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for T being Noetherian sup-Semilattice for I being Ideal of T holds ex_sup_of I, T & sup I in I
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th1
|
On semilattice structure of {M}izar types
|
for a1,a2 being set st a1 <> a2 for A being AdjectiveStr st the adjectives of A = {a1,a2} & (the non-op of A).a1 = a2 & (the non-op of A).a2 = a1 holds A is non void involutive without_fixpoints
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th4
|
On semilattice structure of {M}izar types
|
for A1,A2 being AdjectiveStr st the AdjectiveStr of A1 = the AdjectiveStr of A2 holds A1 is involutive implies A2 is involutive
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th5
|
On semilattice structure of {M}izar types
|
for A1,A2 being AdjectiveStr st the AdjectiveStr of A1 = the AdjectiveStr of A2 holds A1 is without_fixpoints implies A2 is without_fixpoints
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th6
|
On semilattice structure of {M}izar types
|
for T1,T2 being TA-structure st the TA-structure of T1 = the TA-structure of T2 holds T1 is consistent implies T2 is consistent
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th8
|
On semilattice structure of {M}izar types
|
for T1,T2 being non empty TA-structure st the TA-structure of T1 = the TA-structure of T2 holds T1 is adj-structured implies T2 is adj-structured
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th9
|
On semilattice structure of {M}izar types
|
for T being reflexive transitive antisymmetric with_suprema TA-structure st T is adj-structured for t1,t2 being type of T st t1 <= t2 holds adjs t2 c= adjs t1
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th10
|
On semilattice structure of {M}izar types
|
for T1,T2 being TA-structure st the TA-structure of T1 = the TA-structure of T2 for a1 being adjective of T1, a2 being adjective of T2 st a1 = a2 holds types a1 = types a2
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th11
|
On semilattice structure of {M}izar types
|
for T being TA-structure for t being type of T, a being adjective of T holds a in adjs t iff t in types a
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th13
|
On semilattice structure of {M}izar types
|
for T being non empty TA-structure for t being type of T, A being Subset of the adjectives of T holds A c= adjs t iff t in types A
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th14
|
On semilattice structure of {M}izar types
|
for T being non empty TA-structure holds types ({} the adjectives of T) = the carrier of T
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th16
|
On semilattice structure of {M}izar types
|
for T1,T2 being TA-structure st the TA-structure of T1 = the TA-structure of T2 holds T1 is adjs-typed implies T2 is adjs-typed
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th17
|
On semilattice structure of {M}izar types
|
for T being adj-structured reflexive transitive antisymmetric with_suprema TA-structure for a being adjective of T for t being type of T st a is_applicable_to t holds types a /\ downarrow t is Ideal of T
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th19
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a ast t <= t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th20
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a in adjs(a ast t)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th21
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a ast t in types a
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th22
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for a being adjective of T for t9 being type of T st t9 <= t & a in adjs t9 holds a is_applicable_to t & t9 <= a ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th23
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for a being adjective of T st a in adjs t holds a is_applicable_to t & a ast t = t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th24
|
On semilattice structure of {M}izar types
|
for T being adj-structured reflexive transitive antisymmetric with_suprema TA-structure for A being Subset of the adjectives of T for t being type of T st A is_applicable_to t holds types A /\ downarrow t is Ideal of T
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th26
|
On semilattice structure of {M}izar types
|
for T being non empty reflexive transitive antisymmetric TA-structure for t being type of T holds ({} the adjectives of T) ast t = t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th27
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T, a be adjective of T holds apply(<*a*>, t) = <*t, a ast t *>
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th29
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for a being adjective of T holds <*a*> ast t = a ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th31
|
On semilattice structure of {M}izar types
|
for p being non empty FinSequence, q being FinSequence for i being Nat st i < len q holds (p$^q).(len p+i) = q.(i+1)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th33
