HTT2
+ elim (IHT12 … HT12 … HL12) -T1 -HL12 <minus_plus_m_m /3 width=3/
+| #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
+ elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
+ elim (IHT12 … HTT1 … HL12) -T1 -HL12 /3 width=3/
+]
+qed.
+
+lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
+ ⋆. ⓑ{I} V1 ⊢ T1 [0, 1] ▶ U1 →
+ ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ U2.
+#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
+elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ /3 width=3/
+qed.
+
+lemma tpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
+ ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
+#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
+[ /2 width=3/
+| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
+ elim (tpr_tps_ltpr … HUT … HU1 … HL12) -U -HL12 #U2 #HU12 #HTU2
+ lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* CONTEXT-FREE REDUCIBLE TERMS *********************************************)
+
+(* reducible terms *)
+inductive trf: predicate term ≝
+| trf_abst_sn: ∀V,T. trf V → trf (ⓛV. T)
+| trf_abst_dx: ∀V,T. trf T → trf (ⓛV. T)
+| trf_appl_sn: ∀V,T. trf V → trf (ⓐV. T)
+| trf_appl_dx: ∀V,T. trf T → trf (ⓐV. T)
+| trf_abbr : ∀V,T. trf (ⓓV. T)
+| trf_cast : ∀V,T. trf (ⓣV. T)
+| trf_beta : ∀V,W,T. trf (ⓐV. ⓛW. T)
+.
+
+interpretation
+ "context-free reducibility (term)"
+ 'Reducible T = (trf T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact trf_inv_atom_aux: ∀I,T. 𝐑[T] → T = ⓪{I} → False.
+#I #T * -T
+[ #V #T #_ #H destruct
+| #V #T #_ #H destruct
+| #V #T #_ #H destruct
+| #V #T #_ #H destruct
+| #V #T #H destruct
+| #V #T #H destruct
+| #V #W #T #H destruct
+]
+qed.
+
+lemma trf_inv_atom: ∀I. 𝐑[⓪{I}] → False.
+/2 width=4/ qed-.
+
+fact trf_inv_abst_aux: ∀W,U,T. 𝐑[T] → T = ⓛW. U → 𝐑[W] ∨ 𝐑[U].
+#W #U #T * -T
+[ #V #T #HV #H destruct /2 width=1/
+| #V #T #HT #H destruct /2 width=1/
+| #V #T #_ #H destruct
+| #V #T #_ #H destruct
+| #V #T #H destruct
+| #V #T #H destruct
+| #V #W0 #T #H destruct
+]
+qed.
+
+lemma trf_inv_abst: ∀V,T. 𝐑[ⓛV.T] → 𝐑[V] ∨ 𝐑[T].
+/2 width=3/ qed-.
+
+fact trf_inv_appl_aux: ∀W,U,T. 𝐑[T] → T = ⓐW. U →
+ ∨∨ 𝐑[W] | 𝐑[U] | (𝐒[U] → False).
+#W #U #T * -T
+[ #V #T #_ #H destruct
+| #V #T #_ #H destruct
+| #V #T #HV #H destruct /2 width=1/
+| #V #T #HT #H destruct /2 width=1/
+| #V #T #H destruct
+| #V #T #H destruct
+| #V #W0 #T #H destruct
+ @or3_intro2 #H elim (simple_inv_bind … H)
+]
+qed.
+
+lemma trf_inv_appl: ∀W,U. 𝐑[ⓐW.U] → ∨∨ 𝐑[W] | 𝐑[U] | (𝐒[U] → False).
+/2 width=3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/tshf.ma".
+include "Basic_2/reducibility/tpr.ma".
+
+(* CONTEXT-FREE WEAK HEAD NORMAL TERMS **************************************)
+
+definition twhnf: predicate term ≝ NF … tpr tshf.
+
+interpretation
+ "context-free weak head normality (term)"
+ 'WHdNormal T = (twhnf T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma twhnf_inv_tshf: ∀T. 𝐖𝐇𝐍[T] → T ≈ T.
+normalize /2 width=1/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tpr_tshf: ∀T1,T2. T1 ➡ T2 → T1 ≈ T1 → T1 ≈ T2.
+#T1 #T2 #H elim H -T1 -T2 //
+[ #I #V1 #V2 #T1 #T2 #_ #_ #_ #IHT12 #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #HT1U2 #HT1 #_ #H1 #H2 destruct
+ lapply (IHT12 HT1U2) -IHT12 -HT1U2 #HUT2
+ lapply (simple_tshf_repl_dx … HUT2 HT1) /2 width=1/
+| #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H destruct //
+| #V2 #V1 #V #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #U1 #U2 #_ #H
+ elim (simple_inv_bind … H)
+| #V #T #T1 #T2 #_ #_ #_ #H
+ elim (tshf_inv_bind1 … H) -H #W2 #U2 #H destruct
+| #V #T1 #T2 #_ #_ #H
+ elim (tshf_inv_flat1 … H) -H #W2 #U2 #_ #_ #_ #H destruct
+]
+qed.
+
+lemma twhnf_tshf: ∀T. T ≈ T → 𝐖𝐇𝐍[T].
+/2 width=1/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/aarity.ma".
+include "Basic_2/substitution/ldrop.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+inductive aaa: lenv → term → predicate aarity ≝
+| aaa_sort: ∀L,k. aaa L (⋆k) ⓪
+| aaa_lref: ∀I,L,K,V,B,i. ⇩[0, i] L ≡ K. ⓑ{I} V → aaa K V B → aaa L (#i) B
+| aaa_abbr: ∀L,V,T,B,A.
+ aaa L V B → aaa (L. ⓓV) T A → aaa L (ⓓV. T) A
+| aaa_abst: ∀L,V,T,B,A.
+ aaa L V B → aaa (L. ⓛV) T A → aaa L (ⓛV. T) (②B. A)
+| aaa_appl: ∀L,V,T,B,A. aaa L V B → aaa L T (②B. A) → aaa L (ⓐV. T) A
+| aaa_cast: ∀L,V,T,A. aaa L V A → aaa L T A → aaa L (ⓣV. T) A
+.
+
+interpretation "atomic arity assignment (term)"
+ 'AtomicArity L T A = (aaa L T A).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact aaa_inv_sort_aux: ∀L,T,A. L ⊢ T ÷ A → ∀k. T = ⋆k → A = ⓪.
+#L #T #A * -L -T -A
+[ //
+| #I #L #K #V #B #i #_ #_ #k #H destruct
+| #L #V #T #B #A #_ #_ #k #H destruct
+| #L #V #T #B #A #_ #_ #k #H destruct
+| #L #V #T #B #A #_ #_ #k #H destruct
+| #L #V #T #A #_ #_ #k #H destruct
+]
+qed.
+
+lemma aaa_inv_sort: ∀L,A,k. L ⊢ ⋆k ÷ A → A = ⓪.
+/2 width=5/ qed-.
+
+fact aaa_inv_lref_aux: ∀L,T,A. L ⊢ T ÷ A → ∀i. T = #i →
+ ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ÷ A.
+#L #T #A * -L -T -A
+[ #L #k #i #H destruct
+| #I #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5/
+| #L #V #T #B #A #_ #_ #i #H destruct
+| #L #V #T #B #A #_ #_ #i #H destruct
+| #L #V #T #B #A #_ #_ #i #H destruct
+| #L #V #T #A #_ #_ #i #H destruct
+]
+qed.
+
+lemma aaa_inv_lref: ∀L,A,i. L ⊢ #i ÷ A →
+ ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ÷ A.
+/2 width=3/ qed-.
+
+fact aaa_inv_abbr_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓓW. U →
+ ∃∃B. L ⊢ W ÷ B & L. ⓓW ⊢ U ÷ A.
+#L #T #A * -L -T -A
+[ #L #k #W #U #H destruct
+| #I #L #K #V #B #i #_ #_ #W #U #H destruct
+| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=2/
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #A #_ #_ #W #U #H destruct
+]
+qed.
+
+lemma aaa_inv_abbr: ∀L,V,T,A. L ⊢ ⓓV. T ÷ A →
+ ∃∃B. L ⊢ V ÷ B & L. ⓓV ⊢ T ÷ A.
+/2 width=3/ qed-.
+
+fact aaa_inv_abst_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓛW. U →
+ ∃∃B1,B2. L ⊢ W ÷ B1 & L. ⓛW ⊢ U ÷ B2 & A = ②B1. B2.
+#L #T #A * -L -T -A
+[ #L #k #W #U #H destruct
+| #I #L #K #V #B #i #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=5/
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #A #_ #_ #W #U #H destruct
+]
+qed.
+
+lemma aaa_inv_abst: ∀L,W,T,A. L ⊢ ⓛW. T ÷ A →
+ ∃∃B1,B2. L ⊢ W ÷ B1 & L. ⓛW ⊢ T ÷ B2 & A = ②B1. B2.
+/2 width=3/ qed-.
+
+fact aaa_inv_appl_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓐW. U →
+ ∃∃B. L ⊢ W ÷ B & L ⊢ U ÷ ②B. A.
+#L #T #A * -L -T -A
+[ #L #k #W #U #H destruct
+| #I #L #K #V #B #i #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3/
+| #L #V #T #A #_ #_ #W #U #H destruct
+]
+qed.
+
+lemma aaa_inv_appl: ∀L,V,T,A. L ⊢ ⓐV. T ÷ A →
+ ∃∃B. L ⊢ V ÷ B & L ⊢ T ÷ ②B. A.
+/2 width=3/ qed-.
+
+fact aaa_inv_cast_aux: ∀L,T,A. L ⊢ T ÷ A → ∀W,U. T = ⓣW. U →
+ L ⊢ W ÷ A ∧ L ⊢ U ÷ A.
+#L #T #A * -L -T -A
+[ #L #k #W #U #H destruct
+| #I #L #K #V #B #i #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #B #A #_ #_ #W #U #H destruct
+| #L #V #T #A #HV #HT #W #U #H destruct /2 width=1/
+]
+qed.
+
+lemma aaa_inv_cast: ∀L,W,T,A. L ⊢ ⓣW. T ÷ A →
+ L ⊢ W ÷ A ∧ L ⊢ T ÷ A.
+/2 width=3/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop_ldrop.ma".
+include "Basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem aaa_mono: ∀L,T,A1. L ⊢ T ÷ A1 → ∀A2. L ⊢ T ÷ A2 → A1 = A2.
+#L #T #A1 #H elim H -L -T -A1
+[ #L #k #A2 #H
+ >(aaa_inv_sort … H) -H //
+| #I1 #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H
+ elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2
+ lapply (ldrop_mono … HLK1 … HLK2) -L #H destruct /2 width=1/
+| #L #V #T #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
+ elim (aaa_inv_abbr … H) -H /2 width=1/
+| #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H
+ elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1/
+| #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
+ elim (aaa_inv_appl … H) -H #B2 #_ #HA2
+ lapply (IHA1 … HA2) -L #H destruct //
+| #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H
+ elim (aaa_inv_cast … H) -H /2 width=1/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop_ldrop.ma".
