--- /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/rt_transition/cpr.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* A reduction cycle in two steps: the term Omega ***************************)
+
+definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
+
+definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
+
+definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
+
+(* Basic properties *********************************************************)
+
+lemma Delta_lifts: ∀W1,W2,f. ⬆*[f] W1 ≡ W2 →
+ ⬆*[f] (Delta W1) ≡ (Delta W2).
+/4 width=1 by lifts_lref, lifts_bind, lifts_flat/ qed.
+
+(* Main properties **********************************************************)
+
+theorem cpr_Omega_12: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega1 W ➡[h] Omega2 W.
+/2 width=1 by cpm_beta/ qed.
+
+theorem cpr_Omega_21: ∀h,G,L,W. ⦃G, L⦄ ⊢ Omega2 W ➡[h] Omega1 W.
+#h #G #L #W1 elim (lifts_total W1 (𝐔❴1❵))
+/5 width=5 by lifts_flat, cpm_zeta, cpm_eps, cpm_appl, cpm_delta, Delta_lifts/
+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/fpbg_fpbs.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
+
+definition ApplDelta: term → nat → term ≝ λW,s. +ⓛW.ⓐ⋆s.ⓐ#0.#0.
+
+definition ApplOmega1: term → nat → term ≝ λW,s. ⓐ(ApplDelta W s).(ApplDelta W s).
+
+definition ApplOmega2: term → nat → term ≝ λW,s. +ⓓⓝW.(ApplDelta W s).ⓐ⋆s.ⓐ#0.#0.
+
+definition ApplOmega3: term → nat → term ≝ λW,s. ⓐ⋆s.(ApplOmega1 W s).
+
+(* Basic properties *********************************************************)
+
+lemma ApplDelta_lift: ∀W1,W2,s,l,k. ⬆[l, k] W1 ≡ W2 →
+ ⬆[l, k] (ApplDelta W1 s) ≡ (ApplDelta W2 s).
+/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
+
+lemma cpr_ApplOmega_12: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡ ApplOmega2 W s.
+/2 width=1 by cpr_beta/ qed.
+
+lemma cpr_ApplOmega_23: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega2 W s ➡ ApplOmega3 W s.
+#G #L #W1 #s elim (lift_total W1 0 1) #W2 #HW12
+@(cpr_zeta … (ApplOmega3 W2 s)) /4 width=1 by ApplDelta_lift, lift_flat/
+@cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 s) ? 0)
+[3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
+qed.
+
+lemma cpxs_ApplOmega_13: ∀h,o,G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡*[h,o] ApplOmega3 W s.
+/4 width=3 by cpxs_strap1, cpr_cpx/ qed.
+
+lemma fqup_ApplOmega_13: ∀G,L,W,s. ⦃G, L, ApplOmega3 W s⦄ ⊐+ ⦃G, L, ApplOmega1 W s⦄.
+/2 width=1 by/ qed.
+
+(* Main properties **********************************************************)
+
+theorem fpbg_refl: ∀h,o,G,L,W,s. ⦃G, L, ApplOmega1 W s⦄ >≡[h,o] ⦃G, L, ApplOmega1 W s⦄.
+/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ 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/dynamic/snv.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Extended validy (basic?_2) vs. restricted validity (basic_1) *************)
+
+(* extended validity of a closure, last arg of snv_appl > 1 *)
+lemma snv_extended: ∀h,o,a,G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{a}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 ¡[h, o].
+#h #o #a #G #L #s elim (deg_total h o s)
+#d #Hd @(snv_appl … a … (⋆s) … (⋆s) (0+1+1))
+[ /4 width=5 by snv_lref, drop_drop_lt/
+| /4 width=13 by snv_bind, snv_lref/
+| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
+| @(lstas_scpds … (d+1+1))
+ /5 width=11 by lstas_bind, lstas_succ, da_bind, da_ldec, da_sort, lift_bind/
+]
+qed.
+
+(* restricted validity of the η-expanded closure, last arg of snv_appl = 1 **)
+lemma snv_restricted: ∀h,o,a,G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{a}⋆s.⋆s.ⓛⓛ{a}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ¡[h, o].
