(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
drop_refl_atom_O2 drop_drop_lt drop_skip_lt
*)
-inductive drops (s:bool): trace → relation lenv ≝
-| drops_atom: ∀t. (s = Ⓕ → 𝐈⦃t⦄) → drops s (t) (⋆) (⋆)
-| drops_drop: ∀I,L1,L2,V,t. drops s t L1 L2 → drops s (Ⓕ@t) (L1.ⓑ{I}V) L2
-| drops_skip: ∀I,L1,L2,V1,V2,t.
- drops s t L1 L2 → ⬆*[t] V2 ≡ V1 →
- drops s (Ⓣ@t) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+inductive drops (c:bool): rtmap → relation lenv ≝
+| drops_atom: ∀f. (c = Ⓣ → 𝐈⦃f⦄) → drops c (f) (⋆) (⋆)
+| drops_drop: ∀I,L1,L2,V,f. drops c f L1 L2 → drops c (⫯f) (L1.ⓑ{I}V) L2
+| drops_skip: ∀I,L1,L2,V1,V2,f.
+ drops c f L1 L2 → ⬆*[f] V2 ≡ V1 →
+ drops c (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
interpretation "general slicing (local environment)"
- 'RDropStar s t L1 L2 = (drops s t L1 L2).
+ 'RDropStar c f L1 L2 = (drops c f L1 L2).
definition d_liftable1: relation2 lenv term → predicate bool ≝
- λR,s. ∀L,K,t. ⬇*[s, t] L ≡ K →
- ∀T,U. ⬆*[t] T ≡ U → R K T → R L U.
+ λR,c. ∀L,K,f. ⬇*[c, f] L ≡ K →
+ ∀T,U. ⬆*[f] T ≡ U → R K T → R L U.
definition d_liftable2: predicate (lenv → relation term) ≝
- λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,t. ⬇*[s, t] L ≡ K →
- ∀U1. ⬆*[t] T1 ≡ U1 →
- ∃∃U2. ⬆*[t] T2 ≡ U2 & R L U1 U2.
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,c,f. ⬇*[c, f] L ≡ K →
+ ∀U1. ⬆*[f] T1 ≡ U1 →
+ ∃∃U2. ⬆*[f] T2 ≡ U2 & R L U1 U2.
definition d_deliftable2_sn: predicate (lenv → relation term) ≝
- λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,t. ⬇*[s, t] L ≡ K →
- ∀T1. ⬆*[t] T1 ≡ U1 →
- ∃∃T2. ⬆*[t] T2 ≡ U2 & R K T1 T2.
-
-definition dropable_sn: predicate (relation lenv) ≝
- λR. ∀L1,K1,s,t. ⬇*[s, t] L1 ≡ K1 → ∀L2. R L1 L2 →
- ∃∃K2. R K1 K2 & ⬇*[s, t] L2 ≡ K2.
-(*
-definition dropable_dx: predicate (relation lenv) ≝
- λR. ∀L1,L2. R L1 L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
- ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & R K1 K2.
-*)
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,c,f. ⬇*[c, f] L ≡ K →
+ ∀T1. ⬆*[f] T1 ≡ U1 →
+ ∃∃T2. ⬆*[f] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: predicate (rtmap → relation lenv) ≝
+ λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀L2,u2. R u2 L1 L2 →
+ ∀u1. f ⊚ u1 ≡ u2 →
+ ∃∃K2. R u1 K1 K2 & ⬇*[c, f] L2 ≡ K2.
+
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_atom1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → X = ⋆ →
- Y = ⋆ ∧ (s = Ⓕ → 𝐈⦃t⦄).
-#X #Y #s #t * -X -Y -t
+fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ →
+ Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
+#X #Y #c #f * -X -Y -f
[ /3 width=1 by conj/
-| #I #L1 #L2 #V #t #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #t #_ #_ #H destruct
+| #I #L1 #L2 #V #f #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
]
qed-.
(* Basic_1: includes: drop_gen_sort *)
(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀Y,s,t. ⬇*[s, t] ⋆ ≡ Y → Y = ⋆ ∧ (s = Ⓕ → 𝐈⦃t⦄).
+lemma drops_inv_atom1: ∀Y,c,f. ⬇*[c, f] ⋆ ≡ Y → Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
/2 width=3 by drops_inv_atom1_aux/ qed-.
-fact drops_inv_drop1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K,V,tl. X = K.ⓑ{I}V → t = Ⓕ@tl →
- ⬇*[s, tl] K ≡ Y.
-#X #Y #s #t * -X -Y -t
-[ #t #Ht #J #K #W #tl #H destruct
-| #I #L1 #L2 #V #t #HL #J #K #W #tl #H1 #H2 destruct //
-| #I #L1 #L2 #V1 #V2 #t #_ #_ #J #K #W #tl #_ #H2 destruct
+fact drops_inv_drop1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K,V,g. X = K.ⓑ{I}V → f = ⫯g →
+ ⬇*[c, g] K ≡ Y.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K #W #g #H destruct
+| #I #L1 #L2 #V #f #HL #J #K #W #g #H1 #H2 <(injective_next … H2) -g destruct //
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K #W #g #_ #H2 elim (discr_push_next … H2)
]
qed-.
