]
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/
/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: ∀L,i. Decidable (∃∃K,W. ⇩*[i] L ≘ K.ⓛW).
+lemma drops_ldec_dec: ∀L,i. Decidable (∃∃K,W. ⇩[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
(* Basic_2A1: includes: drop_conf_lt *)
lemma drops_conf_skip1: ∀b2,f,L,L2. ⇩*[b2,f] L ≘ L2 →
- ∀b1,f1,I1,K1. ⇩*[b1,f1] L ≘ K1.ⓘ{I1} →
+ ∀b1,f1,I1,K1. ⇩*[b1,f1] L ≘ K1.ⓘ[I1] →
∀f2. f1 ⊚ ⫯f2 ≘ f →
- ∃∃I2,K2. L2 = K2.ⓘ{I2} &
+ ∃∃I2,K2. L2 = K2.ⓘ[I2] &
⇩*[b2,f2] K1 ≘ K2 & ⇧*[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/
(* Basic_2A1: includes: drop_trans_lt *)
lemma drops_trans_skip2: ∀b1,f1,L1,L. ⇩*[b1,f1] L1 ≘ L →
- ∀b2,f2,I2,K2. ⇩*[b2,f2] L ≘ K2.ⓘ{I2} →
+ ∀b2,f2,I2,K2. ⇩*[b2,f2] L ≘ K2.ⓘ[I2] →
∀f. f1 ⊚ f2 ≘ ⫯f →
- ∃∃I1,K1. L1 = K1.ⓘ{I1} &
+ ∃∃I1,K1. L1 = K1.ⓘ[I1] &
⇩*[b1∧b2,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1.
#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
lapply (drops_trans … H1 … H2 … Hf) -L -Hf
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 isuni_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