+include "Basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties concerning basic relocation ***********************************)
+
+lemma aaa_lift: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2,d,e. ⇩[d, e] L2 ≡ L1 →
+ ∀T2. ⇧[d, e] T1 ≡ T2 → L2 ⊢ T2 ÷ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #L2 #d #e #_ #T2 #H
+ >(lift_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #d #e #HL21 #T2 #H
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ elim (ldrop_trans_le … HL21 … HLK1 ?) -L1 /2 width=2/ #X #HLK2 #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=8/
+ | lapply (ldrop_trans_ge … HL21 … HLK1 ?) -L1 // -Hid /3 width=8/
+ ]
+| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=4/
+| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=4/
+| #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #d #e #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=4/
+| #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #d #e #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=4/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrops.ma".
+include "Basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties concerning generic relocation *********************************)
+
+lemma aaa_lifts: ∀L1,L2,T2,A,des. ⇩*[des] L2 ≡ L1 → ∀T1. ⇧*[des] T1 ≡ T2 →
+ L1 ⊢ T1 ÷ A → L2 ⊢ T2 ÷ A.
+#L1 #L2 #T2 #A #des #H elim H -L1 -L2 -des
+[ #L #T1 #H #HT1
+ <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1
+ elim (lifts_inv_cons … H) -H /3 width=9/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+inductive lsuba: relation lenv ≝
+| lsuba_atom: lsuba (⋆) (⋆)
+| lsuba_pair: ∀I,L1,L2,V. lsuba L1 L2 → lsuba (L1. ⓑ{I} V) (L2. ⓑ{I} V)
+| lsuba_abbr: ∀L1,L2,V,W,A. L1 ⊢ V ÷ A → L2 ⊢ W ÷ A →
+ lsuba L1 L2 → lsuba (L1. ⓓV) (L2. ⓛW)
+.
+
+interpretation
+ "local environment refinement (atomic arity assigment)"
+ 'CrSubEqA L1 L2 = (lsuba L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuba_inv_atom1_aux: ∀L1,L2. L1 ÷⊑ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsuba_inv_atom1: ∀L2. ⋆ ÷⊑ L2 → L2 = ⋆.
+/2 width=3/ qed-.
+
+fact lsuba_inv_pair1_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+#L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K1 #V #H destruct /3 width=7/
+]
+qed.
+
+lemma lsuba_inv_pair1: ∀I,K1,L2,V. K1. ⓑ{I} V ÷⊑ L2 →
+ (∃∃K2. K1 ÷⊑ K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
+ L2 = K2. ⓛW & I = Abbr.
+/2 width=3/ qed-.
+
+fact lsuba_inv_atom2_aux: ∀L1,L2. L1 ÷⊑ L2 → L2 = ⋆ → L1 = ⋆.
+#L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #H destruct
+]
+qed.
+
+lemma lsubc_inv_atom2: ∀L1. L1 ÷⊑ ⋆ → L1 = ⋆.
+/2 width=3/ qed-.
+
+fact lsuba_inv_pair2_aux: ∀L1,L2. L1 ÷⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
+ L1 = K1. ⓓV & I = Abst.
+#L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
+| #L1 #L2 #V1 #W2 #A #HV1 #HW2 #HL12 #I #K2 #W #H destruct /3 width=7/
+]
+qed.
+
+lemma lsuba_inv_pair2: ∀I,L1,K2,W. L1 ÷⊑ K2. ⓑ{I} W →
+ (∃∃K1. K1 ÷⊑ K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. K1 ⊢ V ÷ A & K2 ⊢ W ÷ A & K1 ÷⊑ K2 &
+ L1 = K1. ⓓV & I = Abst.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuba_refl: ∀L. L ÷⊑ L.
+#L elim L -L // /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/aaa_aaa.ma".
+include "Basic_2/static/lsuba_ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Properties concerning atomic arity assignment ****************************)
+
+lemma lsuba_aaa_conf: ∀L1,V,A. L1 ⊢ V ÷ A → ∀L2. L1 ÷⊑ L2 → L2 ⊢ V ÷ A.
+#L1 #V #A #H elim H -L1 -V -A
+[ //
+| #I #L1 #K1 #V1 #B #i #HLK1 #HV1B #IHV1 #L2 #HL12
+ elim (lsuba_ldrop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+ elim (lsuba_inv_pair1 … H) -H * #K2
+ [ #HK12 #H destruct /3 width=5/
+ | #V2 #A1 #HV1A1 #HV2 #_ #H1 #H2 destruct
+ >(aaa_mono … HV1B … HV1A1) -B -HV1A1 /2 width=5/
+ ]
+| /4 width=2/
+| /4 width=1/
+| /3 width=3/
+| /3 width=1/
+]
+qed.
+
+lemma lsuba_aaa_trans: ∀L2,V,A. L2 ⊢ V ÷ A → ∀L1. L1 ÷⊑ L2 → L1 ⊢ V ÷ A.
+#L2 #V #A #H elim H -L2 -V -A
+[ //
+| #I #L2 #K2 #V2 #B #i #HLK2 #HV2B #IHV2 #L1 #HL12
+ elim (lsuba_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsuba_inv_pair2 … H) -H * #K1
+ [ #HK12 #H destruct /3 width=5/
+ | #V1 #A1 #HV1 #HV2A1 #_ #H1 #H2 destruct
+ >(aaa_mono … HV2B … HV2A1) -B -HV2A1 /2 width=5/
+ ]
+| /4 width=2/
+| /4 width=1/
+| /3 width=3/
+| /3 width=1/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/lsuba.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+(* Note: the constant 0 cannot be generalized *)
+
+lemma lsuba_ldrop_O1_conf: ∀L1,L2. L1 ÷⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. K1 ÷⊑ K2 & ⇩[0, e] L2 ≡ K2.
+#L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ [ destruct
+ elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK1) -L1 /3 width=3/
+ ]
+]
+qed-.
+
+lemma lsuba_ldrop_O1_trans: ∀L1,L2. L1 ÷⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. K1 ÷⊑ K2 & ⇩[0, e] L1 ≡ K1.
+#L1 #L2 #H elim H -L1 -L2
+[ /2 width=3/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ [ destruct
+ elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ | elim (IHL12 … HLK2) -L2 /3 width=3/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/static/lsuba_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+
+(* Main properties **********************************************************)
+
+theorem lsuba_trans: ∀L1,L. L1 ÷⊑ L → ∀L2. L ÷⊑ L2 → L1 ÷⊑ L2.
+#L1 #L #H elim H -L1 -L
+[ #X #H >(lsuba_inv_atom1 … H) -H //
+| #I #L1 #L #V #HL1 #IHL1 #X #H
+ elim (lsuba_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=1/
+ | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/
+ ]
+| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H
+ elim (lsuba_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=5/
+ | #V #A2 #_ #_ #_ #_ #H destruct
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* 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 *)
+}.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/genv.ma".
+
+(* GLOBAL ENVIRONMENT SLICING ***********************************************)
+
+inductive gdrop (e:nat): relation genv ≝
+| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆)
+| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G
+| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2
+.
+
+interpretation "global slicing"
+ 'RDrop e G1 G2 = (gdrop e G1 G2).
+
+(* basic inversion lemmas ***************************************************)
+
+lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H >H -H >commutative_plus #H
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ #H2
+ lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+]
+qed-.
+
+lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ normalize #H2
+ <(injective_plus_l … H2) in H1; -H2 #H
+ elim (lt_refl_false … H)
+]
+qed-.
+
+fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
+ e < |G1| → ⇩[e] G1 ≡ G2.
+#I #G #G1 #G2 #V #e * -G -G2
+[ #G #H1 #H destruct #H2
+ lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
+ elim (lt_refl_false … H)
+| #J #G #G2 #W #_ #HG2 #H destruct //
+]
+qed.
+
+lemma gdrop_inv_lt: ∀I,G1,G2,V,e.
+ ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
+/2 width=5/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
+#e #G1 elim G1 -G1 /3 width=2/
+#I #V #G1 * #G2 #HG12
+elim (lt_or_eq_or_gt e (|G1|)) #He
+[ /3 width=2/
+| destruct /3 width=2/
+| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/gdrop.ma".
+
+(* GLOBAL ENVIRONMENT SLICING ***********************************************)
+
+(* Main properties **********************************************************)
+
+theorem gdrop_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
+#G #G1 #e #H elim H -G -G1
+[ #G #He #G2 #H
+ >(gdrop_inv_gt … H He) -H -He //
+| #G #He #G2 #H
+ >(gdrop_inv_eq … H He) -H -He //
+| #I #G #G1 #V #He #_ #IHG1 #G2 #H
+ lapply (gdrop_inv_lt … H He) -H -He /2 width=1/
+]
+qed-.
+
+lemma gdrop_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
+#G1 #G2 #e
+elim (gdrop_total e G1) #G #HG1
+elim (genv_eq_dec G G2) #HG2
+[ destruct /2 width=1/
+| @or_intror #HG12
+ lapply (gdrop_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/lsubs.ma".
+include "Basic_2/substitution/lift.ma".
+
+(* LOCAL ENVIRONMENT SLICING ************************************************)
+
+(* Basic_1: includes: drop_skip_bind *)
+inductive ldrop: nat → nat → relation lenv ≝
+| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
+| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2
+| ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
+ ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 →
+ ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| //
+| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+(* Basic_1: was: drop_gen_refl *)
+lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5/ qed-.
+
+fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V #e #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: drop_gen_sort *)
+lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆.
+/2 width=5/ qed-.
+
+fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. ⓑ{I} V →
+ (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
+ (0 < e ∧ ⇩[d, e - 1] K ≡ L2).
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #K #I #V #H destruct
+| #L #I #V #_ #K #J #W #HX destruct /3 width=1/
+| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ldrop_inv_O1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
+ (0 < e ∧ ⇩[0, e - 1] K ≡ L2).
+/2 width=3/ qed-.
+
+(* Basic_1: was: drop_gen_drop *)
+lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
+ ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
+#e #K #I #V #L2 #H #He
+elim (ldrop_inv_O1 … H) -H * // #H destruct
+elim (lt_refl_false … He)
+qed-.
+
+fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
+ ⇧[d - 1, e] V2 ≡ V1 &
+ L2 = K2. ⓑ{I} V2.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/
+]
+qed.
+
+(* Basic_1: was: drop_gen_skip_l *)
+lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
+ ⇧[d - 1, e] V2 ≡ V1 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ qed-.
+
+fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
+ ⇧[d - 1, e] V2 ≡ V1 &
+ L1 = K1. ⓑ{I} V1.
+#d #e #L1 #L2 * -d -e -L1 -L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/
+]
+qed.
+
+(* Basic_1: was: drop_gen_skip_r *)
+lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was by definition: drop_refl *)
+lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
+#L elim L -L //
+qed.