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for v1,v2 being FinSequence of the adjectives of T holds apply(v1^v2, t) = apply(v1, t) $^ apply(v2, v1 ast t)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th34
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for v1,v2 being FinSequence of the adjectives of T for i being Nat st i in dom v1 holds apply(v1^v2, t).i = apply(v1, t).i
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th35
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for v1,v2 being FinSequence of the adjectives of T holds apply(v1^v2, t).(len v1+1) = v1 ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th36
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for v1,v2 being FinSequence of the adjectives of T holds v2 ast (v1 ast t) = v1^v2 ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th37
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TA-structure for t being type of T for v1,v2 being FinSequence of the adjectives of T st v1^ v2 is_applicable_to t holds v1 is_applicable_to t & v2 is_applicable_to v1 ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th40
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t for i1,i2 being Nat st 1 <= i1 & i1 <= i2 & i2 <= len v+1 for t1,t2 being type of T st t1 = apply(v,t).i1 & t2 = apply(v,t).i2 holds t2 <= t1
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th41
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t for s being type of T st s in rng apply(v, t) holds v ast t <= s & s <= t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th42
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds v ast t <= t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th43
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds rng v c= adjs (v ast t)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th44
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t for A being Subset of the adjectives of T st A = rng v holds A is_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th45
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 is_applicable_to t & rng v2 c= rng v1 for s being type of T st s in rng apply(v2,t) holds v1 ast t <= s
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th46
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1^v2 is_applicable_to t holds v2^v1 is_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th47
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for A being Subset of the adjectives of T st A is_applicable_to t holds A ast t <= t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th49
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for A being Subset of the adjectives of T st A is_applicable_to t holds A c= adjs(A ast t)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th50
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema TA-structure for t being type of T for A being Subset of the adjectives of T for t9 being type of T st t9 <= t & A c= adjs t9 holds A is_applicable_to t & t9 <= A ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th52
|
On semilattice structure of {M}izar types
|
for T being TA-structure, t being type of T for A,B being Subset of the adjectives of T st A is_applicable_to t & B c= A holds B is_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th54
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T, a being adjective of T for A,B being Subset of the adjectives of T st B = A \/ {a } & B is_applicable_to t holds a ast (A ast t) = B ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th55
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t for A being Subset of the adjectives of T st A = rng v holds v ast t = A ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th56
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T, v being FinSequence of the adjectives of T st v is_properly_applicable_to t holds v is_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th57
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T, v1,v2 being FinSequence of the adjectives of T st v1^v2 is_properly_applicable_to t holds v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th60
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T, v1,v2 being FinSequence of the adjectives of T st v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t holds v1^v2 is_properly_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th61
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T, A being Subset of the adjectives of T st A is_properly_applicable_to t holds A is finite
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th62
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T holds {} the adjectives of T is_properly_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th63
|
On semilattice structure of {M}izar types
|
for T being non empty non void reflexive transitive TAS-structure for t being type of T, A being Subset of the adjectives of T st A is_properly_applicable_to t ex B being Subset of the adjectives of T st B c= A & B is_properly_applicable_to t & A ast t = B ast t & for C being Subset of the adjectives of T st C c= B & C is_properly_applicable_to t & A ast t = C ast t holds C = B
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th64