+#h #o #a #G #L #s elim (deg_total h o s)
+#d #Hd @(snv_appl … a … (⋆s) … (ⓐ#0.#2) (0+1))
+[ /4 width=5 by snv_lref, drop_drop_lt/
+| @snv_lref [4: // |1,2,3: skip ]
+ @snv_bind //
+ @(snv_appl … a … (⋆s) … (⋆s) (0+1))
+ [ @snv_lref [4: // |1,2,3: skip ] //
+ | @snv_lref [4: /2 width=1 by drop_drop_lt/ |1,2,3: skip ] @snv_bind //
+ | @(lstas_scpds … (d+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
+ | @(lstas_scpds … (d+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
+ @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ] /3 width=1 by da_sort, da_bind/
+ ]
+| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
+| @(lstas_scpds … (d+1+1)) //
+ [ @da_ldec [3: // |1,2: skip ]
+ @da_bind @da_flat @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ]
+ /3 width=1 by da_sort, da_bind/
+ | @lstas_succ [4: // |1,2: skip ]
+ [2: @lstas_bind | skip ]
+ [2: @lstas_appl | skip ]
+ [2: @lstas_zero
+ [4: /2 width=1 by drop_drop_lt/ |5: /2 width=2 by lstas_bind/ |*: skip ]
+ |1: skip ]
+ /4 width=2 by lift_flat, lift_bind, lift_lref_ge_minus, lift_lref_lt/
+ ]
+]
+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/lstas.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Static type assignment (iterated vs primitive): the declared variable ****)
+
+(* basic_1: we have "L.ⓛⓝ⋆k1.⋆k2⦄ ⊢ #0 • ⓝ⋆k1.⋆k2". *)
+theorem sta_ldec: ∀h,G,L,s1,s2. ⦃G, L.ⓛⓝ⋆s1.⋆s2⦄ ⊢ #0 •*[h, 1] ⋆s2.
+/3 width=6 by lstas_sort, lstas_succ, lstas_cast, drop_pair/ qed-.
}
]
(*
- class "orange"
+ class "yellow"
[ { "MLTT1" * } {
[ { "" * } {
[ "genv_primitive" "judgment" * ]
}
]
*)
- class "red"
+ class "orange"
[ { "functional" * } {
[ { "reduction and type machine" * } {
[ "rtm" "rtm_step ( ? ⇨ ? )" * ]
]
}
]
+ class "red"
+ [ { "examples" * } {
+ [ { "terms with special features" * } {
+ [ "ex_sta_ldec" + "ex_cpr_omega" + "ex_fpbg_refl" + "ex_snv_eta" * ]
+ }
+ ]
+ }
+ ]
}
class "top" { * }
cpr_cpx/
qed.
-lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ⬈*[h] U → ⦃G, L⦄ ⊢ ⬈[h]
-𝐍⦃T⦄ → T = U.
+lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ⬈*[h] U → ⦃G, L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → T = U.
#h #o #G #L #T #U #H @(cpxs_ind_dx … H) -T //
#T0 #T #H1T0 #_ #IHT #H2T0
lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
+++ /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/relocation/rtmap_id.ma".
-include "basic_2/notation/relations/relationstar_4.ma".
-include "basic_2/relocation/lexs.ma".
-include "basic_2/static/frees.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition lfxs (R) (T): relation lenv ≝
- λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2.
-
-interpretation "generic extension on referred entries (local environment)"
- 'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-
-definition R_frees_confluent: predicate (relation3 lenv term term) ≝
- λRN.
- ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
- ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
-
-definition lexs_frees_confluent: relation (relation3 lenv term term) ≝
- λRN,RP.
- ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
- ∀L2. L1 ⦻*[RN, RP, f1] L2 →
- ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
-
-definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
- (relation3 lenv term term) … ≝
- λR1,R2,RP1,RP2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-(* Basic properties ***********************************************************)
-
-lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
-/3 width=3 by lexs_atom, frees_atom, ex2_intro/
-qed.
-
-lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
-qed.
-
-lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
-qed.
-
-lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
-qed.
-
-lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
-qed.
