(* *)
(**************************************************************************)
-include "basic_2/notation/relations/topredtysnstrong_5.ma".
+include "ground_2/xoa/ex_4_3.ma".
+include "basic_2/notation/relations/topredtysnstrong_4.ma".
include "basic_2/rt_computation/rsx.ma".
(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
(* Note: this should be an instance of a more general sex *)
(* Basic_2A1: uses: lcosx *)
-inductive jsx (h) (G): rtmap → relation lenv ≝
-| jsx_atom: ∀f. jsx h G f (⋆) (⋆)
-| jsx_push: ∀f,I,K1,K2. jsx h G f K1 K2 →
- jsx h G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
-| jsx_unit: ∀f,I,K1,K2. jsx h G f K1 K2 →
- jsx h G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| jsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ →
- jsx h G f K1 K2 → jsx h G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
+inductive jsx (h) (G): relation lenv ≝
+| jsx_atom: jsx h G (⋆) (⋆)
+| jsx_bind: ∀I,K1,K2. jsx h G K1 K2 →
+ jsx h G (K1.ⓘ[I]) (K2.ⓘ[I])
+| jsx_pair: ∀I,K1,K2,V. jsx h G K1 K2 →
+ G ⊢ ⬈*[h,V] 𝐒❪K2❫ → jsx h G (K1.ⓑ[I]V) (K2.ⓧ)
.
interpretation
"strong normalization for unbound parallel rt-transition (compatibility)"
- 'ToPRedTySNStrong h f G L1 L2 = (jsx h G f L1 L2).
+ 'ToPRedTySNStrong h G L1 L2 = (jsx h G L1 L2).
(* Basic inversion lemmas ***************************************************)
fact jsx_inv_atom_sn_aux (h) (G):
- ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #G #g #L1 #L2 * -g -L1 -L2 //
-[ #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #V #_ #_ #H destruct
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 → L1 = ⋆ → L2 = ⋆.
+#h #G #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #_ #H destruct
+| #I #K1 #K2 #V #_ #_ #H destruct
]
qed-.
-lemma jsx_inv_atom_sn (h) (G): ∀g,L2. G ⊢ ⋆ ⊒[h,g] L2 → L2 = ⋆.
-/2 width=7 by jsx_inv_atom_sn_aux/ qed-.
-
-fact jsx_inv_push_sn_aux (h) (G):
- ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
- ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
- ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓘ{I}.
-#h #G #g #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_push … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_next_push … H)
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #H
- elim (discr_next_push … H)
+lemma jsx_inv_atom_sn (h) (G): ∀L2. G ⊢ ⋆ ⊒[h] L2 → L2 = ⋆.
+/2 width=5 by jsx_inv_atom_sn_aux/ qed-.
+
+fact jsx_inv_bind_sn_aux (h) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀I,K1. L1 = K1.ⓘ[I] →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
+ | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & I = BPair J V & L2 = K2.ⓧ.
+#h #G #L1 #L2 * -L1 -L2
+[ #J #L1 #H destruct
+| #I #K1 #K2 #HK12 #J #L1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #I #K1 #K2 #V #HK12 #HV #J #L1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
]
qed-.
-lemma jsx_inv_push_sn (h) (G):
- ∀f,I,K1,L2. G ⊢ K1.ⓘ{I} ⊒[h,⫯f] L2 →
- ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓘ{I}.
-/2 width=5 by jsx_inv_push_sn_aux/ qed-.
-
-fact jsx_inv_unit_sn_aux (h) (G):
- ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
- ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
- ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
-#h #G #g #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #_ #H destruct
-]
-qed-.
+lemma jsx_inv_bind_sn (h) (G):
+ ∀I,K1,L2. G ⊢ K1.ⓘ[I] ⊒[h] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I]
+ | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & I = BPair J V & L2 = K2.ⓧ.
+/2 width=3 by jsx_inv_bind_sn_aux/ qed-.