(* Basic_1: includes: drop_gen_drop *)
(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀I,K,Y,V,s,t. ⬇*[s, Ⓕ@t] K.ⓑ{I}V ≡ Y → ⬇*[s, t] K ≡ Y.
+lemma drops_inv_drop1: ∀I,K,Y,V,c,f. ⬇*[c, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[c, f] K ≡ Y.
/2 width=7 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_skip1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K1,V1,tl. X = K1.ⓑ{I}V1 → t = Ⓣ@tl →
- ∃∃K2,V2. ⬇*[s, tl] K1 ≡ K2 & ⬆*[tl] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
-#X #Y #s #t * -X -Y -t
-[ #t #Ht #J #K1 #W1 #tl #H destruct
-| #I #L1 #L2 #V #t #_ #J #K1 #W1 #tl #_ #H2 destruct
-| #I #L1 #L2 #V1 #V2 #t #HL #HV #J #K1 #W1 #tl #H1 #H2 destruct
+fact drops_inv_skip1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K1,V1,g. X = K1.ⓑ{I}V1 → f = ↑g →
+ ∃∃K2,V2. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K1 #W1 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K1 #W1 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: includes: drop_gen_skip_l *)
(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀I,K1,V1,Y,s,t. ⬇*[s, Ⓣ@t] K1.ⓑ{I}V1 ≡ Y →
- ∃∃K2,V2. ⬇*[s, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+lemma drops_inv_skip1: ∀I,K1,V1,Y,c,f. ⬇*[c, ↑f] K1.ⓑ{I}V1 ≡ Y →
+ ∃∃K2,V2. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
/2 width=5 by drops_inv_skip1_aux/ qed-.
-fact drops_inv_skip2_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K2,V2,tl. Y = K2.ⓑ{I}V2 → t = Ⓣ@tl →
- ∃∃K1,V1. ⬇*[s, tl] K1 ≡ K2 & ⬆*[tl] V2 ≡ V1 & X = K1.ⓑ{I}V1.
-#X #Y #s #t * -X -Y -t
-[ #t #Ht #J #K2 #W2 #tl #H destruct
-| #I #L1 #L2 #V #t #_ #J #K2 #W2 #tl #_ #H2 destruct
-| #I #L1 #L2 #V1 #V2 #t #HL #HV #J #K2 #W2 #tl #H1 #H2 destruct
+fact drops_inv_skip2_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K2,V2,g. Y = K2.ⓑ{I}V2 → f = ↑g →
+ ∃∃K1,V1. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K2 #W2 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K2 #W2 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: includes: drop_gen_skip_r *)
(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀I,X,K2,V2,s,t. ⬇*[s, Ⓣ@t] X ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[s, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1.
/2 width=5 by drops_inv_skip2_aux/ qed-.
(* Basic properties *********************************************************)
+lemma drops_eq_repl_back: ∀L1,L2,c. eq_stream_repl_back … (λf. ⬇*[c, f] L1 ≡ L2).
+#L1 #L2 #c #f1 #H elim H -L1 -L2 -f1
+[ /4 width=3 by drops_atom, isid_eq_repl_back/
+| #I #L1 #L2 #V #f1 #_ #IH #f2 #H elim (next_inv_sn … H) -H
+ /3 width=1 by drops_drop/
+| #I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H elim (push_inv_sn … H) -H
+ /3 width=3 by drops_skip, lifts_eq_repl_back/
+]
+qed-.
+
+lemma drops_eq_repl_fwd: ∀L1,L2,c. eq_stream_repl_fwd … (λf. ⬇*[c, f] L1 ≡ L2).
+#L1 #L2 #c @eq_stream_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
+qed-.
+
+(* Basic_2A1: includes: drop_refl *)
+lemma drops_refl: ∀c,L,f. 𝐈⦃f⦄ → ⬇*[c, f] L ≡ L.
+#c #L elim L -L /2 width=1 by drops_atom/
+#L #I #V #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
+/3 width=1 by drops_skip, lifts_refl/
+qed.
+
(* Basic_2A1: includes: drop_FT *)
-lemma drops_FT: ∀L1,L2,t. ⬇*[Ⓕ, t] L1 ≡ L2 → ⬇*[Ⓣ, t] L1 ≡ L2.
-#L1 #L2 #t #H elim H -L1 -L2 -t
+lemma drops_TF: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
+#L1 #L2 #f #H elim H -L1 -L2 -f
/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: ∀L1,L2,s,t. ⬇*[Ⓕ, t] L1 ≡ L2 → ⬇*[s, t] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drops_FT/
+lemma drops_gen: ∀L1,L2,c,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[c, f] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_TF/
qed-.
(* Basic_2A1: includes: drop_T *)
-lemma drops_T: ∀L1,L2,s,t. ⬇*[s, t] L1 ≡ L2 → ⬇*[Ⓣ, t] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drops_FT/
+lemma drops_F: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_TF/
qed-.