+
+lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
+ ⇩[0, e - 1] L1 ≡ L2 → 0 < e → ⇩[0, e] L1. ⓑ{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
+ ∀K1,V,i. ⇩[0, i] L1 ≡ K1. ⓓV →
+ d ≤ i → i < d + e →
+ ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
+ ⇩[0, i] L2 ≡ K2. ⓓV.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #d #e #K1 #V #i #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #L2 #K1 #V #i #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 -Hie destruct
+ <minus_n_O <minus_plus_m_m // /2 width=3/
+ | -HL12
+ elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /4 width=3/
+ ]
+| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 -Hie -Hi destruct
+ | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/
+ ]
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
+ elim (le_inv_plus_l … Hdi) #Hdim #Hi
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
+ elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/
+]
+qed.
+
+(* Basic forvard lemmas *****************************************************)
+
+(* Basic_1: was: drop_S *)
+lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
+ ⇩[O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1 destruct /2 width=1/
+ | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
+ ]
+]
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
+[ /2 width=3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(tw_lift … HV21) -HV21 /2 width=1/
+]
+qed-.
+
+lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
+ ⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1 destruct //
+ | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/
+ ]
+]
+qed-.
+
+lemma ldrop_fwd_O1_length: ∀L1,L2,e. ⇩[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 elim L1 -L1
+[ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
+| #K1 #I1 #V1 #IHL1 #L2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1 destruct //
+ | 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
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop_mono *)
+theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
+ ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
+#d #e #L #L1 #H elim H -d -e -L -L1
+[ #d #e #L2 #H
+ >(ldrop_inv_atom1 … H) -L2 //
+| #K #I #V #L2 #HL12
+ <(ldrop_inv_refl … HL12) -L2 //
+| #L #K #I #V #e #_ #IHLK #L2 #H
+ lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/
+| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
+ elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
+ >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
+ >(IHLK1 … HLK2) -IHLK1 -HLK2 //
+]
+qed-.
+
+(* Basic_1: was: drop_conf_ge *)
+theorem ldrop_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 -d1 -e1 -L -L1
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -L2 //
+| //
+| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2
+ <minus_plus >minus_minus_comm /3 width=1/
+| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
+ lapply (transitive_le 1 … Hdee2) // #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2
+ lapply (transitive_le (1 + e) … Hdee2) // #Hee2
+ @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *)
+]
+qed.
+
+(* Basic_1: was: drop_conf_lt *)
+theorem ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
+ ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 &
+ ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
+#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
+[ #d #e #e2 #K2 #I #V2 #H
+ lapply (ldrop_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 (ldrop_inv_O1 … H) -H *
+ [ -IHL12 -He2d #H1 #H2 destruct /2 width=5/
+ | -HL12 -HV12 #He #HLK
+ elim (IHL12 … HLK ?) -IHL12 -HLK [ <minus_minus /3 width=5/ | /2 width=1/ ] (**) (* a bit slow *)
+ ]
+]
+qed.
+
+(* Basic_1: was: drop_trans_le *)
+theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
+ ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 →
+ ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2.
+#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -L2 /2 width=3/
+| #K #I #V #e2 #L2 #HL2 #H
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
+| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+ elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
+ elim (ldrop_inv_O1 … H) -H *
+ [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/
+ | -HL12 -HV12 #He2 #HL2
+ elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
+ ]
+]
+qed.
+
+(* Basic_1: was: drop_trans_ge *)
+theorem ldrop_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 -d1 -e1 -L1 -L
+[ #d #e #e2 #L2 #H
+ >(ldrop_inv_atom1 … H) -H -L2 //
+| //
+| /3 width=1/
+| #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 (ldrop_inv_ldrop1 … H ?) -H // #HL2
+ @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *)
+]
+qed.
+
+theorem ldrop_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 ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
+ ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/term_simple.ma".
+
+(* BASIC TERM 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_gref : ∀p,d,e. lift d e (§p) (§p)
+| lift_bind : ∀I,V1,V2,T1,T2,d,e.
+ lift d e V1 V2 → lift (d + 1) e T1 T2 →
+ lift d e (ⓑ{I} V1. T1) (ⓑ{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 (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
+qed.
+
+lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
+/2 width=4/ 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 width=1/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #p #d #e #i #H destruct
+| #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 width=3/ 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_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#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_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
+/2 width=5/ qed-.
+
+fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = ⓑ{I} V1.U1 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T2 = ⓑ{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
+| #p #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] ⓑ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T2 = ⓑ{I} V2. U2.
+/2 width=3/ qed-.
+
+fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T2 = ⓕ{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
+| #p #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] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+/2 width=3/ 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 width=1/
+| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
+| #p #d #e #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed.
+
+(* Basic_1: was: lift_gen_lref *)
+lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+/2 width=3/ qed-.
+
+(* Basic_1: was: lift_gen_lref_lt *)
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → i < d → T1 = #i.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_false *)
+lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+ d ≤ i → i < d + e → False.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
+[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
+lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 #H
+elim (lt_refl_false … H)
+qed-.
+
+(* Basic_1: was: lift_gen_lref_ge *)
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
+elim (lt_inv_plus_l … Hdd) -Hdd #Hdd
+elim (lt_refl_false … Hdd)
+qed-.
+
+fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#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_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
+/2 width=5/ qed-.
+
+fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = ⓑ{I} V2.U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T1 = ⓑ{I} V1. U1.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
+]
+qed.
+
+(* Basic_1: was: lift_gen_bind *)
+lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓑ{I} V2. U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+ T1 = ⓑ{I} V1. U1.
+/2 width=3/ qed-.
+
+fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T1 = ⓕ{I} V1. U1.
+#d #e #T1 #T2 * -d -e -T1 -T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #p #d #e #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
+]
+qed.
+
+(* Basic_1: was: lift_gen_flat *)
+lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⇧[d,e] T1 ≡ ⓕ{I} V2. U2 →
+ ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+ T1 = ⓕ{I} V1. U1.
+/2 width=3/ qed-.
+
+lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → False.
+#d #e #J #V elim V -V
+[ * #i #T #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * #I #W2 #U2 #IHW2 #_ #T #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+ ]
+]
+qed-.
+
+lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → False.
+#J #T elim T -T
+[ * #i #V #d #e #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * #I #W2 #U2 #_ #IHU2 #V #d #e #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+ ]
+]
+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-.
+
+lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒[T1] → 𝐒[T2].
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒[T2] → 𝐒[T1].
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+elim (simple_inv_bind … H)
+qed-.
+
+(* 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 ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
+qed.
+
+lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e] #j ≡ #i.
+/2 width=1/ 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 width=1/
+| * /2 width=1/
+]
+qed.
+
+lemma lift_total: ∀T1,d,e. ∃T2. ⇧[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
+| * #I #V1 #T1 #IHV1 #IHT1 #d #e
+ elim (IHV1 d e) -IHV1 #V2 #HV12
+ [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
+ | elim (IHT1 d e) -IHT1 /3 width=2/
+ ]
+]
+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 -d1 -e2 -T1 -T2
+[ /3 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+ lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
+ >(plus_minus_m_m e2 e1 ?) // /3 width=3/
+| /3 width=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) /2 width=1/ /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_1: was only: dnf_dec2 dnf_dec *)
+lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⇧[d,e] T1 ≡ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2/ ] #i #d #e
+ elim (lt_dec i d) #Hid
+ [ /4 width=2/
+ | lapply (false_lt_to_le … Hid) -Hid #Hid
+ elim (lt_dec i (d + e)) #Hide
+ [ @or_intror * #T1 #H
+ elim (lift_inv_lref2_be … H Hid Hide)
+ | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
+ ]
+ ]
+| * #I #V2 #T2 #IHV2 #IHT2 #d #e
+ [ elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+ [ * #T1 #HT12 @or_introl /3 width=2/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+ [ * #T1 #HT12 /4 width=2/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ ]
+]
+qed.
+
+(* Basic_1: removed theorems 7:
+ lift_head lift_gen_head
+ lift_weight_map lift_weight lift_weight_add lift_weight_add_O
+ lift_tlt_dx
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Main properies ***********************************************************)
+
+(* Basic_1: was: lift_inj *)
+theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
+#d #e #T1 #U #H elim H -d -e -T1 -U
+[ #k #d #e #X #HX
+ lapply (lift_inv_sort2 … HX) -HX //
+| #i #d #e #Hid #X #HX
+ lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/
+| #p #d #e #X #HX
+ lapply (lift_inv_gref2 … HX) -HX //
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lift *)
+theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ⇧[d2 + e1, e2] T2 ≡ T →
+ d1 ≤ d2 →
+ ∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[d2, e2] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
+ [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
+ | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
+ elim (le_inv_plus_l … H) -H #Hide2 #He2i
+ lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
+ >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
+ /4 width=3/
+ ]
+| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=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
+ elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
+ >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /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
+ elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2 ?) // /3 width=5/
+]
+qed.
+
+(* Note: apparently this was missing in Basic_1 *)
+theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
+ e ≤ e1 → e1 ≤ e + e2 →
+ ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+ [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/
+ | >(lift_inv_lref2_ge … H ?) -H //
+ lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
+ elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
+ @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12
+ @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ]
+ ]
+| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2
+ elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2 ? ?) -V //
+ elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /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 -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 //
+| #p #d #e #X #HX
+ lapply (lift_inv_gref1 … 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 /3 width=1/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+]
+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 -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 width=1/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2
+ [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
+ | -Hd21 /2 width=1/
+ ]
+| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_gref1 … HT2) -HT2 //
+| #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
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1/
+| #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
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) // /2 width=1/
+]
+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 -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3/
+| #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 /3 width=3/ /4 width=3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
+ lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
+ >plus_plus_comm_23 /4 width=3/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3/
+| #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
+ elim (IHV12 … HV20 ?) -IHV12 -HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /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
+ 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 -d1 -e1 -T1 -T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3/
+| #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 width=1/ #Hid2e
+ lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+ elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3/
+| #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
+ elim (IHV12 … HV20 ?) -IHV12 -HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T
+ <plus_minus /2 width=2/ /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20 ?) -IHV12 -HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/lift_vector.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+(* Main properies ***********************************************************)
+
+theorem liftv_mono: ∀Ts,U1s,d,e. ⇧[d,e] Ts ≡ U1s →
+ ∀U2s:list term. ⇧[d,e] Ts ≡ U2s → U1s = U2s.
+#Ts #U1s #d #e #H elim H -Ts -U1s
+[ #U2s #H >(liftv_inv_nil1 … H) -H //
+| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
+ elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
+ >(lift_mono … HTU1 … HTU2) -T /3 width=1/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/substitution/lift.ma".
+
+(* BASIC TERM VECTOR RELOCATION *********************************************)
+
+inductive liftv (d,e:nat) : relation (list term) ≝
+| liftv_nil : liftv d e ◊ ◊
+| liftv_cons: ∀T1s,T2s,T1,T2.
+ ⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s →
+ liftv d e (T1 :: T1s) (T2 :: T2s)
+.
+
+interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
+#T1s #T2s #d #e * -T1s -T2s //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed.
+
+lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
+/2 width=5/ qed-.
+
+fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
+ ∀U1,U1s. T1s = U1 :: U1s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 :: U2s.
+#T1s #T2s #d #e * -T1s -T2s
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5/
+]
+qed.
+
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 :: U1s ≡ T2s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 :: U2s.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s.
