:: A Model of Mizar Concepts -- Unification :: by Grzegorz Bancerek environ vocabularies TARSKI, QC_LANG3, PBOOLE, MSUALG_1, CATALG_1, FINSEQ_1, XBOOLE_0, ZFMISC_1, ARYTM_3, CARD_1, NAT_1, NUMBERS, XXREAL_0, ZF_LANG1, ORDINAL1, TREES_A, ABIAN, CARD_3, MEMBER_1, FINSET_1, FUNCOP_1, FUNCT_1, TREES_4, TREES_2, MSATERM, RELAT_1, MCART_1, MSAFREE, ZF_MODEL, AOFA_000, FINSEQ_2, PARTFUN1, QC_LANG1, FUNCT_2, ORDINAL4, CAT_3, TREES_3, ABCMIZ_0, ABCMIZ_1, ABCMIZ_A, STRUCT_0, FACIRC_1, INSTALG1, MSUALG_2, COMPUT_1, BINTREE1, TREES_9, ARYTM_1, FUNCT_6, SUBSET_1, MARGREL1, SETLIM_2; notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, XFAMILY, SUBSET_1, DOMAIN_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCOP_1, XCMPLX_0, XXREAL_0, ORDINAL1, NAT_D, MCART_1, FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, FINSEQ_4, FUNCT_6, TREES_1, TREES_2, TREES_3, TREES_4, TREES_9, PBOOLE, STRUCT_0, MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, INSTALG1, CATALG_1, MSAFREE3, AOFA_000, ABCMIZ_1; constructors RELSET_1, DOMAIN_1, WELLORD2, TREES_9, EQUATION, NAT_D, FINSEQ_4, CATALG_1, FACIRC_1, ABCMIZ_1, PRE_POLY, XTUPLE_0, XFAMILY; registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, FUNCT_1, FINSET_1, STRUCT_0, PBOOLE, MSUALG_2, FINSEQ_1, NAT_1, CARD_1, TREES_3, TREES_2, FUNCOP_1, RELAT_1, INDEX_1, INSTALG1, MSAFREE3, WAYBEL26, FACIRC_1, ABCMIZ_1, MSATERM, RELSET_1, XTUPLE_0; requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL; definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, PBOOLE, ABCMIZ_1; equalities TARSKI, SUBSET_1, FINSEQ_1, MSAFREE, MSAFREE3, MSUALG_1, ABCMIZ_1, ORDINAL1; expansions TARSKI, XBOOLE_0, FUNCT_1, PBOOLE, ABCMIZ_1, ORDINAL1; theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, XREAL_1, ZFMISC_1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, ENUMSET1, FUNCT_6, INSTALG1, NAT_1, MCART_1, PBOOLE, RELAT_1, RELSET_1, CARD_1, CARD_5, ORDINAL1, MSUALG_2, TREES_3, TREES_4, FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, YELLOW11, PARTFUN1, WELLORD2, ABCMIZ_1, TREES_9, XTUPLE_0, XREGULAR; schemes FUNCT_1, NAT_1, RECDEF_1, CLASSES1, FINSEQ_1; begin :: Preliminary reserve i,j for Nat; ::$CT scheme MinimalElement{X() -> finite non empty set, R[set,set]}: ex x being set st x in X() & for y being set st y in X() holds not R[y,x] provided A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z] holds R[x,z] proof assume A3: for x being set st x in X() ex y being set st y in X() & R[y,x]; set n = card X(); set x0 = the Element of X(); defpred P[Nat,set,set] means $2 in X() implies $3 in X() & R[ $3,$2]; A4: for m being Nat st 1 <= m & m < n+1 for x being set ex y being set st P[m,x,y] proof let m be Nat; assume 1 <= m & m < n+1; let x be set; per cases; suppose A5: x nin X(); set y = the set; take y; thus P[m,x,y] by A5; end; suppose x in X(); then consider y being set such that A6: y in X() & R[y,x] by A3; take y; thus thesis by A6; end; end; consider p being FinSequence such that A7: len p = n+1 and A8: p.1 = x0 or n+1 = 0 and A9: for i being Nat st 1 <= i & i < n+1 holds P[i, p.i, p.(i+1)] from RECDEF_1:sch 3(A4); defpred Q[Nat] means $1 in dom p implies p.$1 in X(); A10: Q[ 0] by FINSEQ_3:25; A11: now let i be Nat; assume A12: Q[i]; thus Q[i+1] proof assume i+1 in dom p; then i+1 <= n+1 by A7,FINSEQ_3:25; then A13: i < n+1 by NAT_1:13; per cases; suppose i = 0; hence thesis by A8; end; suppose i > 0; then i >= 0+1 & i is Element of NAT by NAT_1:13,ORDINAL1:def 12; hence thesis by A12,A7,A9,A13,FINSEQ_3:25; end; end; end; A14: for i being Nat holds Q[i] from NAT_1:sch 2(A10,A11); A15: rng p c= X() proof let x be object; assume x in rng p; then ex i being object st i in dom p & x = p.i by FUNCT_1:def 3; hence thesis by A14; end; A16: for i,j being Nat st 1 <= i & i < j & j <= n+1 holds R[p.j, p.i] proof let i,j be Nat; assume A17: 1 <= i; assume A18: i < j; then i+1 <= j by NAT_1:13; then consider k being Nat such that A19: j = i+1+k by NAT_1:10; assume A20: j <= n+1; then i <= n+1 by A18,XXREAL_0:2; then A21: i in dom p by A17,A7,FINSEQ_3:25; defpred S[Nat] means i+1+$1 <= n+1 implies R[p.(i+1+$1), p.i]; A22: S[ 0] proof assume i+1+0 <= n+1; then A23: i < n+1 by NAT_1:13; p.i in X() & i is Element of NAT by A14,A21; hence R[p.(i+1+0), p.i] by A9,A17,A23; end; A24: now let k be Nat; assume A25: S[k]; thus S[k+1] proof assume A26: i+1+(k+1) <= n+1; A27: i+1+(k+1) = i+1+k+1; then A28: i+1+k < n+1 by A26,NAT_1:13; A29: p.i in X() by A14,A21; i+1+k = 1+(i+k); then A30: 1 <= i+1+k by NAT_1:11; then i+1+k in dom p by A7,A28,FINSEQ_3:25; then A31: p.(i+1+k) in X() & i+1+k is Element of NAT by A14; then p.(i+1+(k+1)) in X() & R[p.(i+1+(k+1)), p.(i+1+k)] by A9,A28,A27,A30; hence R[p.