-
-lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
- ∀V2. R L1 V V2 →
- L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
-/3 width=5 by lexs_pair_repl, ex2_intro/
-qed-.
-
-lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxs R T).
-#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
-/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/
-qed-.
-
-lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2.
-#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
-qed-.
-
-lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
-#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
-qed-.
-
-lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| lapply (frees_inv_sort … H1) -H1 #Hf
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| lapply (frees_inv_gref … H1) -H1 #Hf
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
-/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
- L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
-#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
-/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
- Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
-/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
-qed-.
-
-lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
-qed-.
-
-lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
-qed-.
-
-lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
-qed-.
-
-lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
-qed-.
-
-(* Basic_2A1: removed theorems 24:
- llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
- llpx_sn_bind llpx_sn_flat
- llpx_sn_inv_bind llpx_sn_inv_flat
- llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
- llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
- llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_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/relocation/drops_ceq.ma".
-include "basic_2/relocation/drops_lexs.ma".
-include "basic_2/static/frees_drops.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 →
- ∀K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
- ∃∃L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
-
-definition dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
- ∃∃K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
-
-definition dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U →
- ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[R, T] K2.
-
-(* Properties with generic slicing for local environments *******************)
-
-(* Basic_2A1: includes: llpx_sn_lift_le llpx_sn_lift_ge *)
-lemma lfxs_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
- d_liftable2 R → dedropable_sn R.
-#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
-elim (frees_total L1 U) #f2 #Hf2
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
-elim (lexs_liftable_co_dedropable … H1R … H2R … HLK1 … HK12 … Hf) -f1 -K1
-/3 width=6 by cfull_lift, ex3_intro, ex2_intro/
-qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-(* Basic_2A1: restricts: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
-(* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma lfxs_dropable_sn: ∀R. dropable_sn R.
-#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
-elim (frees_total K1 T) #f1 #Hf1
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
-elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1
-/3 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: was: llpx_sn_drop_trans_O *)
-(* Note: the proof might be simplified *)
-lemma lfxs_dropable_dx: ∀R. dropable_dx R.
-#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
-elim (drops_isuni_ex … H1f L1) #K1 #HLK1
-elim (frees_total K1 T) #f1 #Hf1
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f
-elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2
-/4 width=9 by frees_inv_lifts, ex2_intro/
-qed-.
-
-(* Basic_2A1: was: llpx_sn_inv_lift_O *)
-lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 →
- ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
- ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2.
-#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU
-elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
-lapply (drops_mono … HY … HLK2) -L2 -i #H destruct //
-qed-.
-
-lemma lfxs_inv_lref_sn: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2.
-#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
-#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY
-#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lfxs_inv_lref_dx: ∀R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2.
-#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
-#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY
-#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-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/frees_fqup.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Advanced properties ******************************************************)
-
-lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L.
-#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/
-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/relocation/lexs_length.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|.
-#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/
-qed-.
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM 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/relocation/lexs_lexs.ma".
-include "basic_2/static/frees_fqup.ma".
-include "basic_2/static/frees_frees.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Main properties **********************************************************)
-
-theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
- L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
- L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
-#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
-elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2))
-/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
-qed.
-
-theorem lfxs_flat: ∀R,I,L1,L2,V,T.
- L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 →
- L1 ⦻*[R, ⓕ{I}V.T] L2.
-#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
-/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
-qed.
-
theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull →
∀T. Transitive … (lfxs R T).
#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/
qed-.
-
-theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull →
- R_confluent2_lfxs R R R R →
- ∀T. confluent … (lfxs R T).
-#R #H1R #H2R #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
-lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
-lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
-elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
-[ #L #HL1 #HL2
- elim (H1R … Hf … HL01) -HL01 #f1 #Hf1 #H1
- elim (H1R … Hf … HL02) -HL02 #f2 #Hf2 #H2
- lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
- lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
- /3 width=5 by ex2_intro/
-| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02
- elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
- lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01
- lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02
- elim (H2R … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/
-]
-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/reduction/cpr.ma".
-
-(* EXAMPLES *****************************************************************)
-
-(* A reduction cycle in two steps: the term Omega ***************************)
-
-definition Delta: term → term ≝ λW. +ⓛW.ⓐ#0.#0.