+#d #e #T1s elim T1s -T1s
+[ /2 width=2/
+| #T1 #T1s * #T2s #HT12s
+ elim (lift_total T1 d e) /3 width=2/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
+
+(* Basic_1: includes: csubst1_bind *)
+inductive ltps: nat → nat → relation lenv ≝
+| ltps_atom: ∀d,e. ltps d e (⋆) (⋆)
+| ltps_pair: ∀L,I,V. ltps 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ltps_tps2: ∀L1,L2,I,V1,V2,e.
+ ltps 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶ V2 →
+ ltps 0 (e + 1) (L1. ⓑ{I} V1) L2. ⓑ{I} V2
+| ltps_tps1: ∀L1,L2,I,V1,V2,d,e.
+ ltps d e L1 L2 → L2 ⊢ V1 [d, e] ▶ V2 →
+ ltps (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
+.
+
+interpretation "parallel substritution (local environment)"
+ 'PSubst L1 d e L2 = (ltps d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma ltps_tps2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 [0, e - 1] ▶ L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
+ 0 < e → L1. ⓑ{I} V1 [0, e] ▶ L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) /2 width=1/
+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. ⓑ{I} V1 [d, e] ▶ L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) /2 width=1/
+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 width=1/ * /2 width=1/
+qed.
+
+lemma ltps_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → L1 [0, |L2|] ▶ L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+// /3 width=2/ /3 width=3/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltps_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶ L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+normalize //
+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 -d -e -L1 -L2 //
+[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
+ >(IHL12 ?) -IHL12 // >(tps_inv_refl_O2 … HV12) //
+]
+qed.
+
+lemma ltps_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶ L2 → L1 = L2.
+/2 width=4/ 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. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 [0, e - 1] ▶ K2 &
+ K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L2 = K2. ⓑ{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 /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltps_inv_tps21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶ L2 → 0 < e →
+ ∃∃K2,V2. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L2 = K2. ⓑ{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. ⓑ{I} V1 →
+ ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L2 = K2. ⓑ{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 /2 width=5/
+]
+qed.
+
+lemma ltps_inv_tps11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶ L2 → 0 < d →
+ ∃∃K2,V2. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L2 = K2. ⓑ{I} V2.
+/2 width=3/ 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. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
+ K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L1 = K1. ⓑ{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 /2 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L1 = K1. ⓑ{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. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L1 = K1. ⓑ{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 /2 width=5/
+]
+qed.
+
+lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=3/ 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
+
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_ldrop_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 -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltps_ldrop_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 -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+ elim (le_inv_plus_l … He12) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltps_ldrop_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 -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltps_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ▶ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L [0, d1 + e1 - e2] ▶ L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+ lapply (le_n_O_to_eq … He2) -He2 #H destruct
+ lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+ ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+ elim (le_inv_plus_l … Hd1e2) #_ #He2
+ <minus_le_minus_minus_comm //
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+ elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltps_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ▶ L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 [d1 - e2, e1] ▶ L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK01 -HV01 <minus_le_minus_minus_comm //
+ elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
+
+lemma ltps_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ▶ L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L [d1 - e2, e1] ▶ L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+ lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+ [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+ | -HK10 -HV10 <minus_le_minus_minus_comm //
+ elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+ ]
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/substitution/ltps_ldrop.ma".
+
+(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
+
+lemma ltps_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
+ ∀L1,d1,e1. L0 [d1, e1] ▶ L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 [d2, e2] ▶ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ //
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
+ lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
+ lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+qed.
+
+(* Basic_1: was: subst1_subst1_back *)
+lemma ltps_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
+ ∀L1,d1,e1. L0 [d1, e1] ▶ L1 →
+ ∃∃T. L1 ⊢ T2 [d2, e2] ▶ T &
+ L1 ⊢ U2 [d1, e1] ▶ T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltps_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+ elim (ltps_inv_tps11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltps_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+ elim (ltps_inv_tps21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK1) #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // normalize #HW01
+ lapply (tps_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
+ ]
+ ]
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+ elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -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 -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_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
+ @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
+]
+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 -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+ elim (lt_or_ge i2 d1) #Hi2d1
+ [ elim (ltps_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
+ elim (ltps_inv_tps12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
+ | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+ [ elim (ltps_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
+ elim (ltps_inv_tps22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
+ elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
+ lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // normalize #HW01
+ lapply (tps_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+ | lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
+ ]
+ ]
+| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+ elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
+ elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop.ma".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+inductive tps: nat → nat → lenv → relation term ≝
+| tps_atom : ∀L,I,d,e. tps d e L (⓪{I}) (⓪{I})
+| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
+ ⇩[0, i] L ≡ K. ⓓ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. ⓑ{I} V2) T1 T2 →
+ tps d e L (ⓑ{I} V1. T1) (ⓑ{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 (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+.
+
+interpretation "parallel substritution (term)"
+ 'PSubst L T1 d e T2 = (tps d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tps_lsubs_conf: ∀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 -L1 -T1 -T2 -d -e
+[ //
+| #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+ elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /2 width=4/
+| /4 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ▶ T.
+#T elim T -T //
+#I elim I -I /2 width=1/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma tps_full: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
+ ∃∃T2,T. L ⊢ T1 [d, 1] ▶ T2 & ⇧[d, 1] T ≡ T2.
+#K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK /2 width=4/
+ elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i ? ? ?) // /3 width=4/
+| * #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W2) (d+1) ?) -IHU1 /2 width=1/ -HLK /3 width=8/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
+ ]
+]
+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 -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 width=4/
+| /4 width=3/
+| /4 width=1/
+]
+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 -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+ lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
+ lapply (le_to_lt_to_lt … Hdi Hi) /3 width=4/
+| normalize /2 width=1/
+| /2 width=1/
+]
+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 -L -T1 -T2 -d -e
+[ /2 width=3/
+| #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 j d) in ⊢ (% → ?); // -Hdj /3 width=4/
+ | -Hdi -Hdj #Hid
+ generalize in match Hide; -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
+ >(plus_minus_m_m … Hjde) in ⊢ (% → ?); -Hjde /4 width=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 width=1/
+ -Hdi -Hide >arith_c1x #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L. ⓑ{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 = ⓪{I} →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓ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 /2 width=1/
+| #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #I #H destruct /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 ⊢ ⓪{I} [d, e] ▶ T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV &
+ ⇧[O, i + 1] V ≡ T2 &
+ I = LRef i.
+/2 width=3/ 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. ⓓV &
+ ⇧[O, i + 1] V ≡ T2.
+#L #T2 #i #d #e #H
+elim (tps_inv_atom1 … H) -H /2 width=1/
+* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/
+qed-.
+
+lemma tps_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶ T2 → T2 = §p.
+#L #T2 #p #d #e #H
+elim (tps_inv_atom1 … H) -H //
+* #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
+ ∀I,V1,T1. U1 = ⓑ{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 &
+ L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
+ U2 = ⓑ{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 ⊢ ⓑ{I} V1. T1 [d, e] ▶ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 &
+ L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶ T2 &
+ U2 = ⓑ{I} V2. T2.
+/2 width=3/ qed-.
+
+fact tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ▶ U2 →
+ ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
+ U2 = ⓕ{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 ⊢ ⓕ{I} V1. T1 [d, e] ▶ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶ V2 & L ⊢ T1 [d, e] ▶ T2 &
+ U2 = ⓕ{I} V2. T2.
+/2 width=3/ 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 -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
+ elim (lt_refl_false … Hdd)
+| /3 width=1/
+| /3 width=1/
+]
+qed.
+
+lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ▶ T2 → T1 = T2.
+/2 width=6/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → #[T1] ≤ #[T2].
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e normalize
+/3 by monotonic_le_plus_l, le_plus/ (**) (* just /3 width=1/ is too slow *)
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop_ldrop.ma".
+include "Basic_2/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → e = 1 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
+ >(le_to_le_to_eq … Hdi ?) /2 width=1/ -d #K #V #HLK
+ lapply (ldrop_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 width=1/
+| #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. ⓛ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 -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 #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
+| #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
+ @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /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 /3 width=6/
+]
+qed.
+
+lemma tps_lift_be: ∀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 ≤ d → d ≤ dt + et →
+ L ⊢ U1 [dt, et + e] ▶ U2.
+#K #T1 #T2 #dt #et #H elim 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 #Hdtd #_
+ elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+ | -Hdti
+ lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+ ]
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ]
+ ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
+| #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 /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 -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
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
+| #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
+ @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 /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 -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=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
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+| #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
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_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
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
+ ]
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+ [ -Hdtd -Hidet
+ lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+ | -Hdti -Hdedet
+ lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+ elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+ ]
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
+ /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+ elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /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 -L -U1 -U2 -dt -et
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+ | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=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
+ elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+ elim (le_inv_plus_l … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+ @ex2_1_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
+| #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
+ elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
+ <plus_minus // /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
+ elim (IHV12 … HLK … HWV1 ?) -V1 //
+ elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
+]
+qed.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma tps_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 elim H -L -U1 -U2 -d -e
+[ //
+| #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
+ elim (lift_inv_lref2 … H) -H * #H
+ [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
+ elim (lt_refl_false … H)
+ | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
+ elim (lt_refl_false … H)
+ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
+ >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
+ >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 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
+qed.
+
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 [dt, d - dt] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem tps_conf_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.
+#L #T0 #T1 #d1 #e1 #H elim H -L -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (tps_inv_lref1 … H) -H
+ [ #HX destruct /4 width=4/
+ | -Hd1 -Hde1 * #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3/
+ ]
+| #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
+ lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V1) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV01 … HV02) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … HT2 (L. ⓑ{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
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /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 -L1 -T0 -T1 -d1 -e1
+[ /2 width=3/
+| #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 /4 width=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
+ elim (IHV01 … HV02 H) -V0 #V #HV1 #HV2
+ elim (IHT01 … HT02 ?) -T0
+ [ -H #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L2. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … HT2 (L1. ⓑ{I} V) ?) -HT2 /2 width=1/ /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
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -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 //
+ | * #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (lt_to_le_to_lt … (d + e) Hide2 ?) /2 width=4/
+ ]
+| #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (tps_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1/ #HVT2
+ <(tps_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=4/
+| #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
+ lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ lapply (IHT10 … HT02 He) -T0 #HT12
+ lapply (tps_lsubs_conf … HT12 (L. ⓑ{I} V2) ?) -HT12 /2 width=1/ /3 width=1/
+| #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 /3 width=1/
+]
+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 width=3/
+| #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 width=1/ -Hde2i1 #HWT2
+ <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4 width=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
+ lapply (tps_lsubs_conf … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1/ #HT02
+ elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02 ?) -T0 /2 width=1/ #T #HT1 #HT2
+ lapply (tps_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/
+ lapply (tps_lsubs_conf … HT2 (L. ⓑ{I} V2) ?) -HT2 /2 width=1/ /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
+ elim (IHV10 … HV02 ?) -V0 //
+ elim (IHT10 … HT02 ?) -T0 // /3 width=6/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+
+(* DELIFT ON TERMS **********************************************************)
+
+definition delift: nat → nat → lenv → relation term ≝
+ λd,e,L,T1,T2. ∃∃T. L ⊢ T1 [d, e] ▶* T & ⇧[d, e] T2 ≡ T.