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TAS-structure for t being type of T, A being Subset of the adjectives of T st A is_properly_applicable_to t ex s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th65
|
On semilattice structure of {M}izar types
|
for T being adj-structured antisymmetric non void reflexive transitive with_suprema Noetherian TAS-structure holds T@--> c= the InternalRel of T
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th66
|
On semilattice structure of {M}izar types
|
for T being adj-structured antisymmetric non void reflexive transitive with_suprema Noetherian TAS-structure for t1,t2 being type of T st T@--> reduces t1,t2 holds t1 <= t2
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th67
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric non void with_suprema TAS-structure holds T@--> is irreflexive
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th68
|
On semilattice structure of {M}izar types
|
for T being adj-structured antisymmetric non void reflexive transitive with_suprema Noetherian TAS-structure holds T@--> is strongly-normalizing
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th69
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TAS-structure for t being type of T, A being finite Subset of the adjectives of T st for C being Subset of the adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t holds C = A for s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t for i being Nat st 1 <= i & i <= len s holds [apply(s, t).(i+1), apply(s, t).i] in T@-->
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th70
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TAS-structure for t being type of T, A being finite Subset of the adjectives of T st for C being Subset of the adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t holds C = A for s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t holds Rev apply(s, t) is RedSequence of T @-->
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th71
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TAS-structure for t being type of T, A being finite Subset of the adjectives of T st A is_properly_applicable_to t holds T@--> reduces A ast t, t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th72
|
On semilattice structure of {M}izar types
|
for X being non empty set for R being Relation of X for r being RedSequence of R st r.1 in X holds r is FinSequence of X
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th73
|
On semilattice structure of {M}izar types
|
for X being non empty set for R being Relation of X for x be Element of X, y being set st R reduces x,y holds y in X
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th74
|
On semilattice structure of {M}izar types
|
for X being non empty set for R being Relation of X st R is with_UN_property weakly-normalizing for x be Element of X holds nf(x, R) in X
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th75
|
On semilattice structure of {M}izar types
|
for T being Noetherian adj-structured reflexive transitive antisymmetric with_suprema non void TAS-structure for t1, t2 being type of T st T@--> reduces t1, t2 ex A being finite Subset of the adjectives of T st A is_properly_applicable_to t2 & t1 = A ast t2
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th76
|
On semilattice structure of {M}izar types
|
for T being adj-structured antisymmetric commutative non void reflexive transitive with_suprema Noetherian TAS-structure holds T@--> is with_Church-Rosser_property with_UN_property
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th77
|
On semilattice structure of {M}izar types
|
for T being adj-structured with_suprema antisymmetric commutative non empty non void reflexive transitive Noetherian TAS-structure for t being type of T holds T@--> reduces t, radix t
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th78
|
On semilattice structure of {M}izar types
|
for T being adj-structured with_suprema antisymmetric commutative non empty non void reflexive transitive Noetherian TAS-structure for t being type of T for X being set st X = {t9 where t9 is type of T: ex A being finite Subset of the adjectives of T st A is_properly_applicable_to t9 & A ast t9 = t} holds ex_sup_of X, T & radix t = "\/"(X, T)
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th80
|
On semilattice structure of {M}izar types
|
for T being adj-structured with_suprema antisymmetric commutative non empty non void reflexive transitive Noetherian TAS-structure for t1,t2 being type of T, a being adjective of T st a is_properly_applicable_to t1 & a ast t1 <= radix t2 holds t1 <= radix t2
|
theorem
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:Th81
|
On semilattice structure of {M}izar types
|
MinimalFiniteSet { P[set] } : ex A being finite set st P[A] & for B being set st B c= A & P[B] holds B = A provided A1: ex A being finite set st P[A]
|
scheme
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:sch:MinimalFiniteSet
|
On semilattice structure of {M}izar types
|