-
-definition Omega1: term → term ≝ λW. ⓐ(Delta W).(Delta W).
-
-definition Omega2: term → term ≝ λW. +ⓓⓝW.(Delta W).ⓐ#0.#0.
-
-(* Basic properties *********************************************************)
-
-lemma Delta_lift: ∀W1,W2,l,k. ⬆[l, k] W1 ≡ W2 →
- ⬆[l, k] (Delta W1) ≡ (Delta W2).
-/4 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
-
-(* Main properties **********************************************************)
-
-theorem cpr_Omega_12: ∀G,L,W. ⦃G, L⦄ ⊢ Omega1 W ➡ Omega2 W.
-/2 width=1 by cpr_beta/ qed.
-
-theorem cpr_Omega_21: ∀G,L,W. ⦃G, L⦄ ⊢ Omega2 W ➡ Omega1 W.
-#G #L #W1 elim (lift_total W1 0 1) #W2 #HW12
-@(cpr_zeta … (Omega1 W2)) /3 width=1 by Delta_lift, lift_flat/
-@cpr_flat @(cpr_delta … (Delta W1) ? 0)
-[3,5,8,10: /2 width=2 by Delta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
-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/fpbg_fpbs.ma".
-
-(* EXAMPLES *****************************************************************)
-
-(* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
-
-definition ApplDelta: term → nat → term ≝ λW,s. +ⓛW.ⓐ⋆s.ⓐ#0.#0.
-
-definition ApplOmega1: term → nat → term ≝ λW,s. ⓐ(ApplDelta W s).(ApplDelta W s).
-
-definition ApplOmega2: term → nat → term ≝ λW,s. +ⓓⓝW.(ApplDelta W s).ⓐ⋆s.ⓐ#0.#0.
-
-definition ApplOmega3: term → nat → term ≝ λW,s. ⓐ⋆s.(ApplOmega1 W s).
-
-(* Basic properties *********************************************************)
-
-lemma ApplDelta_lift: ∀W1,W2,s,l,k. ⬆[l, k] W1 ≡ W2 →
- ⬆[l, k] (ApplDelta W1 s) ≡ (ApplDelta W2 s).
-/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
-
-lemma cpr_ApplOmega_12: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡ ApplOmega2 W s.
-/2 width=1 by cpr_beta/ qed.
-
-lemma cpr_ApplOmega_23: ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega2 W s ➡ ApplOmega3 W s.
-#G #L #W1 #s elim (lift_total W1 0 1) #W2 #HW12
-@(cpr_zeta … (ApplOmega3 W2 s)) /4 width=1 by ApplDelta_lift, lift_flat/
-@cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 s) ? 0)
-[3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
-qed.
-
-lemma cpxs_ApplOmega_13: ∀h,o,G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡*[h,o] ApplOmega3 W s.
-/4 width=3 by cpxs_strap1, cpr_cpx/ qed.
-
-lemma fqup_ApplOmega_13: ∀G,L,W,s. ⦃G, L, ApplOmega3 W s⦄ ⊐+ ⦃G, L, ApplOmega1 W s⦄.
-/2 width=1 by/ qed.
-
-(* Main properties **********************************************************)
-
-theorem fpbg_refl: ∀h,o,G,L,W,s. ⦃G, L, ApplOmega1 W s⦄ >≡[h,o] ⦃G, L, ApplOmega1 W s⦄.
-/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ 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/dynamic/snv.ma".
-
-(* EXAMPLES *****************************************************************)
-
-(* Extended validy (basic?_2) vs. restricted validity (basic_1) *************)
-
-(* extended validity of a closure, last arg of snv_appl > 1 *)
-lemma snv_extended: ∀h,o,a,G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{a}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 ¡[h, o].
-#h #o #a #G #L #s elim (deg_total h o s)
-#d #Hd @(snv_appl … a … (⋆s) … (⋆s) (0+1+1))
-[ /4 width=5 by snv_lref, drop_drop_lt/
-| /4 width=13 by snv_bind, snv_lref/
-| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
-| @(lstas_scpds … (d+1+1))
- /5 width=11 by lstas_bind, lstas_succ, da_bind, da_ldec, da_sort, lift_bind/
-]
-qed.
