:: WAYBEL16 semantic presentation begin theorem :: WAYBEL16:1 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "y")) ")" ) ")" ) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "y")))))) ; theorem :: WAYBEL16:2 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "y")) ")" ) ")" ) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y")))))) ; theorem :: WAYBEL16:3 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "y")) ")" )))) "holds" (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "y")) ")" ))))) ; theorem :: WAYBEL16:4 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r5_waybel_4 :::"is_minimal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "y")) ")" )))) "holds" (Bool (Set (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "y")) ")" ))))) ; theorem :: WAYBEL16:5 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X9")) "," (Set (Var "Y9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "X9"))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set (Var "Y9")))) "holds" (Bool (Set (Set (Var "X")) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X9")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "Y9"))))))) ; theorem :: WAYBEL16:6 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X9")) "," (Set (Var "Y9")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "X9"))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set (Var "Y9")))) "holds" (Bool (Set (Set (Var "X")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X9")) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "Y9"))))))) ; theorem :: WAYBEL16:7 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k8_waybel_0 :::"Filt"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k7_waybel_0 :::"Ids"::: ) (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" )))) ; theorem :: WAYBEL16:8 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k8_waybel_0 :::"Filt"::: ) (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" )))) ; begin definitionlet "S", "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::); mode :::"CLHomomorphism"::: "of" "S" "," "T" -> ($#m1_subset_1 :::"Function":::) "of" "S" "," "T" means :: WAYBEL16:def 1 (Bool "(" (Bool it "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool it "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) ")" ); end; :: deftheorem defines :::"CLHomomorphism"::: WAYBEL16:def 1 : (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "S")) "," (Set (Var "T"))) "iff" (Bool "(" (Bool (Set (Var "b3")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "b3")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) ")" ) ")" ))); definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"Poset":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); pred "A" :::"is_FG_set"::: means :: WAYBEL16:def 2 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "A" "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "ex" (Set (Var "h")) "being" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" "S" "," (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "h")) ($#k5_relset_1 :::"|"::: ) "A") ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool "(" "for" (Set (Var "h9")) "being" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" "S" "," (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "h9")) ($#k5_relset_1 :::"|"::: ) "A") ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "h9")) ($#r2_funct_2 :::"="::: ) (Set (Var "h"))) ")" ) ")" )))); end; :: deftheorem defines :::"is_FG_set"::: WAYBEL16:def 2 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r1_waybel16 :::"is_FG_set"::: ) ) "iff" (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "ex" (Set (Var "h")) "being" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "h")) ($#k5_relset_1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool "(" "for" (Set (Var "h9")) "being" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "h9")) ($#k5_relset_1 :::"|"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "h9")) ($#r2_funct_2 :::"="::: ) (Set (Var "h"))) ")" ) ")" )))) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Poset":::); cluster (Set ($#k8_waybel_0 :::"Filt"::: ) "L") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: WAYBEL16:9 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "holds" (Bool (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "Y"))) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" )))) ; theorem :: WAYBEL16:10 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "Y")) "," (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ))) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "Y")) "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL16:11 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_setfam_1 :::"bool"::: ) (Set (Var "X"))) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ))) ; theorem :: WAYBEL16:12 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "X")) ($#k1_tarski :::"}"::: ) ) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ))) ; theorem :: WAYBEL16:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" )) "is" ($#v2_yellow_0 :::"upper-bounded"::: ) )) ; theorem :: WAYBEL16:14 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" )) "is" ($#v1_yellow_0 :::"lower-bounded"::: ) )) ; theorem :: WAYBEL16:15 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set (Var "X"))))) ; theorem :: WAYBEL16:16 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "X")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: WAYBEL16:17 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "Y")) "," (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set (Var "X"))))) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "Y"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "Y")))))) ; theorem :: WAYBEL16:18 (Bool "for" (Set (Var "L")) "being" ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Semilattice":::) "holds" (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) (Set (Var "L")) ")" )) "is" ($#v3_lattice3 :::"complete"::: ) )) ; registrationlet "L" be ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Semilattice":::); cluster (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k8_waybel_0 :::"Filt"::: ) "L" ")" )) -> ($#v3_lattice3 :::"complete"::: ) ; end; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "p" is :::"completely-irreducible"::: means :: WAYBEL16:def 3 (Bool ($#r1_waybel_1 :::"ex_min_of"::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) "p" ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) "p" ($#k6_domain_1 :::"}"::: ) )) "," "L"); end; :: deftheorem defines :::"completely-irreducible"::: WAYBEL16:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) "iff" (Bool ($#r1_waybel_1 :::"ex_min_of"::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "p")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) )) "," (Set (Var "L"))) ")" ))); theorem :: WAYBEL16:19 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "p")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"Irr"::: "L" -> ($#m1_subset_1 :::"Subset":::) "of" "L" means :: WAYBEL16:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) ")" )); end; :: deftheorem defines :::"Irr"::: WAYBEL16:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) ")" )) ")" ))); theorem :: WAYBEL16:20 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) "iff" (Bool "ex" (Set (Var "q")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "q"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Var "q")) ($#r3_orders_2 :::"<="::: ) (Set (Var "s"))) ")" ) & (Bool (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "q")) ")" ))) ")" )) ")" ))) ; theorem :: WAYBEL16:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool (Bool "not" (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L")))))) ; theorem :: WAYBEL16:22 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) "holds" (Bool (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_waybel_6 :::"IRR"::: ) (Set (Var "L"))))) ; theorem :: WAYBEL16:23 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool (Set (Var "x")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ))) ; theorem :: WAYBEL16:24 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: WAYBEL16:25 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) )) "holds" (Bool (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) ; theorem :: WAYBEL16:26 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "p")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "k")) ")" ))))) "holds" (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ))) ; theorem :: WAYBEL16:27 (Bool "for" (Set (Var "L")) "being" ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "q"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Var "q")) ($#r1_orders_2 :::"<="::: ) (Set (Var "s"))) ")" ) & (Bool (Bool "not" (Set (Var "u")) ($#r1_orders_2 :::"<="::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "u")))))) ; theorem :: WAYBEL16:28 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "q"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Var "q")) ($#r3_orders_2 :::"<="::: ) (Set (Var "s"))) ")" ) & (Bool (Bool "not" (Set (Var "u")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))))) "holds" (Bool "not" (Bool (Set (Set (Var "u")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "q"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "p")))))) ; theorem :: WAYBEL16:29 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" )) "is" ($#v1_waybel_6 :::"Open"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L"))))) ; theorem :: WAYBEL16:30 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel_8 :::"CompactSublatt"::: ) (Set (Var "L")) ")" ))) & (Bool (Set (Var "p")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "k")) ")" ))) ")" )))) ; theorem :: WAYBEL16:31 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set (Var "x"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) & (Bool (Bool "not" (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p")))) ")" )))) ; theorem :: WAYBEL16:32 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L"))) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) & (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) )) "holds" (Bool (Set ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) ")" ) ")" )) ; theorem :: WAYBEL16:33 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "s")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "s")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_waybel16 :::"Irr"::: ) (Set (Var "L")) ")" ) ")" ) "," (Set (Var "L")) ")" )))) ; theorem :: WAYBEL16:34 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: WAYBEL16:35 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) )) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "p")))) & (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel_8 :::"CompactSublatt"::: ) (Set (Var "L")) ")" ))) & (Bool (Set (Var "x")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "k")) ")" ))) ")" )) ")" ) "}" "," (Set (Var "L")) ")" )))) ; theorem :: WAYBEL16:36 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel_8 :::"CompactSublatt"::: ) (Set (Var "L")) ")" ))) & (Bool (Set (Var "p")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "k")) ")" ))) ")" )) "iff" (Bool (Set (Var "p")) "is" ($#v1_waybel16 :::"completely-irreducible"::: ) ) ")" ))) ;