RedInd { X() -> non empty set, P[set,set], R() -> Relation of X() } : for x,y being Element of X() st R() reduces x,y holds P[x,y] provided A1: for x,y being Element of X() st [x,y] in R() holds P[x,y] and A2: for x being Element of X() holds P[x,x] and A3: for x,y,z being Element of X() st P[x,y] & P[y,z] holds P[x,z]
|
scheme
|
abcmiz_0
|
[
"vocabularies NUMBERS, ZFMISC_1, RELAT_2, REWRITE1, XBOOLE_0, ORDERS_2,",
"PRELAMB, SUBSET_1, IDEAL_1, TARSKI, RELAT_1, STRUCT_0, ARYTM_3, XXREAL_0,",
"WAYBEL_0, LATTICE3, LATTICES, EQREL_1, CARD_FIL, YELLOW_0, ORDINAL2,",
"BINOP_1, FUNCT_1, OPOSET_1, CARD_1, FUNCOP_1, FINSUB_1, YELLOW_1,",
"ARYTM_0, WELLORD2, FINSEQ_1, FUNCT_7, NAT_1, ORDINAL4, FINSET_1,",
"FINSEQ_5, ARYTM_1, ABCMIZ_0, ABIAN",
"notations TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, RELAT_2, SUBSET_1, ORDINAL1,",
"FINSUB_1, CARD_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, BINOP_1,",
"FUNCOP_1, FINSET_1, FINSEQ_1, FUNCT_4, ALG_1, FINSEQ_5, NUMBERS,",
"XCMPLX_0, NAT_1, DOMAIN_1, STRUCT_0, ORDERS_2, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, XXREAL_0",
"constructors FINSUB_1, NAT_1, FINSEQ_5, REWRITE1, BORSUK_1, LATTICE3,",
"WAYBEL_0, YELLOW_1, FUNCOP_1, XREAL_0",
"registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_2, FINSET_1, FINSUB_1,",
"XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, FINSEQ_5, REWRITE1, STRUCT_0,",
"LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, YELLOW_9, ORDINAL1,",
"CARD_1, RELSET_1",
"requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM",
"definitions TARSKI, XBOOLE_0, RELAT_2, FUNCT_1, FINSEQ_1, LATTICE3, REWRITE1,",
"YELLOW_0, WAYBEL_0, RELSET_1",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, SUBSET_1, FINSUB_1, NAT_1, FINSEQ_1,",
"CARD_1, TREES_1, FINSEQ_5, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCT_4,",
"FUNCOP_1, STRUCT_0, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1,",
"YELLOW_4, YELLOW_7, WAYBEL_6, WAYBEL_8, ZFMISC_1, FINSEQ_2, FINSEQ_3,",
"HILBERT2, REWRITE1, ORDINAL1, XREAL_1, XXREAL_0, PRE_POLY",
"schemes XBOOLE_0, NAT_1, FUNCT_1, FUNCT_2, RECDEF_1, RELSET_1, ORDERS_1"
] |
abcmiz_0.miz
|
abcmiz_0:sch:RedInd
|
On semilattice structure of {M}izar types
|
for f being Function holds f.x c= Union f
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th1
|
Towards the construction of a model of Mizar concepts
|
for f being Function for x,y being set st f = [x,y] holds x = y
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th3
|
Towards the construction of a model of Mizar concepts
|
(id X).:Y c= Y
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th4
|
Towards the construction of a model of Mizar concepts
|
for S being non void Signature for X being non-empty ManySortedSet of the carrier of S for t being Term of S, X holds t is non pair
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th5
|
Towards the construction of a model of Mizar concepts
|
for x,y,z being set st x in {z}* & y in {z}* & card x = card y holds x = y
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th6
|
Towards the construction of a model of Mizar concepts
|
for S being non void Signature for X being non empty-yielding ManySortedSet of the carrier of S for t being Element of Free(S,X) holds (ex s being SortSymbol of S, v being set st t = root-tree [v,s] & v in X.s) or ex o being OperSymbol of S, p being FinSequence of Free(S,X) st t = [o,the carrier of S]-tree p & len p = len the_arity_of o & p is DTree-yielding & p is ArgumentSeq of Sym(o, X\/((the carrier of S)-->{0}))
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th7
|
Towards the construction of a model of Mizar concepts
|
varcl {} = {}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th8
|
Towards the construction of a model of Mizar concepts
|
for A,B being set st A c= B holds varcl A c= varcl B
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th9
|
Towards the construction of a model of Mizar concepts
|
for A being set holds varcl union A = union {varcl a where a is Element of A: not contradiction}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th10
|
Towards the construction of a model of Mizar concepts
|
varcl (X \/ Y) = (varcl X) \/ (varcl Y)
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th11
|
Towards the construction of a model of Mizar concepts
|
for A being non empty set st for a being Element of A holds varcl a = a holds varcl meet A = meet A
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th12
|
Towards the construction of a model of Mizar concepts
|
varcl ((varcl X) /\ (varcl Y)) = (varcl X) /\ (varcl Y)
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th13
|
Towards the construction of a model of Mizar concepts
|
for V being ManySortedSet of NAT st V.0 = {[{}, i] where i is Element of NAT: not contradiction} & for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} for i,j being Element of NAT st i <= j holds V.i c= V.j
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th14
|
Towards the construction of a model of Mizar concepts
|
for V being ManySortedSet of NAT st V.0 = {[{}, i] where i is Element of NAT: not contradiction} & for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} for A being finite Subset of Vars ex i being Element of NAT st A c= V.i