(i+1+(k+1)), p.i] by A2,A28,A25,A31,A29; end; end; for k being Nat holds S[k] from NAT_1:sch 2(A22,A24); hence R[p.j, p.i] by A19,A20; end; A32: dom p = Seg(n+1) & card Seg(n+1) = n+1 by A7,FINSEQ_1:57,def 3; Segm card rng p c= Segm card X() by A15,CARD_1:11; then card rng p <= n & n < n+1 by NAT_1:19,39; then not dom p, rng p are_equipotent by A32,CARD_1:5; then p is not one-to-one by WELLORD2:def 4; then consider i,j being object such that A33: i in dom p & j in dom p & p.i = p.j & i <> j; reconsider i,j as Nat by A33; A34: 1 <= i & 1 <= j & i <= n+1 & j <= n+1 by A7,A33,FINSEQ_3:25; p.i in rng p by A33,FUNCT_1:def 3; then A35: p.i in X() by A15; i < j or j < i by A33,XXREAL_0:1; then R[p.i,p.i] by A16,A33,A34; hence contradiction by A1,A35; end; scheme FiniteC{X() -> finite set, P[set]}: P[X()] provided A1 : for A being Subset of X() st for B being set st B c< A holds P[B] holds P[A] proof defpred Q[Nat] means for A being Subset of X() st card A = $1 holds P[A]; A2: for n being Nat st for i being Nat st i < n holds Q[i] holds Q[n] proof let n be Nat such that A3: for i being Nat st i < n holds Q[i]; let A be Subset of X() such that A4: card A = n; now let B be set such that A5: B c< A; B c= A by A5; then reconsider B9 = B as Subset of X() by XBOOLE_1:1; card B9 < n by A4,A5,TREES_1:6; hence P[B] by A3; end; hence thesis by A1; end; for n being Nat holds Q[n] from NAT_1:sch 4(A2); then Q[card X()] & [#]X() = X(); hence thesis; end; scheme Numeration{X() -> finite set, R[set, set]}: ex s being one-to-one FinSequence st rng s = X() & for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j provided A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z] holds R[x,z] proof defpred P[set] means ex s being one-to-one FinSequence st rng s = $1 & for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j; A3: P[{}] proof reconsider s = {} as one-to-one FinSequence; take s; thus thesis; end; A4: for A being Subset of X() st for B being set st B c< A holds P[B] holds P[A] proof let A be Subset of X() such that A5: for B being set st B c< A holds P[B]; per cases; suppose A is empty; hence P[A] by A3; end; suppose A is non empty; then reconsider A9 = A as non empty finite set; A6: for x,y being set st x in A9 & y in A9 & R[x,y] holds not R[y,x] by A1; A7: for x,y,z being set st x in A9 & y in A9 & z in A9 & R[x,y] & R[y,z] holds R[x,z] by A2; consider x being set such that A8: x in A9 & for y being set st y in A9 holds not R[y,x] from MinimalElement(A6,A7); set B = A\{x}; A9: x nin B & B c= A by ZFMISC_1:56; then B c< A by A8; then consider s being one-to-one FinSequence such that A10: rng s = B and A11: for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j by A5; <*x*> is one-to-one & rng <*x*> = {x} & {x} misses B by FINSEQ_1:39,FINSEQ_3:93,XBOOLE_1:79; then reconsider s9 = <*x*>^s as one-to-one FinSequence by A10,FINSEQ_3:91; A12: {x} c= A by A8,ZFMISC_1:31; A13: len <*x*> = 1 by FINSEQ_1:40; thus P[A] proof take s9; thus rng s9 = (rng <*x*>)\/ rng s by FINSEQ_1:31 .= {x} \/ B by A10,FINSEQ_1:38 .= A by A12,XBOOLE_1:45; let i,j such that A14: i in dom s9 & j in dom s9 & R[s9.i, s9.j]; A15: dom <*x*> = Seg 1 by FINSEQ_1:38; per cases by A13,A14,FINSEQ_1:25; suppose i in dom <*x*> & j in dom <*x*>; then i = 1 & j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then s9.i = x & s9.j = x by FINSEQ_1:41; hence i < j by A8,A14; end; suppose A16: i in dom <*x*> & ex n being Nat st n in dom s & j = 1 + n; then A17: i = 1 by A15,FINSEQ_1:2,TARSKI:def 1; consider n being Nat such that A18: n in dom s & j = 1+n by A16; 1 <= n by A18,FINSEQ_3:25; hence i < j by A17,A18,NAT_1:13; end; suppose A19: j in dom <*x*> & ex n being Nat st n in dom s & i = 1 + n; then j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then A20: s9.j = x by FINSEQ_1:41; consider n being Nat such that A21: n in dom s & i = 1+n by A19; s9.i = s.n by A13,A21,FINSEQ_1:def 7; then s9.i in rng s by A21,FUNCT_1:def 3; hence i < j by A8,A14,A20,A9,A10; end; suppose (ex n being Nat st n in dom s & i = 1 + n) & ex n being Nat st n in dom s & j = 1 + n; then consider ni,nj being Nat such that A22: ni in dom s & i = 1+ni & nj in dom s & j = 1+nj; s9.i = s.ni & s9.j = s.nj by A13,A22,FINSEQ_1:def 7; then ni < nj by A11,A14,A22; hence i < j by A22,XREAL_1:6; end; end; end; end; thus P[X()] from FiniteC(A4); end; theorem Th2: for x being variable holds varcl vars x = vars x proof let x be variable; x in Vars; then consider A being Subset of Vars, j being Element of NAT such that A1: x = [varcl A,j] & A is finite by ABCMIZ_1:18; vars x = varcl A by A1; hence thesis; end; theorem Th3: for C being initialized ConstructorSignature for e being expression of C holds e is compound iff not ex x being Element of Vars st e = x-term C proof let C be initialized ConstructorSignature; let e be expression of C; (ex x being variable st e = x-term C) or (ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & e = c-trm p) or (ex a being expression of C, an_Adj C st e = (non_op C)term a) or (ex a being expression of C, an_Adj C st ex t being expression of C, a_Type C st e = (ast C)term(a,t)) by ABCMIZ_1:53; hence thesis; end; begin :: Standardized Constructor Signature registration cluster empty for quasi-loci; existence by ABCMIZ_1:29; end; definition let C be ConstructorSignature; attr C is standardized means: Def1: for o being OperSymbol of C st o is constructor holds o in Constructors & o`1 = the_result_sort_of o & card o`2`1 = len the_arity_of o; end; theorem Th4: for C being ConstructorSignature st C is standardized for o being OperSymbol of C holds o is constructor iff o in Constructors proof let C be ConstructorSignature such that A1: C is standardized; let o be OperSymbol of C; thus o is constructor implies o in Constructors by A1; assume o in Constructors; then not o in {*, non_op} by ABCMIZ_1:39,XBOOLE_0:3; hence o <> * & o <> non_op by TARSKI:def 2; end; registration cluster MaxConstrSign -> standardized; coherence proof let o be OperSymbol of MaxConstrSign; A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors by ABCMIZ_1:def 24; assume A2: o is constructor; then A3: (the ResultSort of MaxConstrSign).o = o`1 & card ((the Arity of MaxConstrSign).o) = card o`2`1 by ABCMIZ_1:def 24; o <> * & o <> non_op by A2; then not o in {*, non_op} by TARSKI:def 2; hence thesis by A3,A1,XBOOLE_0:def 3; end; end; registration cluster initialized standardized strict for ConstructorSignature; existence proof take MaxConstrSign; thus thesis; end; end; definition let C be initialized standardized ConstructorSignature; let c be constructor OperSymbol of C; func loci_of c -> quasi-loci equals c`2`1; coherence proof reconsider c as Element of Constructors by Th4; loci_of c is quasi-loci; hence thesis; end; end; registration let C be ConstructorSignature; cluster constructor for Subsignature of C; existence proof reconsider S = C as Subsignature of C by INSTALG1:15; take S; thus thesis; end; end; registration let C be initialized ConstructorSignature; cluster initialized for constructor Subsignature of C; existence proof reconsider S = C as constructor Subsignature of C by INSTALG1:15; take S; thus thesis; end; end; registration let C be standardized ConstructorSignature; cluster -> standardized for constructor Subsignature of C; coherence proof let S be constructor Subsignature of C; let o be OperSymbol of S such that A1: o <> * & o <> non_op; A2: the carrier' of S c= the carrier' of C by INSTALG1:10; reconsider c = o as OperSymbol of C by A2; A3: c is constructor by A1; the Arity of S = (the Arity of C)|the carrier' of S & the ResultSort of S = (the ResultSort of C)|the carrier' of S by INSTALG1:12; then the_result_sort_of c = the_result_sort_of o & the_arity_of c = the_arity_of o by FUNCT_1:49; hence thesis by A3,Def1; end; end; theorem for S1,S2 being standardized ConstructorSignature st the carrier' of S1 = the carrier' of S2 holds the ManySortedSign of S1 = the ManySortedSign of S2 proof let S1,S2 be standardized ConstructorSignature such that A1: the carrier' of S1 = the carrier' of S2; A2: the carrier of S1 = 3 & the carrier of S2 = 3 by ABCMIZ_1:def 9,YELLOW11:1; now let o be OperSymbol of S1; reconsider o2 = o as OperSymbol of S2 by A1; per cases; suppose o = * or o = non_op; then (the Arity of S1).o = <*an_Adj*> & (the Arity of S2).o = <*an_Adj*> or (the Arity of S1).o = <*an_Adj,a_Type*> & (the Arity of S2).o = <*an_Adj,a_Type*> by ABCMIZ_1:def 9; hence (the Arity of S1).o = (the Arity of S2).o; end; suppose o is constructor & o2 is constructor; then card o`2`1 = len the_arity_of o & card o`2`1 = len the_arity_of o2 & the_arity_of o = (len the_arity_of o) |-> a_Term & the_arity_of o2 = (len the_arity_of o2) |-> a_Term by Def1,ABCMIZ_1:37; hence (the Arity of S1).o = the_arity_of o2 .= (the Arity of S2).o; end; end; then A3: the Arity of S1 = the Arity of S2 by A1,A2,FUNCT_2:63; now let o be OperSymbol of S1; reconsider o2 = o as OperSymbol of S2 by A1; per cases; suppose o = * or o = non_op; then (the ResultSort of S1).o = a_Type & (the ResultSort of S2).o = a_Type or (the ResultSort of S1).o = an_Adj & (the ResultSort of S2).o = an_Adj by ABCMIZ_1:def 9; hence (the ResultSort of S1).o = (the ResultSort of S2).o; end; suppose o is constructor & o2 is constructor; then the_result_sort_of o = o`1 & the_result_sort_of o2 = o`1 by Def1; hence (the ResultSort of S1).o = the_result_sort_of o2 .