:: CALCUL_1 semantic presentation begin definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); func :::"Ant"::: "f" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" means :: CALCUL_1:def 1 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool it ($#r2_relset_1 :::"="::: ) (Set "f" ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")) ")" )))) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"Ant"::: CALCUL_1:def 1 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set (Var "b3")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "i")) ")" )))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")))) "iff" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ")" ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); func :::"Suc"::: "f" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") equals :: CALCUL_1:def 2 (Set "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" )) if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "f") ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Set ($#k5_cqc_lang :::"VERUM"::: ) "Al"); end; :: deftheorem defines :::"Suc"::: CALCUL_1:def 2 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_cqc_lang :::"VERUM"::: ) (Set (Var "Al")))) ")" ")" ))); definitionlet "f" be ($#m1_hidden :::"Relation":::); let "p" be ($#m1_hidden :::"set"::: ) ; pred "p" :::"is_tail_of"::: "f" means :: CALCUL_1:def 3 (Bool "p" ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f")); end; :: deftheorem defines :::"is_tail_of"::: CALCUL_1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set (Var "f"))) "iff" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f", "g" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); pred "f" :::"is_Subsequence_of"::: "g" means :: CALCUL_1:def 4 (Bool "ex" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "f" ($#r1_relset_1 :::"c="::: ) (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" "g" ($#k2_partfun1 :::"|"::: ) (Set (Var "N")) ")" )))); end; :: deftheorem defines :::"is_Subsequence_of"::: CALCUL_1:def 4 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set (Var "g"))) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "f")) ($#r1_relset_1 :::"c="::: ) (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "N")) ")" )))) ")" ))); theorem :: CALCUL_1:1 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set (Var "f")) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g")))) & (Bool "ex" (Set (Var "N")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "g")) ($#k2_partfun1 :::"|"::: ) (Set (Var "N")) ")" )))) ")" ))) ; theorem :: CALCUL_1:2 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ))) ; theorem :: CALCUL_1:3 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ")" ))) ; theorem :: CALCUL_1:4 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: CALCUL_1:5 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Var "f"))) ")" )))) ; theorem :: CALCUL_1:6 (Bool "for" (Set (Var "fin")) "," (Set (Var "fin1")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fin"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "fin1")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fin1"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "fin1")) ")" ))) & "(" (Bool (Bool (Set (Var "fin")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fin")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fin1"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "fin1")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "fin")) ")" ))) ")" ) ")" ")" )) ; theorem :: CALCUL_1:7 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))))) ; theorem :: CALCUL_1:8 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set (Var "f")) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")))))) ; theorem :: CALCUL_1:9 (Bool "for" (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "fin")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "b")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "c")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ))))) ; theorem :: CALCUL_1:10 (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "fin")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "b")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "b")) ($#k9_finseq_1 :::"*>"::: ) ) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "fin")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "b")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ))) ")" ))) ; theorem :: CALCUL_1:11 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "m")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: CALCUL_1:12 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "m")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: CALCUL_1:13 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set (Set "(" (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: CALCUL_1:14 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Num 2) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k10_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "d"))) ")" )))) ; begin definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); func :::"still_not-bound_in"::: "f" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_qc_lang1 :::"bound_QC-variables"::: ) "Al" ")" ) means :: CALCUL_1:def 5 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")))) ")" ))) ")" )); end; :: deftheorem defines :::"still_not-bound_in"::: CALCUL_1:def 5 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_qc_lang1 :::"bound_QC-variables"::: ) (Set (Var "Al")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")))) ")" ))) ")" )) ")" )))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; func :::"set_of_CQC-WFF-seq"::: "Al" -> ($#m1_hidden :::"set"::: ) means :: CALCUL_1:def 6 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al")) ")" )); end; :: deftheorem defines :::"set_of_CQC-WFF-seq"::: CALCUL_1:def 6 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))) ")" )) ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "PR" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Const "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); let "n" be ($#m1_hidden :::"Nat":::); pred "PR" "," "n" :::"is_a_correct_step"::: means :: CALCUL_1:def 7 (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" )) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k5_cqc_lang :::"VERUM"::: ) "Al" ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1)) (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" ))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 2)) (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 3)) (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al")(Bool "ex" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 