-lemma jsx_inv_unit_sn (h) (G):
- ∀f,I,K1,L2. G ⊢ K1.ⓤ{I} ⊒[h,↑f] L2 →
- ∃∃K2. G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
-/2 width=6 by jsx_inv_unit_sn_aux/ qed-.
-
-fact jsx_inv_pair_sn_aux (h) (G):
- ∀g,L1,L2. G ⊢ L1 ⊒[h,g] L2 →
- ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
- ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
-#h #G #g #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #W #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #_ #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #V #HV #HK12 #g #J #L1 #W #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=4 by ex3_intro/
-]
-qed-.
+(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: uses: lcosx_inv_pair *)
lemma jsx_inv_pair_sn (h) (G):
- ∀f,I,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊒[h,↑f] L2 →
- ∃∃K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & G ⊢ K1 ⊒[h,f] K2 & L2 = K2.ⓧ.
-/2 width=6 by jsx_inv_pair_sn_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma jsx_inv_pair_sn_gen (h) (G): ∀g,I,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊒[h,g] L2 →
- ∨∨ ∃∃f,K2. G ⊢ K1 ⊒[h,f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
- | ∃∃f,K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & G ⊢ K1 ⊒[h,f] K2 & g = ↑f & L2 = K2.ⓧ.
-#h #G #g #I #K1 #L2 #V #H
-elim (pn_split g) * #f #Hf destruct
-[ elim (jsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
-| elim (jsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
+ ∀I,K1,L2,V. G ⊢ K1.ⓑ[I]V ⊒[h] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓑ[I]V
+ | ∃∃K2. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒❪K2❫ & L2 = K2.ⓧ.
+#h #G #I #K1 #L2 #V #H elim (jsx_inv_bind_sn … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #J #K2 #X #HK12 #HX #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
]
qed-.
+lemma jsx_inv_void_sn (h) (G):
+ ∀K1,L2. G ⊢ K1.ⓧ ⊒[h] L2 →
+ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓧ.
+#h #G #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
+/2 width=3 by ex2_intro/
+qed-.
+
(* Advanced forward lemmas **************************************************)
lemma jsx_fwd_bind_sn (h) (G):
- ∀g,I1,K1,L2. G ⊢ K1.ⓘ{I1} ⊒[h,g] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊒[h,⫱g] K2 & L2 = K2.ⓘ{I2}.
-#h #G #g #I1 #K1 #L2
-elim (pn_split g) * #f #Hf destruct
-[ #H elim (jsx_inv_push_sn … H) -H
-| cases I1 -I1 #I1
- [ #H elim (jsx_inv_unit_sn … H) -H
- | #V #H elim (jsx_inv_pair_sn … H) -H
- ]
-]
+ ∀I1,K1,L2. G ⊢ K1.ⓘ[I1] ⊒[h] L2 →
+ ∃∃I2,K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ[I2].
+#h #G #I1 #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
/2 width=4 by ex2_2_intro/
qed-.
-(* Basic properties *********************************************************)
-
-lemma jsx_eq_repl_back (h) (G): ∀L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊒[h,f] L2).
-#h #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-[ #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_px … H) -H /3 width=3 by jsx_push/
-| #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by jsx_unit/
-| #f #I #L1 #L2 #V #HV #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by jsx_pair/
-]
-qed-.
-
-lemma jsx_eq_repl_fwd (h) (G): ∀L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊒[h,f] L2).
-#h #G #L1 #L2 @eq_repl_sym /2 width=3 by jsx_eq_repl_back/
-qed-.
-
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lcosx_O *)
-lemma jsx_refl (h) (G): ∀f. 𝐈⦃f⦄ → reflexive … (jsx h G f).
-#h #G #f #Hf #L elim L -L
-/3 width=3 by jsx_eq_repl_back, jsx_push, eq_push_inv_isid/
+lemma jsx_refl (h) (G): reflexive … (jsx h G).
+#h #G #L elim L -L /2 width=1 by jsx_atom, jsx_bind/
qed.
(* Basic_2A1: removed theorems 2:
projects / helm.git / blobdiff