= (the ResultSort of S2).o; end; end; hence thesis by A1,A2,A3,FUNCT_2:63; end; theorem for C being ConstructorSignature holds C is standardized iff C is Subsignature of MaxConstrSign proof let C be ConstructorSignature; A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors by ABCMIZ_1:def 24; A2: dom the Arity of MaxConstrSign = the carrier' of MaxConstrSign by FUNCT_2:def 1; A3: dom the ResultSort of MaxConstrSign = the carrier' of MaxConstrSign by FUNCT_2:def 1; thus C is standardized implies C is Subsignature of MaxConstrSign proof assume A4: for o being OperSymbol of C st o is constructor holds o in Constructors & o`1 = the_result_sort_of o & card o`2`1 = len the_arity_of o; A5: the carrier of C = 3 & the carrier of MaxConstrSign = 3 by ABCMIZ_1:def 9,YELLOW11:1; A6: the Arity of C c= the Arity of MaxConstrSign proof let x,y be object; assume A7: [x,y] in the Arity of C; then reconsider x as OperSymbol of C by ZFMISC_1:87; x = * or x = non_op or x is constructor; then x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3; A8: y = (the Arity of C).x by A7,FUNCT_1:1; per cases; suppose x = * or x = non_op; then c = * & y = <*an_Adj,a_Type*> or c = non_op & y = <*an_Adj*> by A8,ABCMIZ_1:def 9; then y = (the Arity of MaxConstrSign).c by ABCMIZ_1:def 9; hence thesis by A2,FUNCT_1:def 2; end; suppose A9: x is constructor; then A10: x <> * & x <> non_op; then A11: c is constructor; reconsider y as set by TARSKI:1; card x`2`1 = len the_arity_of x by A4,A9 .= card y by A7,FUNCT_1:1; then A12: card y = card ((the Arity of MaxConstrSign).c) by A11,ABCMIZ_1:def 24; y in {a_Term}* & (the Arity of MaxConstrSign).c in {a_Term}* by A8,A10,ABCMIZ_1:def 9; then y = (the Arity of MaxConstrSign).c by A12,ABCMIZ_1:6; hence thesis by A2,FUNCT_1:def 2; end; end; the ResultSort of C c= the ResultSort of MaxConstrSign proof let x,y be object; assume A13: [x,y] in the ResultSort of C; then reconsider x as OperSymbol of C by ZFMISC_1:87; x is constructor or x = * or x = non_op; then x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3; A14: y = (the ResultSort of C).x by A13,FUNCT_1:1; per cases; suppose x = * or x = non_op; then c = * & y = a_Type or c = non_op & y = an_Adj by A14,ABCMIZ_1:def 9; then y = (the ResultSort of MaxConstrSign).c by ABCMIZ_1:def 9; hence thesis by A3,FUNCT_1:def 2; end; suppose A15: x is constructor & c is constructor; then x`1 = the_result_sort_of x by A4 .= y by A13,FUNCT_1:1; then y = the_result_sort_of c by A15,Def1 .= (the ResultSort of MaxConstrSign).c; hence thesis by A3,FUNCT_1:def 2; end; end; hence thesis by A5,A6,INSTALG1:13; end; assume A16: C is Subsignature of MaxConstrSign; let o be OperSymbol of C such that A17: o <> * & o <> non_op; the carrier' of C c= the carrier' of MaxConstrSign & o in the carrier' of C by A16,INSTALG1:10; then reconsider c = o as OperSymbol of MaxConstrSign; A18: c is constructor by A17; not c in {*, non_op} by A17,TARSKI:def 2; hence o in Constructors by A1,XBOOLE_0:def 3; thus o`1 = (the ResultSort of MaxConstrSign).c by A18,ABCMIZ_1:def 24 .= ((the ResultSort of MaxConstrSign)|the carrier' of C).o by FUNCT_1:49 .= (the ResultSort of C).o by A16,INSTALG1:12 .= the_result_sort_of o; thus card o`2`1 = card ((the Arity of MaxConstrSign).c) by A18,ABCMIZ_1:def 24 .= card (((the Arity of MaxConstrSign)|the carrier' of C).o) by FUNCT_1:49 .= card ((the Arity of C).o) by A16,INSTALG1:12 .= len the_arity_of o; end; registration let C be initialized ConstructorSignature; cluster non compound for quasi-term of C; existence proof set x = the Element of Vars; take x-term C; thus thesis; end; end; registration cluster -> pair for Element of Vars; coherence proof let x be Element of Vars; Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT: A is finite} & x in Vars by ABCMIZ_1:18; then ex A being Subset of Vars, j being Element of NAT st x = [varcl A, j] & A is finite; hence thesis; end; end; theorem Th7: for x being Element of Vars st vars x is natural holds vars x = 0 proof let x be Element of Vars; assume x`1 is natural; then reconsider n = x`1 as Element of NAT; Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT: A is finite} & x in Vars by ABCMIZ_1:18; then consider A being Subset of Vars, j being Element of NAT such that A1: x = [varcl A, j] & A is finite; set i = the Element of n; assume A2: x`1 <> 0; then A3: i in n; reconsider i as Element of NAT by A2,ORDINAL1:10; n = varcl A & vars x c= Vars by A1; then i in Vars by A3; hence thesis; end; theorem Th8: Vars misses Constructors proof assume Vars meets Constructors; then consider x being object such that A1: x in Vars & x in Constructors by XBOOLE_0:3; reconsider x as Element of Vars by A1; consider A being Subset of Vars, j being Element of NAT such that A2: x = [varcl A, j] & A is finite by A1,ABCMIZ_1:18; x in Modes \/ Attrs or x in Funcs by A1,XBOOLE_0:def 3; then x in Modes or x in Attrs or x in Funcs by XBOOLE_0:def 3; then x`2 in [:QuasiLoci,NAT:] & x`2 = j by A2,MCART_1:10; then ex a,b being object st a in QuasiLoci & b in NAT & [a,b] = j by ZFMISC_1:def 2; hence thesis; end; theorem for x being Element of Vars holds x <> * & x <> non_op; theorem Th10: for C being standardized ConstructorSignature holds Vars misses the carrier' of C proof let C be standardized ConstructorSignature; assume Vars meets the carrier' of C; then consider x