-fact sex_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+fact sex_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
∃∃K1. ⇩*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
lemma sex_drops_conf_next: ∀RN,RP.
∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
lemma sex_drops_conf_next: ∀RN,RP.
∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
- ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
- ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
+ ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
lemma sex_drops_conf_push: ∀RN,RP.
∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
lemma sex_drops_conf_push: ∀RN,RP.
∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
- ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀b,f,I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] → 𝐔❪f❫ →
- ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
+ ∃∃I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
qed-.
lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
qed-.
lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
- ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
- ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
+ ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
qed-.
lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
qed-.
lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
- ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀b,f,I2,K2. ⇩*[b,f] L2 ≘ K2.ⓘ[I2] → 𝐔❪f❫ →
- ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
+ ∃∃I1,K1. ⇩*[b,f] L1 ≘ K1.ⓘ[I1] & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
- ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ{I1} ≘ K1 →
+ ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
- ∃∃I2,L2. ⇩*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
- ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ{I1} ≘ K1 →
+ ∀b,f,I1,L1. ⇩*[b,f] L1.ⓘ[I1] ≘ K1 →
- ∃∃I2,L2. ⇩*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⇩*[b,f] L2.ⓘ[I2] ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ[I1] ≡[f] L2.ⓘ[I2].
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
-lemma drops_atom2_sex_conf: â\88\80RN,RP,b,f1,L1. â\87©*[b,f1] L1 â\89\98 â\8b\86 â\86\92 ð\9d\90\94â¦\83f1â¦\84 →
+lemma drops_atom2_sex_conf: â\88\80RN,RP,b,f1,L1. â\87©*[b,f1] L1 â\89\98 â\8b\86 â\86\92 ð\9d\90\94â\9dªf1â\9d« →
∀f,L2. L1 ⪤[RN,RP,f] L2 →
∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
∀f,L2. L1 ⪤[RN,RP,f] L2 →
∀f2. f1 ~⊚ f2 ≘f → ⇩*[b,f1] L2 ≘ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3