+
+interpretation "delift (term)"
+ 'TSubst L T1 d e T2 = (delift d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma delift_lsubs_conf: ∀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 * /3 width=3/
+qed.
+
+lemma delift_bind: ∀I,L,V1,V2,T1,T2,d,e.
+ L ⊢ V1 [d, e] ≡ V2 → L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 →
+ L ⊢ ⓑ{I} V1. T1 [d, e] ≡ ⓑ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
+qed.
+
+lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
+ L ⊢ V1 [d, e] ≡ V2 → L ⊢ T1 [d, e] ≡ T2 →
+ L ⊢ ⓕ{I} V1. T1 [d, e] ≡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma delift_fwd_sort1: ∀L,U2,d,e,k. L ⊢ ⋆k [d, e] ≡ U2 → U2 = ⋆k.
+#L #U2 #d #e #k * #U #HU
+>(tpss_inv_sort1 … HU) -HU #HU2
+>(lift_inv_sort2 … HU2) -HU2 //
+qed-.
+
+lemma delift_fwd_gref1: ∀L,U2,d,e,p. L ⊢ §p [d, e] ≡ U2 → U2 = §p.
+#L #U #d #e #p * #U #HU
+>(tpss_inv_gref1 … HU) -HU #HU2
+>(lift_inv_gref2 … HU2) -HU2 //
+qed-.
+
+lemma delift_fwd_bind1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓑ{I} V1. T1 [d, e] ≡ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
+ L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 &
+ U2 = ⓑ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
+lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+lemma delift_fwd_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓕ{I} V1. T1 [d, e] ≡ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
+ L ⊢ T1 [d, e] ≡ T2 &
+ U2 = ⓕ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
+qed-.
+
+(* Basic Inversion lemmas ***************************************************)
+
+lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≡ T2 → T1 = T2.
+#L #T1 #T2 #d * #T #HT1
+>(tpss_inv_refl_O2 … HT1) -HT1 #HT2
+>(lift_inv_refl_O2 … HT2) -HT2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_lift.ma".
+include "Basic_2/unfold/delift.ma".
+
+(* DELIFT ON TERMS **********************************************************)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma delift_fwd_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 → i < d → U2 = #i.
+#L #U2 #i #d #e * #U #HU #HU2 #Hid
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_lt … HU2) //
+| * #K #V1 #V2 #Hdi
+ lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
+
+lemma delift_fwd_lref1_be: ∀L,U2,d,e,i. L ⊢ #i [d, e] ≡ U2 →
+ d ≤ i → i < d + e →
+ ∃∃K,V1,V2. ⇩[0, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 [0, d + e - i - 1] ≡ V2 &
+ ⇧[0, d] V2 ≡ U2.
+#L #U2 #d #e #i * #U #HU #HU2 #Hdi #Hide
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct elim (lift_inv_lref2_be … HU2 ? ?) //
+| * #K #V1 #V #_ #_ #HLK #HV1 #HVU
+ elim (lift_div_be … HVU … HU2 ? ?) -U // /2 width=1/ /3 width=6/
+]
+qed-.
+
+lemma delift_fwd_lref1_ge: ∀L,U2,i,d,e. L ⊢ #i [d, e] ≡ U2 →
+ d + e ≤ i → U2 = #(i - e).
+#L #U2 #i #d #e * #U #HU #HU2 #Hdei
+elim (tpss_inv_lref1 … HU) -HU
+[ #H destruct >(lift_inv_lref2_ge … HU2) //
+| * #K #V1 #V2 #_ #Hide
+ lapply (lt_to_le_to_lt … Hide Hdei) -Hide -Hdei #Hi
+ elim (lt_refl_false … Hi)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+inductive at: list2 nat nat → relation nat ≝
+| at_nil: ∀i. at ⟠ i i
+| at_lt : ∀des,d,e,i1,i2. i1 < d →
+ at des i1 i2 → at ({d, e} :: des) i1 i2
+| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
+ at des (i1 + e) i2 → at ({d, e} :: des) i1 i2
+.
+
+interpretation "application (generic relocation with pairs)"
+ 'RAt i1 des i2 = (at des i1 i2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact at_inv_nil_aux: ∀des,i1,i2. @[i1] des ≡ i2 → des = ⟠ → i1 = i2.
+#des #i1 #i2 * -des -i1 -i2
+[ //
+| #des #d #e #i1 #i2 #_ #_ #H destruct
+| #des #d #e #i1 #i2 #_ #_ #H destruct
+]
+qed.
+
+lemma at_inv_nil: ∀i1,i2. @[i1] ⟠ ≡ i2 → i1 = i2.
+/2 width=3/ qed-.
+
+fact at_inv_cons_aux: ∀des,i1,i2. @[i1] des ≡ i2 →
+ ∀d,e,des0. des = {d, e} :: des0 →
+ i1 < d ∧ @[i1] des0 ≡ i2 ∨
+ d ≤ i1 ∧ @[i1 + e] des0 ≡ i2.
+#des #i1 #i2 * -des -i1 -i2
+[ #i #d #e #des #H destruct
+| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
+| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
+]
+qed.
+
+lemma at_inv_cons: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
+ i1 < d ∧ @[i1] des ≡ i2 ∨
+ d ≤ i1 ∧ @[i1 + e] des ≡ i2.
+/2 width=3/ qed-.
+
+lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
+ i1 < d → @[i1] des ≡ i2.
+#des #d #e #i1 #e2 #H
+elim (at_inv_cons … H) -H * // #Hdi1 #_ #Hi1d
+lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
+elim (lt_refl_false … Hd)
+qed-.
+
+lemma at_inv_cons_ge: ∀des,d,e,i1,i2. @[i1] {d, e} :: des ≡ i2 →
+ d ≤ i1 → @[i1 + e] des ≡ i2.
+#des #d #e #i1 #e2 #H
+elim (at_inv_cons … H) -H * // #Hi1d #_ #Hdi1
+lapply (le_to_lt_to_lt … Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
+elim (lt_refl_false … Hd)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+(* Main properties **********************************************************)
+
+theorem at_mono: ∀des,i,i1. @[i] des ≡ i1 → ∀i2. @[i] des ≡ i2 → i1 = i2.
+#des #i #i1 #H elim H -des -i -i1
+[ #i #x #H <(at_inv_nil … H) -x //
+| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H
+ lapply (at_inv_cons_lt … H Hid) -H -Hid /2 width=1/
+| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H
+ lapply (at_inv_cons_ge … H Hdi) -H -Hdi /2 width=1/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+inductive minuss: nat → relation (list2 nat nat) ≝
+| minuss_nil: ∀i. minuss i ⟠ ⟠
+| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
+ minuss i ({d, e} :: des1) ({d - i, e} :: des2)
+| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
+ minuss i ({d, e} :: des1) des2
+.
+
+interpretation "minus (generic relocation with pairs)"
+ 'RMinus des1 i des2 = (minuss i des1 des2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ → des2 = ⟠.
+#des1 #des2 #i * -des1 -des2 -i
+[ //
+| #des1 #des2 #d #e #i #_ #_ #H destruct
+| #des1 #des2 #d #e #i #_ #_ #H destruct
+]
+qed.
+
+lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
+/2 width=4/ qed-.
+
+fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
+ ∀d,e,des. des1 = {d, e} :: des →
+ d ≤ i ∧ des ▭ e + i ≡ des2 ∨
+ ∃∃des0. i < d & des ▭ i ≡ des0 &
+ des2 = {d - i, e} :: des0.
+#des1 #des2 #i * -des1 -des2 -i
+[ #i #d #e #des #H destruct
+| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/
+| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/
+]
+qed.
+
+lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+ d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
+ ∃∃des. i < d & des1 ▭ i ≡ des &
+ des2 = {d - i, e} :: des.
+/2 width=3/ qed-.
+
+lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+ d ≤ i → des1 ▭ e + i ≡ des2.
+#des1 #des2 #d #e #i #H
+elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi
+lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+elim (lt_refl_false … Hi)
+qed-.
+
+lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+ i < d →
+ ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} :: des.
+#des1 #des2 #d #e #i #H
+elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid
+lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
+elim (lt_refl_false … Hi)
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/gr2.ma".
+
+(* GENERIC RELOCATION WITH PAIRS ********************************************)
+
+let rec pluss (des:list2 nat nat) (i:nat) on des ≝ match des with
+[ nil2 ⇒ ⟠
+| cons2 d e des ⇒ {d + i, e} :: pluss des i
+].
+
+interpretation "plus (generic relocation with pairs)"
+ 'plus x y = (pluss x y).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma pluss_inv_nil2: ∀i,des. des + i = ⟠ → des = ⟠.
+#i * // normalize
+#d #e #des #H destruct
+qed.
+
+lemma pluss_inv_cons2: ∀i,d,e,des2,des. des + i = {d, e} :: des2 →
+ ∃∃des1. des1 + i = des2 & des = {d - i, e} :: des1.
+#i #d #e #des2 * normalize
+[ #H destruct
+| #d1 #e1 #des1 #H destruct /2 width=3/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop.ma".
+include "Basic_2/unfold/gr2_minus.ma".
+include "Basic_2/unfold/lifts.ma".
+
+(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
+
+inductive ldrops: list2 nat nat → relation lenv ≝
+| ldrops_nil : ∀L. ldrops ⟠ L L
+| ldrops_cons: ∀L1,L,L2,des,d,e.
+ ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} :: des) L1 L2
+.
+
+interpretation "generic local environment slicing"
+ 'RDropStar des T1 T2 = (ldrops des T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2.
+#L1 #L2 #des * -L1 -L2 -des //
+#L1 #L #L2 #d #e #des #_ #_ #H destruct
+qed.
+
+(* Basic_1: was: drop1_gen_pnil *)
+lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2.
+/2 width=3/ qed-.
+
+fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 →
+ ∀d,e,tl. des = {d, e} :: tl →
+ ∃∃L. ⇩*[tl] L1 ≡ L & ⇩[d, e] L ≡ L2.
+#L1 #L2 #des * -L1 -L2 -des
+[ #L #d #e #tl #H destruct
+| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+ /2 width=3/
+qed.
+
+(* Basic_1: was: drop1_gen_pcons *)
+lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} :: des] L1 ≡ L2 →
+ ∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2.
+/2 width=3/ qed-.
+
+lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 →
+ ∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
+ ⇩*[des1] K1 ≡ K2 &
+ ⇧*[des1] V2 ≡ V1 &
+ L1 = K1. ⓑ{I} V1.
+#I #des #i #des2 #H elim H -des -i -des2
+[ #i #L1 #K2 #V2 #H
+ >(ldrops_inv_nil … H) -L1 /2 width=7/
+| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+ elim (ldrops_inv_cons … H) -H #L #HL1 #H
+ elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct
+ elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ]
+ normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *)
+| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+ elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ /4 width=7/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: drop1_skip_bind *)
+lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
+ ∀I. ⇩*[des + 1] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
+#L1 #L2 #des #H elim H -L1 -L2 -des
+[ #L #V1 #V2 #HV12 #I
+ >(lifts_inv_nil … HV12) -HV12 //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
+ elim (lifts_inv_cons … H) -H /3 width=5/
+].