(* Basic forward lemmas *****************************************************)
-fact drops_fwd_drop2_aux: ∀X,Y,s,t2. ⬇*[s, t2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
- ∃∃t1,t. 𝐈⦃t1⦄ & t2 ⊚ Ⓕ@t1 ≡ t & ⬇*[s, t] X ≡ K.
-#X #Y #s #t2 #H elim H -X -Y -t2
-[ #t2 #Ht2 #J #K #W #H destruct
-| #I #L1 #L2 #V #t2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL
- /3 width=5 by after_false, ex3_2_intro, drops_drop/
-| #I #L1 #L2 #V1 #V2 #t2 #HL #_ #_ #J #K #W #H destruct
- elim (isid_after_dx t2) /3 width=5 by after_true, ex3_2_intro, drops_drop/
+fact drops_fwd_drop2_aux: ∀X,Y,c,f2. ⬇*[c, f2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
+#X #Y #c #f2 #H elim H -X -Y -f2
+[ #f2 #Ht2 #J #K #W #H destruct
+| #I #L1 #L2 #V #f2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL
+ /3 width=7 by after_next, ex3_2_intro, drops_drop/
+| #I #L1 #L2 #V1 #V2 #f2 #HL #_ #_ #J #K #W #H destruct
+ lapply (isid_after_dx 𝐈𝐝 f2 ?) /3 width=9 by after_push, ex3_2_intro, drops_drop/
]
qed-.
(* Basic_1: includes: drop_S *)
(* Basic_2A1: includes: drop_fwd_drop2 *)
-lemma drops_fwd_drop2: ∀I,X,K,V,s,t2. ⬇*[s, t2] X ≡ K.ⓑ{I}V →
- ∃∃t1,t. 𝐈⦃t1⦄ & t2 ⊚ Ⓕ@t1 ≡ t & ⬇*[s, t] X ≡ K.
+lemma drops_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
/2 width=5 by drops_fwd_drop2_aux/ qed-.
-fact drops_after_fwd_drop2_aux: ∀X,Y,s,t2. ⬇*[s, t2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
- ∀t1,t. 𝐈⦃t1⦄ → t2 ⊚ Ⓕ@t1 ≡ t → ⬇*[s, t] X ≡ K.
-#X #Y #s #t2 #H elim H -X -Y -t2
-[ #t2 #Ht2 #J #K #W #H destruct
-| #I #L1 #L2 #V #t2 #_ #IHL #J #K #W #H #t1 #t #Ht1 #Ht elim (after_inv_false1 … Ht) -Ht
- /3 width=3 by drops_drop/
-| #I #L1 #L2 #V1 #V2 #t2 #HL #_ #_ #J #K #W #H #t1 #t #Ht1 #Ht elim (after_inv_true1 … Ht) -Ht
- #u1 #u #b #H1 #H2 #Hu destruct >(after_isid_inv_dx … Hu) -Hu /2 width=1 by drops_drop/
-]
+lemma drops_after_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
+ ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[c, f] X ≡ K.
+#I #X #K #V #c #f2 #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
+#g1 #g #Hg1 #Hg #HK lapply (after_mono … Hg Hf ??) -Hg -Hf
+/3 width=3 by drops_eq_repl_back, isid_inv_eq_repl, next_eq_repl/
qed-.
-lemma drops_after_fwd_drop2: ∀I,X,K,V,s,t2. ⬇*[s, t2] X ≡ K.ⓑ{I}V →
- ∀t1,t. 𝐈⦃t1⦄ → t2 ⊚ Ⓕ@t1 ≡ t → ⬇*[s, t] X ≡ K.
-/2 width=9 by drops_after_fwd_drop2_aux/ qed-.
-
(* Basic_1: includes: drop_gen_refl *)
(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀L1,L2,s,t. ⬇*[s, t] L1 ≡ L2 → 𝐈⦃t⦄ → L1 = L2.
-#L1 #L2 #s #t #H elim H -L1 -L2 -t //
-[ #I #L1 #L2 #V #t #_ #_ #H elim (isid_inv_false … H)
-| /5 width=3 by isid_inv_true, lifts_fwd_isid, eq_f3, sym_eq/
+lemma drops_fwd_isid: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → 𝐈⦃f⦄ → L1 = L2.
+#L1 #L2 #c #f #H elim H -L1 -L2 -f //
+[ #I #L1 #L2 #V #f #_ #_ #H elim (isid_inv_next … H)
+| /5 width=3 by isid_inv_push, lifts_fwd_isid, eq_f3, sym_eq/
]
qed-.
-(* Basic_2A1: removed theorems 13:
+(* Basic_2A1: removed theorems 14:
drops_inv_nil drops_inv_cons d1_liftable_liftables
+ drop_refl_atom_O2
drop_inv_O1_pair1 drop_inv_pair1 drop_inv_O1_pair2
drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
drop_fwd_length_minus2 drop_fwd_length_minus4
projects / helm.git / blobdiff