being object such that A1: x in Vars & x in the carrier' of C by XBOOLE_0:3; reconsider x as Element of Vars by A1; reconsider c = x as OperSymbol of C by A1; x = * or x = non_op or c is constructor; then x = * or x = non_op or x in Constructors & Vars misses Constructors by Th8,Def1; hence thesis by XBOOLE_0:3; end; theorem Th11: :: see ABCMIZ_1:51 for C being initialized standardized ConstructorSignature for e being expression of C holds (ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or (ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op )) proof let C be initialized standardized ConstructorSignature; let e be expression of C; set X = MSVars C; set Y = X (\/) ((the carrier of C) --> {0}); reconsider q = e as Term of C,Y by MSAFREE3:8; per cases by MSATERM:2; suppose ex s being SortSymbol of C, v being Element of Y.s st q.{} = [v,s]; then consider s being SortSymbol of C, v being Element of Y.s such that A1: q.{} = [v,s]; consider z being object such that A2: z in dom the Sorts of Free(C,X) & e in (the Sorts of Free(C, X)).z by CARD_5:2; reconsider z as SortSymbol of C by A2; the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then A3: z = a_Type or z = an_Adj or z = a_Term by ENUMSET1:def 1; A4: q = root-tree [v,s] by A1,MSATERM:5; then A5: the_sort_of q = s by MSATERM:14; A6: the Sorts of Free(C, X) = C-Terms(X,Y) by MSAFREE3:24; then the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18; then (the Sorts of Free(C, X)).z c= (the Sorts of FreeMSA Y).z & FreeMSA Y = MSAlgebra(#FreeSort Y, FreeOper Y#); then q in (the Sorts of FreeMSA Y).z & (the Sorts of FreeMSA Y).z = FreeSort(Y, z) by A2,MSAFREE:def 11; then A7: s = z by A5,MSATERM:def 5; then v in (MSVars C).z by A4,A2,A6,MSAFREE3:18; then A8: v in Vars & z = a_Term by A3,ABCMIZ_1:def 25; then reconsider x = v as Element of Vars; e = x-term C by A1,A7,A8,MSATERM:5; hence thesis by A1,A7,A8; end; suppose q.{} in [:the carrier' of C,{the carrier of C}:]; then consider o,s being object such that A9: o in the carrier' of C & s in {the carrier of C} & q.{} = [o,s] by ZFMISC_1:def 2; reconsider o as OperSymbol of C by A9; o is constructor iff o <> * & o <> non_op; then s = the carrier of C & (o in Constructors or o = * or o = non_op) by A9,Def1,TARSKI:def 1; hence thesis by A9; end; end; registration let C be initialized standardized ConstructorSignature; let e be expression of C; cluster e.{} -> pair; coherence proof (ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or (ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op )) by Th11; hence thesis; end; end; theorem Th12: for C being initialized ConstructorSignature for e being expression of C for o being OperSymbol of C st e.{} = [o, the carrier of C] holds e is expression of C, the_result_sort_of o proof let C be initialized ConstructorSignature; let e be expression of C; let o be OperSymbol of C such that A1: e.{} = [o, the carrier of C]; set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0}); reconsider t = e as Term of C, Y by MSAFREE3:8; variables_in t c= X by MSAFREE3:27; then e in {t1 where t1 is Term of C, Y: the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then A2: e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24; the_sort_of t = the_result_sort_of o by A1,MSATERM:17; hence thesis by A2,ABCMIZ_1:def 28; end; theorem Th13: for C being initialized standardized ConstructorSignature for e being expression of C holds ( (e.{})`1 = * implies e is expression of C, a_Type C ) & ( (e.{})`1 = non_op implies e is expression of C, an_Adj C ) proof let C be initialized standardized ConstructorSignature; let e be expression of C; per cases by Th11; suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then consider x being Element of Vars such that A1: e = x-term C & e.{} = [x, a_Term]; thus thesis by A1; end; suppose (ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op )); then consider o being OperSymbol of C such that A2: e.{} = [o, the carrier of C]; set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0}); reconsider t = e as Term of C, Y by MSAFREE3:8; variables_in t c= X by MSAFREE3:27; then e in {t1 where t1 is Term of C, Y: the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then A3: e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24; A4: the_result_sort_of non_op C = an_Adj C & the_result_sort_of ast C = a_Type C by ABCMIZ_1:38; A5: (e.{})`1 = o & non_op C = non_op & ast C = * by A2; the_sort_of t = the_result_sort_of o by A2,MSATERM:17; hence thesis by A3,A4,A5,ABCMIZ_1:def 28; end; end; theorem Th14: for C being initialized standardized ConstructorSignature for e being expression of C holds (e.{})`1 in Vars & (e.{})`2 = a_Term & e is quasi-term of C or (e.{})`2 = the carrier of C & ( (e.{})`1 in Constructors & (e.{})`1 in the carrier' of C or (e.{})`1 = * or (e.