(* Main properties **********************************************************)
(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: â\88\80b1,f1,L1,L. â¬\87*[b1, f1] L1 ≘ L →
- â\88\80b2,f,L2. â¬\87*[b2, f] L1 ≘ L2 →
- â\88\80f2. f1 â\8a\9a f2 â\89\98 f â\86\92 â¬\87*[b2, f2] L ≘ L2.
+theorem drops_conf: â\88\80b1,f1,L1,L. â\87©*[b1,f1] L1 ≘ L →
+ â\88\80b2,f,L2. â\87©*[b2,f] L1 ≘ L2 →
+ â\88\80f2. f1 â\8a\9a f2 â\89\98 f â\86\92 â\87©*[b2,f2] L ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
#H #Hf destruct @drops_atom
- #H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+ #H elim (pr_after_inv_isi … Hf12) -Hf12 /2 width=1 by/
+| #f1 #I1 #K1 #K #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (pr_after_inv_next_sn … Hf) -Hf [2,3: // ]
#g #Hg #H destruct /3 width=3 by drops_inv_drop1/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
+| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (pr_after_inv_push_sn … Hf) -Hf [1,3: * |*:// ]
#g2 #g #Hf #H1 #H2 destruct
[ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_div3/
| /4 width=3 by drops_inv_drop1, drops_drop/
qed-.
(* Basic_1: was: drop1_trans *)
-(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
+(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
drops_drop_trans
*)
-theorem drops_trans: â\88\80b1,f1,L1,L. â¬\87*[b1, f1] L1 ≘ L →
- â\88\80b2,f2,L2. â¬\87*[b2, f2] L ≘ L2 →
- â\88\80f. f1 â\8a\9a f2 â\89\98 f â\86\92 â¬\87*[b1â\88§b2, f] L1 ≘ L2.
+theorem drops_trans: â\88\80b1,f1,L1,L. â\87©*[b1,f1] L1 ≘ L →
+ â\88\80b2,f2,L2. â\87©*[b2,f2] L ≘ L2 →
+ â\88\80f. f1 â\8a\9a f2 â\89\98 f â\86\92 â\87©*[b1â\88§b2,f] L1 ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
#H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
- #H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
- /3 width=3 by isid_eq_repl_back/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
+ #H1 #H2 lapply (pr_after_isi_inv_sn … Hf ?) -Hf
+ /3 width=3 by pr_isi_eq_repl_back/
+| #f1 #I1 #K1 #K #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (pr_after_inv_next_sn … Hf) -Hf
/3 width=3 by drops_drop/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (pr_after_inv_push_sn … Hf) -Hf [1,3: * |*: // ]
#g2 #g #Hg #H1 #H2 destruct
[ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_trans/
| /4 width=3 by drops_inv_drop1, drops_drop/
]
qed-.
-theorem drops_conf_div: ∀f1,L,K. ⬇*[Ⓣ,f1] L ≘ K → ∀f2. ⬇*[Ⓣ,f2] L ≘ K →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2.
+theorem drops_conf_div_isuni:
+ ∀f1,L,K. ⇩*[Ⓣ,f1] L ≘ K → ∀f2. ⇩*[Ⓣ,f2] L ≘ K →
+ 𝐔❪f1❫ → 𝐔❪f2❫ → f1 ≡ f2.
#f1 #L #K #H elim H -f1 -L -K
[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
- /3 width=1 by isid_inv_eq_repl/
-| #f1 #I #L #K #Hf1 #IH #f2 elim (pn_split f2) *
+ /3 width=1 by pr_isi_inv_eq_repl/
+| #f1 #I #L #K #Hf1 #IH #f2 elim (pr_map_split_tl f2) *
#g2 #H2 #Hf2 #HU1 #HU2 destruct
[ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
- lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
+ lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by pr_isu_inv_push/ -HU2
#H destruct elim (drops_inv_x_bind_xy … Hf1)
- | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
+ | /4 width=5 by drops_inv_drop1, pr_isu_inv_next, pr_eq_next/
]
-| #f1 #I1 #I2 #L #K #Hf1 #_ #IH #f2 elim (pn_split f2) *
+| #f1 #I1 #I2 #L #K #Hf1 #_ #IH #f2 elim (pr_map_split_tl f2) *
#g2 #H2 #Hf2 #HU1 #HU2 destruct
[ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
- /4 width=5 by isuni_fwd_push, eq_push/
+ /4 width=5 by pr_isu_fwd_push, pr_eq_push/
| lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
- lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
+ lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by pr_isu_inv_push/ -HU1
#H destruct elim (drops_inv_x_bind_xy … Hg2)
]
]
(* Advanced properties ******************************************************)
(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: â\88\80b1,f,L,L1. â¬\87*[b1, f] L ≘ L1 →
- â\88\80b2,L2. â¬\87*[b2, f] L ≘ L2 → L1 = L2.