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th15
|
Towards the construction of a model of Mizar concepts
|
{[{}, i] where i is Element of NAT: not contradiction} c= Vars
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th16
|
Towards the construction of a model of Mizar concepts
|
for A being finite Subset of Vars, i being Nat holds [varcl A, i] in Vars
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th17
|
Towards the construction of a model of Mizar concepts
|
Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT: A is finite}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th18
|
Towards the construction of a model of Mizar concepts
|
varcl Vars = Vars proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = {[{}, i] where i is Element of NAT: not contradiction} and A3: for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} by Def2; defpred P[Nat] means varcl(V.$1) = V.$1; now let x,y; assume [x,y] in V.0; then ex i being Element of NAT st [x,y] = [{}, i] by A2; then x = {} by XTUPLE_0:1; hence x c= V.0 by XBOOLE_1:2; end; then A4: varcl (V.0) c= V.0 by Def1; V.0 c= varcl (V.0) by Def1; then A5: P[ 0] by A4,XBOOLE_0:def 10; A6: now let i; assume A7: P[i]; reconsider i9 = i as Element of NAT by ORDINAL1:def 12; A8: V.(i+1) = {[varcl A, j] where A is Subset of V.i, j is Element of NAT: A is finite} by A3; now let x,y; assume [x,y] in V.(i+1); then consider A being Subset of V.i, j being Element of NAT such that A9: [x,y] = [varcl A, j] and A is finite by A8; x = varcl A by A9,XTUPLE_0:1; then A10: x c= V.i by A7,Th9; V.i9 c= V.(i9+1) by A2,A3,Th14,NAT_1:11; hence x c= V.(i+1) by A10,XBOOLE_1:1; end; then A11: varcl (V.(i+1)) c= V.(i+1) by Def1; V.(i+1) c= varcl (V.(i+1)) by Def1; hence P[i+1] by A11,XBOOLE_0:def 10; end; A12: P[i] from NAT_1:sch 2(A5,A6); A13: varcl Vars = union {varcl a where a is Element of rng V: not contradiction} by A1,Th10; thus now let x; assume x in varcl Vars; then consider Y such that A14: x in Y and A15: Y in {varcl a where a is Element of rng V: not contradiction} by A13,TARSKI:def 4; consider a being Element of rng V such that A16: Y = varcl a by A15; consider i being set such that A17: i in dom V and A18: a = V.i by FUNCT_1:def 3; reconsider i as Element of NAT by A17; varcl (V.i) = a by A12,A18; hence x in Vars by A1,A14,A16,A17,A18,CARD_5:2; end; thus thesis by Def1; end;
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th19
|
Towards the construction of a model of Mizar concepts
|
for X st the_rank_of X is finite holds X is finite
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th20
|
Towards the construction of a model of Mizar concepts
|
the_rank_of varcl X = the_rank_of X
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th21
|
Towards the construction of a model of Mizar concepts
|
for X being finite Subset of Rank omega holds X in Rank omega
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th22
|
Towards the construction of a model of Mizar concepts
|
Vars c= Rank omega proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = {[{}, i] where i is Element of NAT: not contradiction} and A3: for n being Nat holds V.(n+1) = {[varcl a, j] where a is Subset of V.n, j is Element of NAT: a is finite} by Def2; let x; assume x in Vars; then consider i being set such that A4: i in dom V and A5: x in V.i by A1,CARD_5:2; reconsider i as Element of NAT by A4; defpred P[Nat] means V.$1 c= Rank omega; A6: P[ 0] proof let x; assume x in V.0; then consider i being Element of NAT such that A7: x = [{}, i] by A2; A8: i+1 = succ i by NAT_1:38; A9: {} c= i; A10: i in i+1 by A8,ORDINAL1:6; A11: {} in i+1 by A8,A9,ORDINAL1:6,12; A12: the_rank_of {} = {} by CLASSES1:73; A13: the_rank_of i = i by CLASSES1:73; A14: {} in Rank (i+1) by A11,A12,CLASSES1:66; i in Rank (i+1) by A10,A13,CLASSES1:66; then A15: x in Rank succ succ (i+1) by A7,A14,CLASSES1:45; succ succ (i+1) c= omega; then Rank succ succ (i+1) c= Rank omega by CLASSES1:37; hence thesis by A15; end; A16: now let n be Element of NAT such that A17: P[n]; A18: V.(n+1) = {[varcl a, j] where a is Subset of V.n, j is Element of NAT: a is finite} by A3; thus P[n+1] proof let x; assume x in V.(n+1); then consider a being Subset of V.n, j being Element of NAT such that A19: x = [varcl a, j] and A20: a is finite by A18; a c= Rank omega by A17,XBOOLE_1:1; then a in Rank omega by A20,Th22; then reconsider i = the_rank_of a as Element of NAT by CLASSES1:66; reconsider k = j \/ i as Element of NAT by ORDINAL3:12; A21: the_rank_of varcl a = i by Th21; A22: the_rank_of j = j by CLASSES1:73; A23: k in succ k by ORDINAL1:6; then A24: i in succ k by ORDINAL1:12,XBOOLE_1:7; A25: j in succ k by A23,ORDINAL1:12,XBOOLE_1:7; A26: succ k = k+1 by NAT_1:38; then A27: varcl a in Rank (k+1) by A21,A24,CLASSES1:66; j in Rank (k+1) by A22,A25,A26,CLASSES1:66; then A28: x in Rank succ succ (k+1) by A19,A27,CLASSES1:45; succ succ (k+1) c= omega; then Rank succ succ (k+1) c= Rank omega by CLASSES1:37; hence thesis by A28; end; end; for n being Element of NAT holds P[n] from NAT_1:sch 1(A6,A16); then V.i c= Rank omega; hence thesis by A5; end;
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th23