{})`1 = non_op ) proof let C be initialized standardized ConstructorSignature; let e be expression of C; per cases by Th11; suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then consider x being Element of Vars such that A1: e = x-term C & e.{} = [x, a_Term]; thus thesis by A1; end; suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ); then consider o being OperSymbol of C such that A2: e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ); thus thesis by A2; end; end; theorem for C being initialized standardized ConstructorSignature for e being expression of C st (e.{})`1 in Constructors holds e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1 proof let C be initialized standardized ConstructorSignature; let e be expression of C; assume A1: (e.{})`1 in Constructors; per cases by Th11; suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then consider x being Element of Vars such that A2: e = x-term C & e.{} = [x, a_Term]; (e.{})`1 = x by A2; hence thesis by A1,Th8,XBOOLE_0:3; end; suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ); then consider o being OperSymbol of C such that A3: e.{} = [o, the carrier of C]; A4: (e.{})`1 = o by A3; * in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then A5: o is constructor; set X = MSVars C; reconsider t = e as Term of C, X (\/) ((the carrier of C) --> {0}) by MSAFREE3:8; A6: the_sort_of t = the_result_sort_of o by A3,MSATERM:17 .= o`1 by A5,Def1; variables_in t c= X by MSAFREE3:27; then e in {t1 where t1 is Term of C, X (\/) ((the carrier of C) --> {0}): the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then e in C-Terms(X, X (\/) ((the carrier of C)-->{0})).the_sort_of t by MSAFREE3:def 5; hence e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1 by A4,A6,MSAFREE3:23; end; end; theorem for C being initialized standardized ConstructorSignature for e being expression of C holds not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C proof let C be initialized standardized ConstructorSignature; let e be expression of C; A1: (e.{})`1 in Vars or (e.{})`1 in the carrier' of C or (e.{})`1 = ast C or (e.{})`1 = non_op C by Th14; Vars misses the carrier' of C by Th10; hence not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C by A1,XBOOLE_0:3; end; theorem Th17: for C being initialized standardized ConstructorSignature for e being expression of C st (e.{})`1 in Vars ex x being Element of Vars st x = (e.{})`1 & e = x-term C proof let C be initialized standardized ConstructorSignature; let t be expression of C such that A1: (t.{})`1 in Vars; set X = MSVars C; set V = X (\/) ((the carrier of C) --> {0}); reconsider q = t as Term of C, V by MSAFREE3:8; per cases by MSATERM:2; suppose q.{} in [:the carrier' of C, {the carrier of C}:]; then (q.{})`1 in the carrier' of C & the carrier' of C misses Vars by Th10,MCART_1:10; hence thesis by A1,XBOOLE_0:3; end; suppose ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then consider s being SortSymbol of C, v being Element of V.s such that A2: t.{} = [v,s]; A3: q = root-tree [v,s] by A2,MSATERM:5; reconsider x = v as Element of Vars by A1,A2; take x; the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then A4: s = a_Term or s = a_Type or s = an_Adj by ENUMSET1:def 1; ((the carrier of C) --> {0}).s = {0}; then V.s = X.s \/ {0} by PBOOLE:def 4; then A5: s = a_Term or V.s = {} \/ {0} by A4,ABCMIZ_1:def 25; v in V.s & x <> 0; hence thesis by A2,A3,A5; end; end; theorem Th18: for C being initialized standardized ConstructorSignature for e being expression of C st (e.{})`1 = * ex a being expression of C, an_Adj C, q being expression of C, a_Type C st e = [*,3]-tree(a,q) proof let C be initialized standardized ConstructorSignature; let e be expression of C such that A1: (e.{})`1 = *; not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term] by A1; then consider o being OperSymbol of C such that A2: e.{} = [o, the carrier of C] and o in Constructors or o = * or o = non_op by Th11; set Y = (MSVars C) (\/) ((the carrier of C) --> {0}); reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0}) by MSAFREE3:8; consider aa being ArgumentSeq of Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10; A4: * = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o; A5: the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38; A6: len aa = len the_arity_of o by MSATERM:22 .= 2 by A4,A5,FINSEQ_1:44; then dom aa = Seg 2 by FINSEQ_1:def 3; then A7: 1 in dom aa & 2 in dom aa; then reconsider t1 = aa.1, t2 = aa.2 as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22; A8: len doms aa = len aa by TREES_3:38; (doms aa).1 = dom t1 & (doms aa).2 = dom t2 by A7,FUNCT_6:22; then A9: 0 < 2 & 0+1 = 1 & 1 < 2 & 1+1 = 2 & {} in (doms aa).1 & {} in (doms aa).2 & <* 0*>^<*>NAT = <* 0*> & <* 1*>^<*>NAT = <* 1*> by FINSEQ_1:34,TREES_1:22; dom t = tree doms aa by A3,TREES_4:10; then reconsider 00 = <* 0*>, 01 = <* 1*> as Element of dom t by A6,A8,A9,TREES_3:def 15; 0 < 2 & 1 = 0+1 & 1 < 2 & 2 = 1+1 & aa is DTree-yielding; then t1 = t|00 & t2 = t|01 by A3,A6,TREES_4:def 4; then A10: t1 is expression of C & t2 is expression of C & variables_in t1 c= variables_in t & variables_in t2 c= variables_in t by MSAFREE3:32,33; then A11: variables_in t1 c= MSVars C & variables_in t2 c= MSVars C by MSAFREE3:27; the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23 .