4)) (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k7_cqc_lang :::"'&'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 5)) (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al")(Bool "ex" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )))) if (Bool (Set (Set "(" "PR" ($#k1_funct_1 :::"."::: ) "n" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 6)) (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al")(Bool "ex" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) "n") & (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) 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:::"QC-alphabet"::: ) (Bool "for" (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" (Set (Var 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:::"="::: ) (Num 5))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k7_cqc_lang :::"'&'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ))) ")" ) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 6))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )))) ")" ) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 7))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )))) ")" ) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 8))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ))))) ")" ) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 9))) "implies" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")))(Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" )))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ))))) ")" ) ")" ")" )))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "PR" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Const "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ); attr "PR" is :::"a_proof"::: means :: CALCUL_1:def 8 (Bool "(" (Bool "PR" ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "PR"))) "holds" (Bool "PR" "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) ")" ) ")" ); end; :: deftheorem defines :::"a_proof"::: CALCUL_1:def 8 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "PR")) "is" ($#v1_calcul_1 :::"a_proof"::: ) ) "iff" (Bool "(" (Bool (Set (Var "PR")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR"))))) "holds" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) ")" ) ")" ) ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); pred :::"|-"::: "f" means :: CALCUL_1:def 9 (Bool "ex" (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) "Al" ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "PR")) "is" ($#v1_calcul_1 :::"a_proof"::: ) ) & (Bool "f" ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR")) ")" ) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )); end; :: deftheorem defines :::"|-"::: CALCUL_1:def 9 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool "(" (Bool (Set (Var "PR")) "is" ($#v1_calcul_1 :::"a_proof"::: ) ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR")) ")" ) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )) ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "p" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al")) ")" ); pred "p" :::"is_formal_provable_from"::: "X" means :: CALCUL_1:def 10 (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) "X") & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "p") & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) ")" )); end; :: deftheorem defines :::"is_formal_provable_from"::: CALCUL_1:def 10 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r5_calcul_1 :::"is_formal_provable_from"::: ) (Set (Var "X"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) ")" )) ")" )))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al")) ")" ); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Const "Al")) "," (Set (Const "A")); let "v" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Const "Al")) "," (Set (Const "A")) ")" ); pred "J" "," "v" :::"|="::: "X" means :: CALCUL_1:def 11 (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al") "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "J" "," "v" ($#r1_valuat_1 :::"|="::: ) (Set (Var "p")))); end; :: deftheorem defines :::"|="::: CALCUL_1:def 11 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")) ")" ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "holds" (Bool "(" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r6_calcul_1 :::"|="::: ) (Set (Var "X"))) "iff" (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set (Var "p")))) ")" )))))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al")) ")" ); let "p" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); pred "X" :::"|="::: "p" means :: CALCUL_1:def 12 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" "Al" "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" "Al" "," (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r6_calcul_1 :::"|="::: ) "X")) "holds" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) "p")))); end; :: deftheorem defines :::"|="::: CALCUL_1:def 12 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r7_calcul_1 :::"|="::: ) (Set (Var "p"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r6_calcul_1 :::"|="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set (Var "p")))))) ")" )))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "p" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); pred :::"|="::: "p" means :: CALCUL_1:def 13 (Bool (Set ($#k1_subset_1 :::"{}"::: ) (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) "Al" ")" )) ($#r7_calcul_1 :::"|="::: ) "p"); end; :: deftheorem defines :::"|="::: CALCUL_1:def 13 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool ($#r8_calcul_1 :::"|="::: ) (Set (Var "p"))) "iff" (Bool (Set ($#k1_subset_1 :::"{}"::: ) (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")) ")" )) ($#r7_calcul_1 :::"|="::: ) (Set (Var "p"))) ")" ))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Const "Al")) "," (Set (Const "A")); let "v" be ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Const "Al")) "," (Set (Const "A")) ")" ); pred "J" "," "v" :::"|="::: "f" means :: CALCUL_1:def 14 (Bool "J" "," "v" ($#r6_calcul_1 :::"|="::: ) (Set ($#k2_relset_1 :::"rng"::: ) "f")); end; :: deftheorem defines :::"|="::: CALCUL_1:def 14 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "holds" (Bool "(" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) (Set (Var "f"))) "iff" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r6_calcul_1 :::"|="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) ")" )))))); definitionlet "Al" be ($#m1_qc_lang1 :::"QC-alphabet"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); let "p" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Const "Al"))); pred "f" :::"|="::: "p" means :: CALCUL_1:def 15 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" "Al" "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" "Al" "," (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) "f")) "holds" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) "p")))); end; :: deftheorem defines :::"|="::: CALCUL_1:def 15 : (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r10_calcul_1 :::"|="::: ) (Set (Var "p"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set (Var "p")))))) ")" )))); theorem :: CALCUL_1:15 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))))) ; theorem :: CALCUL_1:16 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))))) ; theorem :: CALCUL_1:17 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")))) & (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) ")" ) "iff" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) (Set (Var "f"))) ")" )))))) ; theorem :: CALCUL_1:18 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))))) ; theorem :: CALCUL_1:19 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r10_calcul_1 :::"|="::: ) (Set (Var "p")))))) ; theorem :: CALCUL_1:20 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k7_cqc_lang :::"'&'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")) ")" ))))) ; theorem :: CALCUL_1:21 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Var "p")))))) ; theorem :: CALCUL_1:22 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Var "q")))))) ; theorem :: CALCUL_1:23 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) (Bool "for" (Set (Var "Sub")) "being" ($#m1_subset_1 :::"CQC_Substitution":::) "of" (Set (Var "Al")) "holds" (Bool "(" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_sublemma :::"|="::: ) (Set ($#k2_substut2 :::"["::: ) (Set (Var "p")) "," (Set (Var "Sub")) ($#k2_substut2 :::"]"::: ) )) "iff" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set (Var "p"))) ")" ))))))) ; theorem :: CALCUL_1:24 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "holds" (Bool "(" (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r1_valuat_1 :::"|="::: ) (Set (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Var "J")) "," (Set (Set (Var "v")) ($#k1_sublemma :::"."::: ) (Set "(" (Set (Var "x")) ($#k12_sublemma :::"|"::: ) (Set (Var "a")) ")" )) ($#r1_valuat_1 :::"|="::: ) (Set (Var "p"))) ")" )) ")" ))))))) ; theorem :: CALCUL_1:25 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" )) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ))))))) ; theorem :: CALCUL_1:26 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k3_qc_lang1 :::"bound_QC-variables"::: ) (Set (Var "Al")))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Set "(" (Set (Var "v")) ($#k1_sublemma :::"."::: ) (Set "(" (Set (Var "x")) ($#k12_sublemma :::"|"::: ) (Set (Var "a")) ")" ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "v")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")))))))))) ; theorem :: CALCUL_1:27 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_valuat_1 :::"interpretation"::: ) "of" (Set (Var "Al")) "," (Set (Var "A")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Set (Var "v")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "w")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "J")) "," (Set (Var "v")) ($#r9_calcul_1 :::"|="::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "J")) "," (Set (Var "w")) ($#r9_calcul_1 :::"|="::: ) (Set (Var "f")))))))) ; theorem :: CALCUL_1:28 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "v")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k2_valuat_1 :::"Valuations_in"::: ) "(" (Set (Var "Al")) "," (Set (Var "A")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ))))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "v")) ($#k1_sublemma :::"."::: ) (Set "(" (Set (Var "y")) ($#k12_sublemma :::"|"::: ) (Set (Var "a")) ")" ) ")" ) ($#k1_sublemma :::"."::: ) (Set "(" (Set (Var "x")) ($#k12_sublemma :::"|"::: ) (Set (Var "a")) ")" ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k1_sublemma :::"."::: ) (Set "(" (Set (Var "x")) ($#k12_sublemma :::"|"::: ) (Set (Var "a")) ")" ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")) ")" ))))))))) ; theorem :: CALCUL_1:29 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ))))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r10_calcul_1 :::"|="::: ) (Set ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" )))))) ; theorem :: CALCUL_1:30 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k5_cqc_lang :::"VERUM"::: ) (Set (Var "Al")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" )) ($#r10_calcul_1 :::"|="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k5_cqc_lang :::"VERUM"::: ) (Set (Var "Al")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ))))) ; theorem :: CALCUL_1:31 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR")))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 1))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 2))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 3))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 4))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 5))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 6))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 7))) & (Bool (Bool "not" (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 8)))) "holds" (Bool (Set (Set "(" (Set (Var "PR")) ($#k1_funct_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Num 9))))) ; theorem :: CALCUL_1:32 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r5_calcul_1 :::"is_formal_provable_from"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "X")) ($#r7_calcul_1 :::"|="::: ) (Set (Var "p")))))) ; begin theorem :: CALCUL_1:33 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))))) ; theorem :: CALCUL_1:34 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "PR")) "," (Set (Var "PR1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR"))))) "holds" (Bool "(" (Bool (Set (Var "PR")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) "iff" (Bool (Set (Set (Var "PR")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "PR1"))) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) ) ")" )))) ; theorem :: CALCUL_1:35 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "PR1")) "," (Set (Var "PR")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k4_calcul_1 :::"set_of_CQC-WFF-seq"::: ) (Set (Var "Al")) ")" ) "," (Set ($#k2_cqc_the1 :::"Proof_Step_Kinds"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR1")))) & (Bool (Set (Var "PR1")) "," (Set (Var "n")) ($#r3_calcul_1 :::"is_a_correct_step"::: ) )) "holds" (Bool (Set (Set (Var "PR")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "PR1"))) "," (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "PR")) ")" )) ($#r3_calcul_1 :::"is_a_correct_step"::: ) )))) ; theorem :: CALCUL_1:36 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_calcul_1 :::"is_Subsequence_of"::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "g"))))) ; theorem :: CALCUL_1:37 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "g")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: CALCUL_1:38 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")))) & (Bool (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "g")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:39 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "g")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "g")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ($#k7_cqc_lang :::"'&'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "g")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) ; theorem :: CALCUL_1:40 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:41 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_cqc_lang :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:42 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" )) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:43 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Set ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" )) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" )))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" )))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f")))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:44 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k2_calcul_1 :::"Suc"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:45 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Var "f"))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" ($#k1_calcul_1 :::"Ant"::: ) (Set (Var "f")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:46 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:47 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:48 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:49 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:50 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k9_cqc_lang :::"'or'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:51 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k9_cqc_lang :::"'or'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:52 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k9_cqc_lang :::"'or'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:53 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k9_cqc_lang :::"'or'"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:54 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:55 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:56 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k8_cqc_lang :::"=>"::: ) (Set (Var "q")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) )))))) ; theorem :: CALCUL_1:57 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "holds" (Bool (Set (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_cqc_lang :::"'not'"::: ) (Set "(" (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" )))))) ; theorem :: CALCUL_1:58 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool "ex" (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "st" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_cqc_lang :::"Ex"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:59 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "g")) ")" ))))) ; theorem :: CALCUL_1:60 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")))))) ; theorem :: CALCUL_1:61 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k4_substut2 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_cqc_lang :::"Ex"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ))))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_cqc_lang :::"Ex"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "q")) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:62 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k24_qc_lang1 :::"still_not-bound_in"::: ) (Set (Var "p")) ")" ) where p "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) : (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" )) "}" )))) ; theorem :: CALCUL_1:63 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f"))) "is" ($#v1_finset_1 :::"finite"::: ) ))) ; theorem :: CALCUL_1:64 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) "holds" (Bool "(" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_qc_lang1 :::"bound_QC-variables"::: ) (Set (Var "Al")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k1_qc_lang1 :::"QC-symbols"::: ) (Set (Var "Al")) ")" ))) & (Bool (Bool "not" (Set ($#k3_qc_lang1 :::"bound_QC-variables"::: ) (Set (Var "Al"))) "is" ($#v1_finset_1 :::"finite"::: ) )) ")" )) ; theorem :: CALCUL_1:65 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "not" (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_calcul_1 :::"still_not-bound_in"::: ) (Set (Var "f")))))))) ; theorem :: CALCUL_1:66 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ")" ) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:67 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "st" (Bool (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ")" ) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) "holds" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: CALCUL_1:68 (Bool "for" (Set (Var "Al")) "being" ($#m1_qc_lang1 :::"QC-alphabet"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) (Bool "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"bound_QC-variable":::) "of" (Set (Var "Al")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_cqc_lang :::"CQC-WFF"::: ) (Set (Var "Al"))) "holds" (Bool "(" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_cqc_lang :::"All"::: ) "(" (Set (Var "x")) "," (Set (Var "p")) ")" ")" ) ($#k12_finseq_1 :::"*>"::: ) ))) "iff" (Bool ($#r4_calcul_1 :::"|-"::: ) (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set "(" ($#k12_cqc_lang :::"Ex"::: ) "(" (Set (Var "x")) "," (Set "(" ($#k6_cqc_lang :::"'not'"::: ) (Set (Var "p")) ")" ) ")" ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))) ")" ))))) ; definitionlet "f" be ($#m1_hidden :::"FinSequence":::); let "p" be ($#m1_hidden :::"set"::: ) ; redefine pred "p" :::"is_tail_of"::: "f" means :: CALCUL_1:def 16 (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) "p") ")" )); end; :: deftheorem defines :::"is_tail_of"::: CALCUL_1:def 16 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_calcul_1 :::"is_tail_of"::: ) (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )) ")" )));