-
-(* restricted validity of the η-expanded closure, last arg of snv_appl = 1 **)
-lemma snv_restricted: ∀h,o,a,G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{a}⋆s.⋆s.ⓛⓛ{a}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ¡[h, o].
-#h #o #a #G #L #s elim (deg_total h o s)
-#d #Hd @(snv_appl … a … (⋆s) … (ⓐ#0.#2) (0+1))
-[ /4 width=5 by snv_lref, drop_drop_lt/
-| @snv_lref [4: // |1,2,3: skip ]
- @snv_bind //
- @(snv_appl … a … (⋆s) … (⋆s) (0+1))
- [ @snv_lref [4: // |1,2,3: skip ] //
- | @snv_lref [4: /2 width=1 by drop_drop_lt/ |1,2,3: skip ] @snv_bind //
- | @(lstas_scpds … (d+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
- | @(lstas_scpds … (d+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
- @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ] /3 width=1 by da_sort, da_bind/
- ]
-| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
-| @(lstas_scpds … (d+1+1)) //
- [ @da_ldec [3: // |1,2: skip ]
- @da_bind @da_flat @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ]
- /3 width=1 by da_sort, da_bind/
- | @lstas_succ [4: // |1,2: skip ]
- [2: @lstas_bind | skip ]
- [2: @lstas_appl | skip ]
- [2: @lstas_zero
- [4: /2 width=1 by drop_drop_lt/ |5: /2 width=2 by lstas_bind/ |*: skip ]
- |1: skip ]
- /4 width=2 by lift_flat, lift_bind, lift_lref_ge_minus, lift_lref_lt/
- ]
-]
-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/lstas.ma".
-
-(* EXAMPLES *****************************************************************)
-
-(* Static type assignment (iterated vs primitive): the declared variable ****)
-
-(* basic_1: we have "L.ⓛⓝ⋆k1.⋆k2⦄ ⊢ #0 • ⓝ⋆k1.⋆k2". *)
-theorem sta_ldec: ∀h,G,L,s1,s2. ⦃G, L.ⓛⓝ⋆s1.⋆s2⦄ ⊢ #0 •*[h, 1] ⋆s2.
-/3 width=6 by lstas_sort, lstas_succ, lstas_cast, drop_pair/ 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/notation/relations/relationstarstar_4.ma".
+include "basic_2/static/lfxs.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition lfxss (R) (T): relation lenv ≝ TC … (lfxs R T).
+
+interpretation "tc of generic extension on referred entries (local environment)"
+ 'RelationStarStar R T L1 L2 = (lfxss R T L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma lfxss_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
+/2 width=1 by inj/ qed.
+
+lemma lfxss_sort: ∀R,I,L1,L2,V1,V2,s.
+ L1 ⦻**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
+/3 width=4 by lfxs_sort, step, inj/
+qed.
+
+lemma lfxss_lref: ∀R,I,L1,L2,V1,V2,i.
+ L1 ⦻**[R, #i] L2 → L1.ⓑ{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
+/3 width=4 by lfxs_lref, step, inj/
+qed.
+
+lemma lfxss_gref: ∀R,I,L1,L2,V1,V2,l.
+ L1 ⦻**[R, §l] L2 → L1.ⓑ{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
+/3 width=4 by lfxs_gref, step, inj/
+qed.
+
+lemma lfxss_sym: ∀R. lexs_frees_confluent R cfull →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (lfxss R T).
+#R #H1R #H2R #T #L1 #L2 #H elim H -L2
+/4 width=3 by lfxs_sym, TC_strap, inj/
+qed-.
+
+lemma lfxss_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⦻**[R1, T] L2 → L1 ⦻**[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
+/4 width=5 by lfxs_co, step, inj/
+qed-.
+(*
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
+#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
+qed-.
+
+lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
+qed-.
+
+lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
+[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| lapply (frees_inv_sort … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
+[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| lapply (frees_inv_gref … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
+/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
+ L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
+/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
+qed-.
+
+lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
+qed-.