= an_Adj C by A4,A5,FINSEQ_1:44; then t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C & variables_in s c= MSVars C} by A11; then t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28; the_sort_of t2 = (the_arity_of o).2 by A7,MSATERM:23 .= a_Type C by A4,A5,FINSEQ_1:44; then t2 in {s where s is Term of C,Y: the_sort_of s = a_Type C & variables_in s c= MSVars C} by A11; then t2 in (C-Terms(MSVars C, Y)).a_Type C by MSAFREE3:def 5; then t2 in (the Sorts of Free(C, MSVars C)).a_Type C by MSAFREE3:24; then reconsider q = t2 as expression of C, a_Type C by A10,ABCMIZ_1:def 28; take a,q; A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1; aa = <*a,q*> by A6,FINSEQ_1:44; hence thesis by A3,A4,A12,TREES_4:def 6; end; theorem Th19: for C being initialized standardized ConstructorSignature for e being expression of C st (e.{})`1 = non_op ex a being expression of C, an_Adj C st e = [non_op,3]-tree a proof let C be initialized standardized ConstructorSignature; let e be expression of C such that A1: (e.{})`1 = non_op; not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term] by A1; then consider o being OperSymbol of C such that A2: e.{} = [o, the carrier of C] and o in Constructors or o = * or o = non_op by Th11; set Y = (MSVars C) (\/) ((the carrier of C) --> {0}); reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0}) by MSAFREE3:8; consider aa being ArgumentSeq of Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10; A4: non_op = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o; A5: the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; A6: len aa = len the_arity_of o by MSATERM:22 .= 1 by A4,A5,FINSEQ_1:40; then dom aa = Seg 1 by FINSEQ_1:def 3; then A7: 1 in dom aa; then reconsider t1 = aa.1 as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22; A8: len doms aa = len aa by TREES_3:38; (doms aa).1 = dom t1 by A7,FUNCT_6:22; then A9: 0 < 1 & 0+1 = 1 & {} in (doms aa).1 & <* 0*>^<*>NAT = <* 0*> by FINSEQ_1:34,TREES_1:22; dom t = tree doms aa by A3,TREES_4:10; then reconsider 00 = <* 0*> as Element of dom t by A6,A8,A9,TREES_3:def 15; t1 = t|00 by A3,A6,A9,TREES_4:def 4; then A10: t1 is expression of C & variables_in t1 c= variables_in t by MSAFREE3:32,33; then A11: variables_in t1 c= MSVars C by MSAFREE3:27; the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23 .= an_Adj C by A4,A5,FINSEQ_1:40; then t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C & variables_in s c= MSVars C} by A11; then t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28; take a; A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1; aa = <*a*> by A6,FINSEQ_1:40; hence thesis by A3,A4,A12,TREES_4:def 5; end; theorem Th20: for C being initialized standardized ConstructorSignature for e being expression of C st (e.{})`1 in Constructors ex o being OperSymbol of C st o = (e.{})`1 & the_result_sort_of o = o`1 & e is expression of C, the_result_sort_of o proof let C be initialized standardized ConstructorSignature; let e be expression of C such that A1: (e.{})`1 in Constructors; per cases by Th11; suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then consider x being Element of Vars such that A2: e = x-term C & e.{} = [x, a_Term]; (e.{})`1 = x & x in Vars by A2; hence thesis by A1,Th8,XBOOLE_0:3; end; suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ); then consider o being OperSymbol of C such that A3: e.{} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ); take o; A4: (e.{})`1 = o & (e.{})`2 = the carrier of C by A3; * in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then o is constructor; hence o = (e.{})`1 & the_result_sort_of o = o`1 by A3,Def1; set X = MSVars C; set V = X (\/) ((the carrier of C) --> {0}); reconsider q = e as Term of C,V by MSAFREE3:8; A5: variables_in q c= X by MSAFREE3:27; A6: the_sort_of q = the_result_sort_of o by A3,MSATERM:17; (the Sorts of Free(C, MSVars C)).the_result_sort_of o = C-Terms(X,V).the_result_sort_of o by MSAFREE3:24 .= {a where a is Term of C,V: the_sort_of a = the_result_sort_of o & variables_in a c= X} by MSAFREE3:def 5; hence e in (the Sorts of Free(C, MSVars C)).the_result_sort_of o by A5,A6; end; end; theorem Th21: for C being initialized standardized ConstructorSignature for t being quasi-term of C holds t is compound iff (t.{})`1 in Constructors & (t.