+qed.
+
+(* Basic_1: removed theorems 1: drop1_getl_trans
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrop_ldrop.ma".
+include "Basic_2/unfold/ldrops.ma".
+
+(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
+
+(* Properties concerning basic local environment slicing ********************)
+
+lemma ldrops_ldrop_trans: ∀L1,L,des. ⇩*[des] L1 ≡ L → ∀L2,i. ⇩[0, i] L ≡ L2 →
+ ∃∃L0,des0,i0. ⇩[0, i0] L1 ≡ L0 & ⇩*[des0] L0 ≡ L2 &
+ @[i] des ≡ i0 & des ▭ i ≡ des0.
+#L1 #L #des #H elim H -L1 -L -des
+[ /2 width=7/
+| #L1 #L3 #L #des3 #d #e #_ #HL3 #IHL13 #L2 #i #HL2
+ elim (lt_or_ge i d) #Hid
+ [ elim (ldrop_trans_le … HL3 … HL2 ?) -L /2 width=2/ #L #HL3 #HL2
+ elim (IHL13 … HL3) -L3 /3 width=7/
+ | lapply (ldrop_trans_ge … HL3 … HL2 ?) -L // #HL32
+ elim (IHL13 … HL32) -L3 /3 width=7/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/ldrops_ldrop.ma".
+
+(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: drop1_trans *)
+theorem ldrops_trans: ∀L,L2,des2. ⇩*[des2] L ≡ L2 → ∀L1,des1. ⇩*[des1] L1 ≡ L →
+ ⇩*[des2 @ des1] L1 ≡ L2.
+#L #L2 #des2 #H elim H -L -L2 -des2 // /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.ma".
+include "Basic_2/unfold/gr2_plus.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+inductive lifts: list2 nat nat → relation term ≝
+| lifts_nil : ∀T. lifts ⟠ T T
+| lifts_cons: ∀T1,T,T2,des,d,e.
+ ⇧[d,e] T1 ≡ T → lifts des T T2 → lifts ({d, e} :: des) T1 T2
+.
+
+interpretation "generic relocation (term)"
+ 'RLiftStar des T1 T2 = (lifts des T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_nil_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → des = ⟠ → T1 = T2.
+#T1 #T2 #des * -T1 -T2 -des //
+#T1 #T #T2 #d #e #des #_ #_ #H destruct
+qed.
+
+lemma lifts_inv_nil: ∀T1,T2. ⇧*[⟠] T1 ≡ T2 → T1 = T2.
+/2 width=3/ qed-.
+
+fact lifts_inv_cons_aux: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 →
+ ∀d,e,tl. des = {d, e} :: tl →
+ ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[tl] T ≡ T2.
+#T1 #T2 #des * -T1 -T2 -des
+[ #T #d #e #tl #H destruct
+| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+ /2 width=3/
+qed.
+
+lemma lifts_inv_cons: ∀T1,T2,d,e,des. ⇧*[{d, e} :: des] T1 ≡ T2 →
+ ∃∃T. ⇧[d, e] T1 ≡ T & ⇧*[des] T ≡ T2.
+/2 width=3/ qed-.
+
+(* Basic_1: was: lift1_sort *)
+lemma lifts_inv_sort1: ∀T2,k,des. ⇧*[des] ⋆k ≡ T2 → T2 = ⋆k.
+#T2 #k #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_sort1 … H) -H /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+lemma lifts_inv_lref1: ∀T2,des,i1. ⇧*[des] #i1 ≡ T2 →
+ ∃∃i2. @[i1] des ≡ i2 & T2 = #i2.
+#T2 #des elim des -des
+[ #i1 #H <(lifts_inv_nil … H) -H /2 width=3/
+| #d #e #des #IH #i1 #H
+ elim (lifts_inv_cons … H) -H #X #H1 #H2
+ elim (lift_inv_lref1 … H1) -H1 * #Hdi1 #H destruct
+ elim (IH … H2) -IH -H2 /3 width=3/
+]
+qed-.
+
+lemma lifts_inv_gref1: ∀T2,p,des. ⇧*[des] §p ≡ T2 → T2 = §p.
+#T2 #p #des elim des -des
+[ #H <(lifts_inv_nil … H) -H //
+| #d #e #des #IH #H
+ elim (lifts_inv_cons … H) -H #X #H
+ >(lift_inv_gref1 … H) -H /2 width=1/
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+lemma lifts_inv_bind1: ∀I,T2,des,V1,U1. ⇧*[des] ⓑ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des + 1] U1 ≡ U2 &
+ T2 = ⓑ{I} V2. U2.
+#I #T2 #des elim des -des
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5/
+| #d #e #des #IHdes #V1 #U1 #H
+ elim (lifts_inv_cons … H) -H #X #H #HT2
+ elim (lift_inv_bind1 … H) -H #V #U #HV1 #HU1 #H destruct
+ elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+lemma lifts_inv_flat1: ∀I,T2,des,V1,U1. ⇧*[des] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⇧*[des] V1 ≡ V2 & ⇧*[des] U1 ≡ U2 &
+ T2 = ⓕ{I} V2. U2.
+#I #T2 #des elim des -des
+[ #V1 #U1 #H
+ <(lifts_inv_nil … H) -H /2 width=5/
+| #d #e #des #IHdes #V1 #U1 #H
+ elim (lifts_inv_cons … H) -H #X #H #HT2
+ elim (lift_inv_flat1 … H) -H #V #U #HV1 #HU1 #H destruct
+ elim (IHdes … HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ /3 width=5/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lifts_simple_dx: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒[T1] → 𝐒[T2].
+#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_dx/
+qed-.
+
+lemma lifts_simple_sn: ∀T1,T2,des. ⇧*[des] T1 ≡ T2 → 𝐒[T2] → 𝐒[T1].
+#T1 #T2 #des #H elim H -T1 -T2 -des // /3 width=5 by lift_simple_sn/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_bind: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+ ∀T1. ⇧*[des + 1] T1 ≡ T2 →
+ ⇧*[des] ⓑ{I} V1. T1 ≡ ⓑ{I} V2. T2.
+#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3/
+]
+qed.
+
+lemma lifts_flat: ∀I,T2,V1,V2,des. ⇧*[des] V1 ≡ V2 →
+ ∀T1. ⇧*[des] T1 ≡ T2 →
+ ⇧*[des] ⓕ{I} V1. T1 ≡ ⓕ{I} V2. T2.
+#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
+[ #V #T1 #H >(lifts_inv_nil … H) -H //
+| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+ elim (lifts_inv_cons … H) -H /3 width=3/
+]
+qed.
+
+lemma lifts_total: ∀des,T1. ∃T2. ⇧*[des] T1 ≡ T2.
+#des elim des -des /2 width=2/
+#d #e #des #IH #T1
+elim (lift_total T1 d e) #T #HT1
+elim (IH T) -IH /3 width=4/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/unfold/gr2_minus.ma".
+include "Basic_2/unfold/lifts.ma".
+
+(* GENERIC TERM RELOCATION **************************************************)
+
+(* Properties concerning basic term relocation ******************************)
+
+(* Basic_1: was: lift1_xhg (right to left) *)
+lemma lifts_lift_trans_le: ∀T1,T,des. ⇧*[des] T1 ≡ T → ∀T2. ⇧[0, 1] T ≡ T2 →
+ ∃∃T0. ⇧[0, 1] T1 ≡ T0 & ⇧*[des + 1] T0 ≡ T2.
+#T1 #T #des #H elim H -T1 -T -des
+[ /2 width=3/
+| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2
+ elim (IHT13 … HT2) -T #T #HT3 #HT2
+ elim (lift_trans_le … HT13 … HT3 ?) -T3 // /3 width=5/
+]
+qed-.
+
+(* Basic_1: was: lift1_free (right to left) *)
+lemma lifts_lift_trans: ∀des,i,i0. @[i] des ≡ i0 →
+ ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 →
+ ∀T1,T0. ⇧*[des0] T1 ≡ T0 →
+ ∀T2. ⇧[O, i0 + 1] T0 ≡ T2 →
+ ∃∃T. ⇧[0, i + 1] T1 ≡ T & ⇧*[des] T ≡ T2.
+#des elim des -des normalize
+[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2
+ <(at_inv_nil … H1) -x #HT02
+ lapply (minuss_inv_nil1 … H2) -H2 #H
+ >(pluss_inv_nil2 … H) in HT10; -des0 #H
+ >(lifts_inv_nil … H) -T1 /2 width=3/
+| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
+ elim (at_inv_cons … H1) -H1 * #Hid #Hi0
+ [ elim (minuss_inv_cons1_lt … H2 ?) -H2 [2: /2 width=1/ ] #des1 #Hdes1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
+ elim (pluss_inv_cons2 … H) -H #des2 #H1 #H2 destruct
+ elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
+ elim (IHdes … Hi0 … Hdes1 … HT0 … HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02
+ elim (lift_trans_le … HT1 … HT0 ?) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
+ | >commutative_plus in Hi0; #Hi0
+ lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hdes0
+ elim (IHdes … Hi0 … Hdes0 … HT10 … HT02) -IHdes -Hi0 -Hdes0 -T0 #T0 #HT0 #HT02
+ elim (lift_split … HT0 d (i+1) ? ? ?) -HT0 /2 width=1/ /3 width=5/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/unfold/lifts_lift.ma".
+include "Basic_2/unfold/lifts_vector.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lifts1_xhg (right to left) *)
+lemma liftsv_liftv_trans_le: ∀T1s,Ts,des. ⇧*[des] T1s ≡ Ts →
+ ∀T2s:list term. ⇧[0, 1] Ts ≡ T2s →
+ ∃∃T0s. ⇧[0, 1] T1s ≡ T0s & ⇧*[des + 1] T0s ≡ T2s.
+#T1s #Ts #des #H elim H -T1s -Ts
+[ #T1s #H
+ >(liftv_inv_nil1 … H) -T1s /2 width=3/
+| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #H
+ elim (liftv_inv_cons1 … H) -H #T2 #T2s #HT2 #HT2s #H destruct
+ elim (IHT1s … HT2s) -Ts #Ts #HT1s #HT2s
+ elim (lifts_lift_trans_le … HT1 … HT2) -T /3 width=5/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/lifts_lift.ma".
+
+(* GENERIC RELOCATION *******************************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: lift1_lift1 (left to right) *)
+theorem lifts_trans: ∀T1,T,des1. ⇧*[des1] T1 ≡ T → ∀T2:term. ∀des2. ⇧*[des2] T ≡ T2 →
+ ⇧*[des1 @ des2] T1 ≡ T2.
+#T1 #T #des1 #H elim H -T1 -T -des1 // /3 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_vector.ma".
+include "Basic_2/unfold/lifts.ma".