+
+lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
+qed-.
+
+(* Basic_2A1: removed theorems 24:
+ llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
+ llpx_sn_bind llpx_sn_flat
+ llpx_sn_inv_bind llpx_sn_inv_flat
+ llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
+ llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
+ llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
+*)
+*)
--- /dev/null
+(*
+lemma lfxss_zero: ∀R,I,L1,L2,V1. L1 ⦻**[R, V1] L2 → ∀V2.
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⦻**[R, #0] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #H
+
+elim H -L2 /3 width=5 by lfxs_zero, inj/
+#L #L2 #H0 #HL2 #IH #V2 #HV12
+lapply (IH … HV12) -IH #HL1
+@(step … HL1) -HL1 @lfxs_zero
+
+axiom pippo_fwd: ∀R,I,Y1,L2,V2,T. Y1 ⦻*[R, T] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. Y1 = L1.ⓑ{I}V1.
+
+fact pippo: ∀R,I,L1,Y,T,V. L1.ⓑ{I}V ⦻**[R, T] Y →
+ ∀L2,V1. Y = L2.ⓑ{I}V1 →
+ ∀V2. R L1 V V2 →
+ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V2.
+#R #I #L1 #Y #T #V #H elim H -Y
+[ /3 width=2 by lfxs_pair_repl_dx, inj/
+| #L #Y #_ #HL2 #IH #L2 #V1 #H #V2 #HV2 destruct
+ elim (pippo_fwd … HL2) #L0 #V0 #H destruct
+ @step [2: @(IH … HV2) // | skip ] -IH -HV2
+
+lemma lfxss_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
+ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V1 →
+ ∀V2. R L1 V V2 →
+ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
+/3 width=5 by lexs_pair_repl, ex2_intro/
+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 FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ⬈ * [ break term 46 h , break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'PRedTySnStar $h $T $G $L1 $L2 }.
--- /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 ****************************************)
+
+notation "hvbox( L1 ⦻ * * [ break term 46 R , break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RelationStarStar $R $T $L1 $L2 }.
(**************************************************************************)
include "basic_2/syntax/tsts.ma".
-include "basic_2/computation/lpxs_cpxs.ma".
+include "basic_2/rt_computation/lfpxs_cpxs.ma".
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Forward lemmas involving same top term structure *************************)
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* Forward lemmas with same top term structure ******************************)
+(*
lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ⬈*[h, o] U → T ≂ U.
#h #o #G #L #T #HT #U #H
>(cpxs_inv_cnx1 … H HT) -G -L -T //
qed-.
-
-lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h, o] U →
- ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h, o] U.
-#h #o #G #L #U #s #H
-elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus … n) -n
-[ #s #_ #H -d destruct /2 width=1 by or_introl/
-| #n #IHn #s >plus_plus_comm_23 #Hnd #H destruct
- lapply (deg_next_SO … Hnd) -Hnd #Hnd
- elim (IHn … Hnd) -IHn
- [ #H lapply (tsts_inv_atom1 … H) -H #H >H -H /2 width=1 by or_intror/
- | generalize in match Hnd; -Hnd @(nat_ind_plus … n) -n
- /4 width=3 by cpxs_strap2, cpx_st, or_intror/
- | >iter_SO >iter_n_Sm //
- ]
+*)
+lemma cpxs_fwd_sort: ∀h,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
+ ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
+#h #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
+[ #H destruct /2 width=1 by or_introl/
+| #n #H destruct
+ @or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
]
qed-.
(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,o,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ⬈*[h, o] U →
- ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ⬈*[h, o] U.
-#h #o #a #G #L #V #W #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
+lemma cpxs_fwd_beta: ∀h,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
+ ⓐV.ⓛ{p}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
+#h #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
[ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
| #b #W0 #T0 #HT0 #HU
elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
(**************************************************************************)
include "basic_2/syntax/term_vector.ma".
-include "basic_2/computation/csx.ma".
+include "basic_2/rt_computation/csx.ma".
-(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+(* STRONGLY NORMALIZING TERMS VECTORS FOR UNCOUNTED PARALLEL RT-TRANSITION **)
definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
λh,o,G,L. all … (csx h o G L).
interpretation
- "context-sensitive strong normalization (term vector)"
- 'SN h o G L Ts = (csxv h o G L Ts).