|
Towards the construction of a model of Mizar concepts
|
for A being finite Subset of Vars holds varcl A is finite Subset of Vars
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th24
|
Towards the construction of a model of Mizar concepts
|
for A being Subset of Vars holds varcl {[varcl A, j]} = (varcl A) \/ {[varcl A, j]}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th26
|
Towards the construction of a model of Mizar concepts
|
for x being variable holds varcl {x} = (vars x) \/ {x}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th27
|
Towards the construction of a model of Mizar concepts
|
<*>Vars in QuasiLoci
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th29
|
Towards the construction of a model of Mizar concepts
|
for l being one-to-one FinSequence of Vars holds l is quasi-loci iff for i being Nat, x being variable st i in dom l & x = l.i for y being variable st y in vars x ex j being Nat st j in dom l & j < i & y = l.j
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th30
|
Towards the construction of a model of Mizar concepts
|
for l being quasi-loci, x being variable holds l^<*x*> is quasi-loci iff not x in rng l & vars x c= rng l
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th31
|
Towards the construction of a model of Mizar concepts
|
for p,q being FinSequence st p^q is quasi-loci holds p is quasi-loci & q is FinSequence of Vars
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th32
|
Towards the construction of a model of Mizar concepts
|
for x being variable holds <*x*> is quasi-loci iff vars x = {}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th34
|
Towards the construction of a model of Mizar concepts
|
for x,y being variable holds <*x,y*> is quasi-loci iff vars x = {} & x <> y & vars y c= {x}
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th35
|
Towards the construction of a model of Mizar concepts
|
z is expression of C, s iff z in (the Sorts of Free(C, MSVars C)).s
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th41
|
Towards the construction of a model of Mizar concepts
|
for o st len the_arity_of o = 1 for a being expression of C st ex s st s = (the_arity_of o).1 & a is expression of C, s holds [o, the carrier of C]-tree <*a*> is expression of C, the_result_sort_of o
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th42
|
Towards the construction of a model of Mizar concepts
|
(non_op C)term a is expression of C, an_Adj C & (non_op C)term a = [non_op, the carrier of C]-tree <*a*>
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th43
|
Towards the construction of a model of Mizar concepts
|
(non_op C)term a = (non_op C)term b implies a = b
|
theorem
|
abcmiz_1
|
[
"vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1,",
"XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1,",
"ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4,",
"MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3,",
"XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0,",
"FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000,",
"CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES,",
"YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR,",
"BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1",
"notations TARSKI, XBOOLE_0, ZFMISC_1,",
"XTUPLE_0, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1,",
"FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2,",
"PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, NAT_1, MCART_1,",
"FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3,",
"TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1,",
"ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR,",
"MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3,",
"AOFA_000, PRE_POLY",
"constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1,",
"CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2,",
"RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0",
"registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1,",
"FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1,",
"MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2,",
"TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1,",
"INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2,",
"PARTIT_2, XTUPLE_0",
"requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL",
"definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FACIRC_1, FINSEQ_1, FINSEQ_2,",
"LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, PBOOLE, TREES_3, MSUALG_1,",
"WAYBEL_0, XTUPLE_0",
"theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1,",
"FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1,",
"MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3,",
"CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4,",
"FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3,",
"YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, FUNCT_7,",
"XXREAL_2, CARD_2, XTUPLE_0",
"schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR,",
"CLASSES1, FUNCT_2"
] |
abcmiz_1.miz
|
abcmiz_1:Th44
|
Towards the construction of a model of Mizar concepts
|
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