-#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
+lemma drops_mono: â\88\80b1,f,L,L1. â\87©*[b1,f] L ≘ L1 →
+ â\88\80b2,L2. â\87©*[b2,f] L ≘ L2 → L1 = L2.
+#b1 #f #L #L1 lapply (pr_after_isi_dx 𝐢 … f)
/3 width=8 by drops_conf, drops_fwd_isid/
qed-.
-lemma drops_inv_uni: â\88\80L,i. â¬\87*[â\92», ð\9d\90\94â\9d´iâ\9dµ] L â\89\98 â\8b\86 â\86\92 â\88\80I,K. â¬\87*[i] L â\89\98 K.â\93\98{I} → ⊥.
+lemma drops_inv_uni: â\88\80L,i. â\87©*[â\92»,ð\9d\90\94â\9d¨iâ\9d©] L â\89\98 â\8b\86 â\86\92 â\88\80I,K. â\87©[i] L â\89\98 K.â\93\98[I] → ⊥.
#L #i #H1 #I #K #H2
lapply (drops_F … H2) -H2 #H2
lapply (drops_mono … H2 … H1) -L -i #H destruct
qed-.
-lemma drops_ldec_dec: â\88\80L,i. Decidable (â\88\83â\88\83K,W. â¬\87*[i] L ≘ K.ⓛW).
+lemma drops_ldec_dec: â\88\80L,i. Decidable (â\88\83â\88\83K,W. â\87©[i] L ≘ K.ⓛW).
#L #i elim (drops_F_uni L i) [| * * [ #I #K1 | * #W1 #K1 ] ]
[4: /3 width=3 by ex1_2_intro, or_introl/
|*: #H1L @or_intror * #K2 #W2 #H2L
qed-.
(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: â\88\80b2,f,L,L2. â¬\87*[b2, f] L ≘ L2 →
- â\88\80b1,f1,I1,K1. â¬\87*[b1, f1] L â\89\98 K1.â\93\98{I1} →
+lemma drops_conf_skip1: â\88\80b2,f,L,L2. â\87©*[b2,f] L ≘ L2 →
+ â\88\80b1,f1,I1,K1. â\87©*[b1,f1] L â\89\98 K1.â\93\98[I1] →
∀f2. f1 ⊚ ⫯f2 ≘ f →
- ∃∃I2,K2. L2 = K2.ⓘ{I2} &
- â¬\87*[b2, f2] K1 â\89\98 K2 & â¬\86*[f2] I2 ≘ I1.
+ ∃∃I2,K2. L2 = K2.ⓘ[I2] &
+ â\87©*[b2,f2] K1 â\89\98 K2 & â\87§*[f2] I2 ≘ I1.
#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: â\88\80b1,f1,L1,L. â¬\87*[b1, f1] L1 ≘ L →
- â\88\80b2,f2,I2,K2. â¬\87*[b2, f2] L â\89\98 K2.â\93\98{I2} →
+lemma drops_trans_skip2: â\88\80b1,f1,L1,L. â\87©*[b1,f1] L1 ≘ L →
+ â\88\80b2,f2,I2,K2. â\87©*[b2,f2] L â\89\98 K2.â\93\98[I2] →
∀f. f1 ⊚ f2 ≘ ⫯f →
- ∃∃I1,K1. L1 = K1.ⓘ{I1} &
- â¬\87*[b1â\88§b2, f] K1 â\89\98 K2 & â¬\86*[f] I2 ≘ I1.
+ ∃∃I1,K1. L1 = K1.ⓘ[I1] &
+ â\87©*[b1â\88§b2,f] K1 â\89\98 K2 & â\87§*[f] I2 ≘ I1.
#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
lapply (drops_trans … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: drops_conf_div *)
-lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
- ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
+lemma drops_conf_div_bind_isuni:
+ ∀f1,f2,I1,I2,L,K.
+ ⇩*[Ⓣ,f1] L ≘ K.ⓘ[I1] → ⇩*[Ⓣ,f2] L ≘ K.ⓘ[I2] →
+ 𝐔❪f1❫ → 𝐔❪f2❫ → f1 ≡ f2 ∧ I1 = I2.
#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
-lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
+lapply (drops_conf_div_isuni … H1 … H2 ??) /2 width=3 by pr_isu_next/ -H1 -H2 -HU1 -HU2 #H
lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
lapply (drops_mono … H0 … Hf2) -L #H
projects / helm.git / blobdiff