+ "strong normalization for uncounted context-sensitive parallel rt-transition (term vector)"
+ 'PRedTyStrong h o G L Ts = (csxv h o G L Ts).
(* Basic inversion lemmas ***************************************************)
-lemma csxv_inv_cons: â\88\80h,o,G,L,T,Ts. â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T @ Ts →
- â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] Ts.
+lemma csxv_inv_cons: â\88\80h,o,G,L,T,Ts. â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83T@Tsâ¦\84 →
+ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83Tâ¦\84 â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83Tsâ¦\84.
normalize // qed-.
(* Basic forward lemmas *****************************************************)
-lemma csx_fwd_applv: â\88\80h,o,G,L,T,Vs. â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] â\92¶ Vs.T →
- â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] Vs â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\8a*[h, o] T.
+lemma csx_fwd_applv: â\88\80h,o,G,L,T,Vs. â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83â\92¶Vs.Tâ¦\84 →
+ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83Vsâ¦\84 â\88§ â¦\83G, Lâ¦\84 â\8a¢ â¬\88*[h, o] ð\9d\90\92â¦\83Tâ¦\84.
#h #o #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
#V #Vs #IHVs #HVs
lapply (csx_fwd_pair_sn … HVs) #HV
--- /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/notation/relations/predtysnstar_5.ma".
+include "basic_2/rt_transition/lfpx.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+definition lfpxs: ∀h. relation4 genv term lenv lenv ≝
+ λh,G,T. TC … (lfpx h G T).
+
+interpretation
+ "uncounted parallel rt-computation on referred entries (local environment)"
+ 'PRedTySnStar h T G L1 L2 = (lfpxs h G T L1 L2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: was just: lpx_lpxs *)
+lemma lfpx_lfpxs: ∀h,G,T,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=1 by inj/ qed.
+
+(* Basic_2A1: was just: lpxs_strap1 *)
+lemma lfpxs_strap1: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by step/ qed.
+
+(* Basic_2A1: was just: lpxs_strap2 *)
+lemma lfpxs_strap2: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by TC_strap/ qed.
+(*
+(* Basic_2A1: was just: lpxs_pair_refl *)
+lemma lfpxs_pair_refl: ∀h,G,T,L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈*[h, T] L2.ⓑ{I}V.
+/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: was just: lpxs_inv_atom1 *)
+lemma lfpxs_inv_atom1: ∀h,G,L2.T. ⦃G, ⋆⦄ ⊢ ⬈*[h, T] L2 → L2 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+
+(* Basic_2A1: was just: lpxs_inv_atom2 *)
+lemma lfpxs_inv_atom2: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] ⋆ → L1 = ⋆.
+/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
+*)
\ No newline at end of file
--- /dev/null
+(* Basic_2A1: was just: lprs_lpxs *)
+lemma lprs_lfpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
+/3 width=3 by lpr_lpx, monotonic_TC/ 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/rt_transition/lfpx_fqup.ma".
+include "basic_2/rt_computation/lfpxs.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+(* Advanced eliminators *****************************************************)
+
+(* Basic_2A1: was just: lpxs_ind *)
+lemma lfpxs_ind: ∀h,G,L1,T. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → R L2.
+#h #G #L1 #T #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+(* Basic_2A1: was just: lpxs_ind_dx *)
+lemma lfpxs_ind_dx: ∀h,G,L2,T. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → R L1.
+#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: was just: lpxs_refl *)
+lemma lfpxs_refl: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, T] L.
+/2 width=1 by lfpx_lfpxs/ qed.
--- /dev/null
+(* Basic forward lemmas *****************************************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: was just: lpxs_fwd_length *)
+lemma lfpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h, o] L2 → |L1| = |L2|.
+/2 width=2 by TC_lpx_sn_fwd_length/ 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/notation/relations/predsnstar_5.ma".
-include "basic_2/reduction/lpx.ma".
-include "basic_2/computation/lprs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
-
-definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
- λh,o,G. TC … (lpx h o G).