+
+(* GENERIC TERM VECTOR RELOCATION *******************************************)
+
+inductive liftsv (des:list2 nat nat) : relation (list term) ≝
+| liftsv_nil : liftsv des ◊ ◊
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+ ⇧*[des] T1 ≡ T2 → liftsv des T1s T2s →
+ liftsv des (T1 :: T1s) (T2 :: T2s)
+.
+
+interpretation "generic relocation (vector)"
+ 'RLiftStar des T1s T2s = (liftsv des T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lifts_inv_applv1: ∀V1s,U1,T2,des. ⇧*[des] Ⓐ V1s. U1 ≡ T2 →
+ ∃∃V2s,U2. ⇧*[des] V1s ≡ V2s & ⇧*[des] U1 ≡ U2 &
+ T2 = Ⓐ V2s. U2.
+#V1s elim V1s -V1s normalize
+[ #T1 #T2 #des #HT12
+ @(ex3_2_intro) [3,4: // |1,2: skip | // ] (**) (* explicit constructor *)
+| #V1 #V1s #IHV1s #T1 #X #des #H
+ elim (lifts_inv_flat1 … H) -H #V2 #Y #HV12 #HY #H destruct
+ elim (IHV1s … HY) -IHV1s -HY #V2s #T2 #HV12s #HT12 #H destruct
+ @(ex3_2_intro) [4: // |3: /2 width=2/ |1,2: skip | // ] (**) (* explicit constructor *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_applv: ∀V1s,V2s,des. ⇧*[des] V1s ≡ V2s →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 →
+ ⇧*[des] Ⓐ V1s. T1 ≡ Ⓐ V2s. T2.
+#V1s #V2s #des #H elim H -V1s -V2s // /3 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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:predicate lenv. 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 width=3/ qed.
+
+lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
+/2 width=1/ qed.
+
+lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
+#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
+#L #L2 #_ #HL2
+>(ltps_fwd_length … HL2) /3 width=5/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
+/2 width=3 by ltps_fwd_length/
+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
+>(ltps_inv_atom1 … HL2) -HL2 //
+qed-.
+
+fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
+#d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
+#L2 #L #_ #HL2 #IHL2 #H destruct
+lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
+qed.
+
+lemma ltpss_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. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
+ K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L1 = K1. ⓑ{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 /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. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
+ L1 = K1. ⓑ{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. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L1 = K1. ⓑ{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 /2 width=5/
+]
+qed.
+
+lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
+ K2 ⊢ V1 [d - 1, e] ▶ V2 &
+ L1 = K1. ⓑ{I} V1.
+/2 width=1/ qed.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_ldrop.ma".
+include "Basic_2/unfold/ltpss.ma".
+
+(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+
+lemma ltpss_ldrop_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 @(ltpss_ind … H) -L1 // /3 width=6/
+qed.
+
+lemma ltpss_ldrop_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 @(ltpss_ind … H) -L0 // /3 width=6/
+qed.
+
+lemma ltpss_ldrop_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 @(ltpss_ind … H) -L1
+[ /2 width=3/
+| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
+ elim (IHL … HL02 Hd1e2 He2de1) -L0 #L0 #HL20 #HL0
+ elim (ltps_ldrop_conf_be … HL1 … HL0 Hd1e2 He2de1) -L /3 width=3/
+]
+qed.
+
+lemma ltpss_ldrop_trans_be: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. L [0, d1 + e1 - e2] ▶* L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
+[ /2 width=3/
+| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #Hd1e2 #He2de1
+ elim (ltps_ldrop_trans_be … HL0 … HL02 Hd1e2 He2de1) -L0 #L0 #HL02 #HL0
+ elim (IHL … HL0 Hd1e2 He2de1) -L /3 width=3/
+]
+qed.
+
+lemma ltpss_ldrop_conf_le: ∀L0,L1,d1,e1. L0 [d1, e1] ▶* L1 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L2 [d1 - e2, e1] ▶* L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H @(ltpss_ind … H) -L1
+[ /2 width=3/
+| #L #L1 #_ #HL1 #IHL #L2 #e2 #HL02 #He2d1
+ elim (IHL … HL02 He2d1) -L0 #L0 #HL20 #HL0
+ elim (ltps_ldrop_conf_le … HL1 … HL0 He2d1) -L /3 width=3/
+]
+qed.
+
+lemma ltpss_ldrop_trans_le: ∀L1,L0,d1,e1. L1 [d1, e1] ▶* L0 →
+ ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. L [d1 - e2, e1] ▶* L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H @(ltpss_ind … H) -L0
+[ /2 width=3/
+| #L #L0 #_ #HL0 #IHL #L2 #e2 #HL02 #He2d1
+ elim (ltps_ldrop_trans_le … HL0 … HL02 He2d1) -L0 #L0 #HL02 #HL0
+ elim (IHL … HL0 He2d1) -L /3 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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_tpss.ma".
+
+(* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
+
+(* Main properties **********************************************************)
+
+theorem ltpss_trans_eq: ∀L1,L,L2,d,e.
+ L1 [d, e] ▶* L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
+/2 width=3/ qed.
+
+lemma ltpss_tpss2: ∀L1,L2,I,V1,V2,e.
+ L1 [0, e] ▶* L2 → L2 ⊢ V1 [0, e] ▶* V2 →
+ L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #H @(tpss_ind … H) -V2
+[ /2 width=1/
+| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2 width=1/
+]
+qed.
+
+lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
+ L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶* V2 →
+ 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
+ L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶* V2 →
+ L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #H @(tpss_ind … H) -V2
+[ /2 width=1/
+| #V #V2 #_ #HV2 #IHV @(ltpss_trans_eq … IHV) /2 width=1/
+]
+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. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) // /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 width=1/
+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 width=1/ (**) (* /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 width=1/
+| #L #L1 #_ #HL1 #IHL #T2 #U2 #HTU2
+ lapply (ltps_tpss_trans_eq … HL1 HTU2) -HL1 -HTU2 /2 width=1/
+]
+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 width=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 width=1/ (**) (* /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 width=3/
+| -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 width=3/
+]
+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 width=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 width=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 width=3/
+]
+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 width=1/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma ltpss_tps2: ∀L1,L2,I,e.
+ L1 [0, e] ▶* L2 → ∀V1,V2. L2 ⊢ V1 [0, e] ▶ V2 →
+ L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #e #H @(ltpss_ind … H) -L2
+[ /3 width=1/
+| #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. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
+>(plus_minus_m_m e 1) // /2 width=1/
+qed.
+
+lemma ltpss_tps1: ∀L1,L2,I,d,e. L1 [d, e] ▶* L2 →
+ ∀V1,V2. L2 ⊢ V1 [d, e] ▶ V2 →
+ L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #d #e #H @(ltpss_ind … H) -L2
+[ /3 width=1/
+| #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. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
+>(plus_minus_m_m d 1) // /2 width=1/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma ltpss_fwd_tpss21: ∀e,K1,I,V1,L2. 0 < e → K1. ⓑ{I} V1 [0, e] ▶* L2 →
+ ∃∃K2,V2. K1 [0, e - 1] ▶* K2 & K1 ⊢ V1 [0, e - 1] ▶* V2 &
+ L2 = K2. ⓑ{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
+ 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. ⓑ{I} V1 [d, e] ▶* L2 →
+ ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
+ K1 ⊢ V1 [d - 1, e] ▶* V2 &
+ L2 = K2. ⓑ{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
+ 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-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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:predicate term. 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 width=3/ qed.
+
+lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶* T2 →
+ ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ▶* T2.
+/3 width=3/ qed.
+
+lemma tpss_refl: ∀d,e,L,T. L ⊢ T [d, e] ▶* T.
+/2 width=1/ qed.
+
+lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 [d, e] ▶* V2 →
+ ∀I,T1,T2. L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 →
+ L ⊢ ⓑ{I} V1. T1 [d, e] ▶* ⓑ{I} V2. T2.
+#L #V1 #V2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #I #T1 #T2 #HT12 elim 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_lsubs_conf … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #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 ⊢ ⓕ{I} V1. T1 [d, e] ▶* ⓕ{I} V2. T2.
+#L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -V2
+[ #V2 #HV12 #HT12 elim HT12 -T2
+ [ /3 width=1/
+ | #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) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 -Hd21 -Hde12 /2 width=3/
+]
+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) -T2
+[ //
+| #T #T2 #_ #HT12 #IHT
+ lapply (tps_weak_top … HT12) -HT12 /2 width=3/
+]
+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) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_sort1 … HT2) -HT2 //
+]
+qed-.
+
+(* Note: this can be derived from tpss_inv_atom1 *)
+lemma tpss_inv_gref1: ∀L,T2,p,d,e. L ⊢ §p [d, e] ▶* T2 → T2 = §p.
+#L #T2 #p #d #e #H @(tpss_ind … H) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT destruct
+ >(tps_inv_gref1 … HT2) -HT2 //
+]
+qed-.
+
+lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 [d, e] ▶* U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 &
+ L. ⓑ{I} V2 ⊢ T1 [d + 1, e] ▶* T2 &
+ U2 = ⓑ{I} V2. T2.
+#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+]
+qed-.
+
+lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 [d, e] ▶* U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ▶* V2 & L ⊢ T1 [d, e] ▶* T2 &
+ U2 = ⓕ{I} V2. T2.
+#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+[ /2 width=5/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ 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) -T2
+[ //
+| #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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. ⓓ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) -U1
+[ /3 width=4/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU … HU0) -IHU #H
+ lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+ lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
+ lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
+]
+qed.
+
+(* Advanced inverion lemmas *************************************************)
+
+lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} [d, e] ▶* T2 →
+ T2 = ⓪{I} ∨
+ ∃∃K,V1,V2,i. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓ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) -T2
+[ /2 width=1/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
+ | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
+ lapply (ldrop_fwd_ldrop2 … HLK) #H
+ elim (tps_inv_lift1_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
+ @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
+ ]
+]
+qed-.
+
+lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ▶* T2 →
+ T2 = #i ∨
+ ∃∃K,V1,V2. d ≤ i & i < d + e &
+ ⇩[O, i] L ≡ K. ⓓV1 &
+ K ⊢ V1 [0, d + e - i - 1] ▶* V2 &
+ ⇧[O, i + 1] V2 ≡ T2.
+#L #T2 #i #d #e #H
+elim (tpss_inv_atom1 … H) -H /2 width=1/
+* #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
+qed-.
+
+lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ▶* T2 →
+ ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶* T2 →
+ ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+ L ⊢ U1 [dt, et] ▶* U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … 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_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+qed.
+
+lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶* T2 →
+ ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 [dt, et + e] ▶* U2.
+#K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … 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_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
+]
+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) -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 width=3/
+]
+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) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
+
+lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶* U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [dt, et - e] ▶* T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=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) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=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) -U2 //
+#U #U2 #_ #HU2 #IHU destruct
+<(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+qed.
+
+lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶* U2 →
+ ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+ dt ≤ d → dt + et ≤ d + e →
+ ∃∃T2. K ⊢ T1 [dt, d - dt] ▶* T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
+[ /2 width=3/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 width=3/
+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 width=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 width=3/
+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 width=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 width=3/ (**) (* /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
+ lapply (ldrop_fwd_lw … HLK1) normalize #H1
+ elim (ltps_ldrop_trans_be … HL10 … HLK1 ? ?) -HL10 -HLK1 // /2 width=2/ #X #H #HLK0
+ elim (ltps_inv_tps22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
+ lapply (tps_fwd_tw … HV01) #H2
+ lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 /2 width=1/ -H2 #H
+ lapply (IH … HV01 … HK01 ? ?) -IH -HV01 -HK01
+ [1,3: // |2,4: skip | normalize /2 width=1/ | /3 width=6/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct
+ lapply (tps_lsubs_conf … HT12 (L. ⓑ{I} V1) ?) -HT12 /2 width=1/ #HT12
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=2/ ] #HV12
+ lapply (IH … HT12 (L0. ⓑ{I} V1) ? ? ?) -IH -HT12 [1,3,5: /2 width=2/ |2,4: skip | normalize // ] -HL0 #HT12
+ lapply (tpss_lsubs_conf … HT12 (L0. ⓑ{I} V2) ?) -HT12 /2 width=1/
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=3/ ]
+ lapply (IH … HT12 … HL0 ? ?) -IH -HT12 [1,3,5: normalize // |2,4: skip ] -HL0 /2 width=1/
+]
+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 width=3/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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) -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 width=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 width=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 width=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 width=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) -T2
+[ /2 width=3/
+| #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 /2 width=1/ ]
+ /3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
+]
+qed.
+
+lemma tpss_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 (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem tpss_conf_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 width=3/ qed.
+
+theorem tpss_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.
+/3 width=3/ qed.
+
+theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 [d, e] ▶* T → L ⊢ T [d, e] ▶* T2 →
+ L ⊢ T1 [d, e] ▶* T2.
+/2 width=3/ qed.
+
+theorem tpss_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.
+/3 width=3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "arithmetics/nat.ma".
+include "Ground_2/star.ma".
+
+(* ARITHMETICAL PROPERTIES **************************************************)
+
+(* Equations ****************************************************************)
+
+lemma le_plus_minus: ∀m,n,p. p ≤ n → m + n - p = m + (n - p).
+/2 by plus_minus/ qed.
+
+lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
+/2 by plus_minus/ qed.
+
+lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
+#a #b #c1 #H >minus_minus_comm >minus_le_minus_minus_comm //
+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 width=1/
+qed.
+
+lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
+/3 by monotonic_le_minus_l, le_to_le_to_eq, le_n/ qed.
+
+lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
+ a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
+#a1 #a2 #b #c1 #H1 #H2 >plus_minus // /2 width=1/
+qed.
+
+(* inversion & forward lemmas ***********************************************)
+
+axiom eq_nat_dec: ∀n1,n2:nat. Decidable (n1 = n2).
+
+axiom lt_dec: ∀n1,n2. Decidable (n1 < n2).
+
+lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
+#m #n elim (lt_or_ge m n) /2 width=1/
+#H elim H -m /2 width=1/
+#m #Hm * #H /2 width=1/ /3 width=1/
+qed-.
+
+lemma lt_refl_false: ∀n. n < n → False.
+#n #H elim (lt_to_not_eq … H) -H /2 width=1/
+qed-.
+
+lemma lt_zero_false: ∀n. n < 0 → False.
+#n #H elim (lt_to_not_le … H) -H /2 width=1/
+qed-.
+
+lemma false_lt_to_le: ∀x,y. (x < y → False) → y ≤ x.
+#x #y #H elim (decidable_lt x y) /2 width=1/
+#Hxy elim (H Hxy)
+qed-.
+
+(*
+lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
+/3 width=2/
+
+lemma le_or_ge: ∀m,n. m ≤ n ∨ n ≤ m.
+#m #n elim (lt_or_ge m n) /2 width=1/ /3 width=2/
+qed-.
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* 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 all A (R:predicate A) (l:list A) on l ≝
+ match l with
+ [ nil ⇒ True
+ | cons hd tl ⇒ R hd ∧ all A R tl
+ ].
+
+inductive list2 (A1,A2:Type[0]) : Type[0] :=
+ | nil2 : list2 A1 A2
+ | cons2: A1 → A2 → list2 A1 A2 → list2 A1 A2.
+
+interpretation "nil (list of pairs)" 'Nil2 = (nil2 ? ?).
+
+interpretation "cons (list of pairs)" 'Cons hd1 hd2 tl = (cons2 ? ? hd1 hd2 tl).
+
+let rec append2 (A1,A2:Type[0]) (l1,l2:list2 A1 A2) on l1 ≝ match l1 with
+[ nil2 ⇒ l2
+| cons2 a1 a2 tl ⇒ {a1, a2} :: append2 A1 A2 tl l2
+].
+
+interpretation "append (list of pairs)"
+ 'Append l1 l2 = (append2 ? ? l1 l2).
+
+let rec length2 (A1,A2:Type[0]) (l:list2 A1 A2) on l ≝ match l with
+[ nil2 ⇒ 0
+| cons2 _ _ l ⇒ length2 A1 A2 l + 1
+].
+
+interpretation "length (list of pairs)"
+ 'card l = (length2 ? ? l).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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( ◊ )"
+ non associative with precedence 90
+ for @{'Nil}.
+
+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 }.
+
+notation "hvbox( ⟠ )"
+ non associative with precedence 90
+ for @{'Nil2}.
+
+notation "hvbox( { hd1 , break hd2 } :: break tl )"
+ non associative with precedence 47
+ for @{'Cons $hd1 $hd2 $tl}.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+include "Ground_2/notation.ma".
+
+(* PROPERTIES OF RELATIONS **************************************************)
+
+definition Decidable: Prop → Prop ≝ λR. R ∨ (R → False).
+
+definition confluent2: ∀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 transitive2: ∀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. confluent2 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 -a1
+[ #a1 #Ha01 #a2 #Ha02
+ elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+ elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+ elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/
+]
+qed.
+
+lemma TC_strip2: ∀A,R1,R2. confluent2 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 -a2
+[ #a2 #Ha02 #a1 #Ha01
+ elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
+| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
+ elim (IHa0 … Ha01) -a0 #a0 #Ha10 #Ha0
+ elim (HR12 … Ha0 … Ha2) -HR12 -a /4 width=3/
+]
+qed.
+
+lemma TC_confluent2: ∀A,R1,R2.
+ confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
+#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
+[ #a1 #Ha01 #a2 #Ha02
+ elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
+| #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+ elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+ elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/
+]
+qed.
+
+lemma TC_strap1: ∀A,R1,R2. transitive2 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 -a0
+[ #a0 #Ha10 #a2 #Ha02
+ elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+ elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
+ elim (IHa … Ha0) -a /4 width=3/
+]
+qed.
+
+lemma TC_strap2: ∀A,R1,R2. transitive2 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 -a2
+[ #a2 #Ha02 #a1 #Ha10
+ elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
+| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
+ elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
+ elim (HR12 … Ha0 … Ha02) -HR12 -a /4 width=3/
+]
+qed.
+
+lemma TC_transitive2: ∀A,R1,R2.
+ transitive2 A R1 R2 → transitive2 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 width=3/
+| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
+ elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
+ elim (IHa … Ha0) -a /4 width=3/
+]
+qed.
+
+definition NF: ∀A. relation A → relation A → predicate A ≝
+ λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
+
+inductive SN (A) (R,S:relation A): predicate A ≝
+| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → False) → SN A R S a2) → SN A R S a1
+.
+
+lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
+#A #R #S #a1 #Ha1
+@SN_intro #a2 #HRa12 #HSa12
+elim (HSa12 ?) -HSa12 /2 width=1/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+(* TRICOTOMY FUNCTION *******************************************************)
+
+let rec tri (A:Type[0]) m n a b c on m : A ≝
+ match m with
+ [ O ⇒ match n with [ O ⇒ b | S n ⇒ a ]
+ | S m ⇒ match n with [ O ⇒ c | S n ⇒ tri A m n a b c ]
+ ].
+
+(* Basic properties *********************************************************)
+
+lemma tri_lt: ∀A,a,b,c,n,m. m < n → tri A m n a b c = a.
+#A #a #b #c #n elim n -n
+[ #m #H elim (lt_zero_false … H)
+| #n #IH #m elim m -m // /3 width=1/
+]
+qed.
+
+lemma tri_eq: ∀A,a,b,c,m. tri A m m a b c = b.
+#A #a #b #c #m elim m -m normalize //
+qed.
+
+lemma tri_gt: ∀A,a,b,c,m,n. n < m → tri A m n a b c = c.
+#A #a #b #c #m elim m -m
+[ #n #H elim (lt_zero_false … H)
+| #m #IH #n elim n -n // /3 width=1/
+]
+qed.
+
--- /dev/null
+<?xml version="1.0" encoding="utf-8"?>
+<helm_registry>
+ <section name="matita">
+ <key name="rt_base_dir">$(MATITA_RT_BASE_DIR)</key>
+<!--
+ <key name="system">false</key>
+ <key name="map_unicode_to_tex">false</key>
+ <key name="do_heavy_checks">true</key>
+ <key name="include_path">lib</key>
+-->
+ </section>
+ <section name="xoa">
+ <key name="output_dir">contribs/lambda_delta/Ground_2/</key>
+ <key name="objects">xoa</key>
+ <key name="notations">xoa_notation</key>
+ <key name="include">basics/pts.ma</key>
+ <key name="ex">1 2</key>
+ <key name="ex">2 1</key>
+ <key name="ex">2 2</key>
+ <key name="ex">2 3</key>
+ <key name="ex">3 1</key>
+ <key name="ex">3 2</key>
+ <key name="ex">3 3</key>
+ <key name="ex">4 2</key>
+ <key name="ex">4 3</key>
+ <key name="ex">4 4</key>
+ <key name="ex">5 2</key>
+ <key name="ex">5 3</key>
+ <key name="ex">5 4</key>
+ <key name="ex">6 4</key>
+ <key name="ex">6 6</key>
+ <key name="ex">7 6</key>
+ <key name="or">3</key>
+ <key name="or">4</key>
+ <key name="and">3</key>
+ </section>
+</helm_registry>
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 (1, 2) *)
+
+inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝
+ | ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ?
+.
+
+interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
+
+(* 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 (2, 3) *)
+
+inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝
+ | ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? 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, 2) *)
+
+inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝
+ | ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
+
+(* 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).
+
+(* multiple conjunction connective (3) *)
+
+inductive and3 (P0,P1,P2:Prop) : Prop ≝
+ | and3_intro: P0 → P1 → P2 → and3 ? ? ?
+.
+
+interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 (1, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }.
+
+(* 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 (2, 3) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
+ 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) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
+ 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) }.
+
+(* 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, 2) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 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}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) (λ${ident x0}.λ${ident x1}.$P3) (λ${ident x0}.λ${ident x1}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$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.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }.
+
+(* 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 }.
+
+(* multiple conjunction connective (3) *)
+
+notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)"
+ non associative with precedence 35
+ for @{ 'And $P0 $P1 $P2 }.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 width=3/
+qed.
projects / helm.git / commitdiff