-
-interpretation "extended parallel computation (local environment, sn variant)"
- 'PRedSnStar h o G L1 L2 = (lpxs h o G L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lpxs_ind: ∀h,o,G,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L2.
-#h #o #G #L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lpxs_ind_dx: ∀h,o,G,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1.
-#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lprs_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/3 width=3 by lpr_lpx, monotonic_TC/ qed.
-
-lemma lpx_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=1 by inj/ qed.
-
-lemma lpxs_refl: ∀h,o,G,L. ⦃G, L⦄ ⊢ ➡*[h, o] L.
-/2 width=1 by lprs_lpxs/ qed.
-
-lemma lpxs_strap1: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by step/ qed.
-
-lemma lpxs_strap2: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by TC_strap/ qed.
-
-lemma lpxs_pair_refl: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpxs_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, o] L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
-
-lemma lpxs_inv_atom2: ∀h,o,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, o] ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ qed-.
cpxs.ma cpxs_tdeq.ma cpxs_cpxs.ma
+lfpxs.ma lfpxs_fqup.ma
csx.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
+csx_vector.ma
#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
qed.
-(*
-theorem lfxs_trans: ∀R. lexs_frees_confluent R cfull →
- ∀T. Transitive … (lfxs R T).
-#R #H1R #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
-elim (H1R … Hf1 … HL1) #f #H0 #H1
-lapply (frees_mono … Hf2 … H0) -Hf2 -H0 #Hf2
-lapply (lexs_eq_repl_back … HL2 … Hf2) -f2 #HL2
-lapply (sle_lexs_trans … HL1 … H1) -HL1 // #Hl1
-@(ex2_intro … f)
-/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/
-qed-.
-*)
theorem lfxs_conf: ∀R. lexs_frees_confluent R cfull →
R_confluent2_lfxs R R R R →
∀T. confluent … (lfxs R T).
]
(*
class "wine"
- [ { "examples" * } {
- [ { "terms with special features" * } {
- [ "ex_sta_ldec" + "ex_cpr_omega" + "ex_fpbg_refl" + "ex_snv_eta" * ]
- }
- ]
- }
- ]
- class "magenta"
[ { "" * } {
[ { "" * } {
[ "" * ]
}
]
*)
- class "prune"
+ class "magenta"
[ { "dynamic typing" * } {
(*
[ { "local env. ref. for native type assignment" * } {
]
}
]
- class "blue"
+ class "prune"
[ { "equivalence" * } {
[ { "decomposed rt-equivalence" * } {
[ "scpes ( ⦃?,?⦄ ⊢ ? •*⬌*[?,?,?,?] ? )" "scpes_aaa" + "scpes_cpcs" + "scpes_scpes" * ]
]
}
]
- class "sky"
+ class "blue"
[ { "conversion" * } {
[ { "context-sensitive conversion" * } {
[ "cpc ( ⦃?,?⦄ ⊢ ? ⬌ ? )" "cpc_cpc" * ]
}
]
*)
- class "cyan"
+ class "sky"
[ { "rt-computation" * } {
(*
[ { "evaluation for context-sensitive rt-reduction" * } {
[ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_drop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ "cpxs_tsts" + "cpxs_tsts_vector" + "cpxs_lreq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
*)
+ [ "csx_vector ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" * ]
[ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
+ [ "lfpxs ( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_fqup" * ]
[ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_cpxs" * ]
}
]
}
]
- class "water"
+ class "cyan"
[ { "rt-transition" * } {
[ { "parallel qrst-transition" * } {
(* [ "fpbq ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_alt ( ⦃?,?,?⦄ ≽≽[?,?] ⦃?,?,?⦄ )" + "fpbq_aaa" * ] *)
]
}
]
+ class "water"
+ [ { "iterated static typing" * } {
+ [ { "generic extension on referred entries" * } {
+ [ "lfxss ( ? ⦻**[?,?] ? )" * ]
+ }
+ ]
+ }
+ ]
class "green"
[ { "static typing" * } {
[ { "generic reducibility" * } {
basic_2/s_transition
basic_2/s_computation
basic_2/static
+basic_2/i_static
projects / helm.git / commitdiff