{})`1`1 = a_Term proof let C be initialized standardized ConstructorSignature; set X = MSVars C; set V = X (\/) ((the carrier of C) --> {0}); let t be quasi-term of C; C-Terms(X, V) c= the Sorts of FreeMSA V & the Sorts of Free(C, X) = C-Terms(X, V) by MSAFREE3:24,PBOOLE:def 18; then A1: FreeMSA V = MSAlgebra(#FreeSort V, FreeOper V#) & (C-Terms(X, V)).a_Term C c= (the Sorts of FreeMSA V).a_Term C & t in C-Terms(X,V).a_Term C by ABCMIZ_1:def 28; then t in (FreeSort V).a_Term C; then A2: t in FreeSort(V,a_Term C) by MSAFREE:def 11; A3: (MSVars C).a_Term = Vars & a_Term C = a_Term & a_Term = 2 by ABCMIZ_1:def 25; reconsider q = t as Term of C, V by MSAFREE3:8; per cases by MSATERM:2; suppose ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then consider s being SortSymbol of C, v being Element of V.s such that A4: t.{} = [v,s]; A5: q = root-tree [v,s] & the_sort_of q = a_Term C by A2,A4,MSATERM:5,def 5; then A6: a_Term C = s & (t.{})`1 = v by A4,MSATERM:14; then reconsider x = v as Element of Vars by A3,A5,A1,MSAFREE3:18; q = x-term C & vars x <> 2 by A5,A6,Th7; hence thesis by A6; end; suppose q.{} in [:the carrier' of C,{the carrier of C}:]; then consider o, k being object such that A7: o in the carrier' of C & k in {the carrier of C} & q.{} = [o,k] by ZFMISC_1:def 2; reconsider o as OperSymbol of C by A7; k = the carrier of C by A7,TARSKI:def 1; then A8: the_result_sort_of o = the_sort_of q by A7,MSATERM:17 .= a_Term C by A1,MSAFREE3:17; then o <> ast C & o <> non_op C by ABCMIZ_1:38; then A9: o is constructor; then A10: a_Term C = o`1 by A8,Def1 .= (q.{})`1`1 by A7; A11: (q.{})`1 = o by A7; now given x being Element of Vars such that A12: q = x-term C; q.{} = [x,a_Term] by A12,TREES_4:3; then k = a_Term by A7,XTUPLE_0:1; then 2 = the carrier of C by A7,TARSKI:def 1; hence contradiction by ABCMIZ_1:def 9,YELLOW11:1; end; hence thesis by A9,A10,A11,Th3,Def1; end; end; theorem Th22: for C being initialized standardized ConstructorSignature for t being expression of C holds t is non compound quasi-term of C iff (t.{})`1 in Vars proof let C be initialized standardized ConstructorSignature; let t be expression of C; thus now assume t is non compound quasi-term of C; then consider x being Element of Vars such that A1: t = x-term C by Th3; t.{} = [x,a_Term] & x in Vars by A1,TREES_4:3; hence (t.{})`1 in Vars; end; assume (t.{})`1 in Vars; then ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17; hence thesis; end; theorem for C being initialized standardized ConstructorSignature for t being expression of C holds t is quasi-term of C iff (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars proof let C be initialized standardized ConstructorSignature; let t be expression of C; hereby assume t is quasi-term of C; then reconsider tr = t as quasi-term of C; tr is compound or tr is non compound; hence (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars by Th21,Th22; end; assume that A1: (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars and A2: t is not quasi-term of C; A3: (t.{})`1 in Vars implies ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17; then (t.{})`1 in Constructors & (t.{})`1`1 = a_Term by A1,A2; then ex o being OperSymbol of C st o = (t.{})`1 & the_result_sort_of o = o`1 & t is expression of C, the_result_sort_of o by Th20; hence thesis by A2,A3,A1; end; theorem Th24: for C being initialized standardized ConstructorSignature for a being expression of C holds a is positive quasi-adjective of C iff (a .{})`1 in Constructors & (a .{})`1`1 = an_Adj proof let C be initialized standardized ConstructorSignature; set X = MSVars C; set V = X (\/) ((the carrier of C) --> {0}); let t be expression of C; consider A being MSSubset of FreeMSA V such that A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1; the Sorts of Free(C, X) is MSSubset of FreeMSA V by A1,MSUALG_2:def 9; then the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then A2: (the Sorts of Free(C, MSVars C)).an_Adj C c= (the Sorts of FreeMSA V).an_Adj C; per cases by Th14; suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C; hence thesis by Th8,ABCMIZ_1:77,XBOOLE_0:3; end; suppose that A3: (t.{})`2 = the carrier of C and A4: (t.{})`1 in Constructors and A5: (t.{})`1 in the carrier' of C; reconsider o = (t.{})`1 as OperSymbol of C by A5; reconsider tt = t as Term of C, V by MSAFREE3:8; not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then o <> * & o <> non_op by TARSKI:def 2; then o is constructor; then A6: o`1 = the_result_sort_of o by Def1; A7: t.{} = [o, (t.{})`2]; then A8: the_sort_of tt = the_result_sort_of o by A3,MSATERM:17; hereby assume t is positive quasi-adjective of C; then A9: t in (the Sorts of Free(C, MSVars C)).an_Adj C by ABCMIZ_1:def 28; thus (t.{})`1 in Constructors by A4; assume (t.{})`1`1 <> an_Adj; hence contradiction by A2,A9,A6,A8,MSAFREE3:7; end; assume (t.{})`1 in Constructors; assume (t.{})`1`1 = an_Adj; then reconsider t as expression of C, an_Adj C by A3,A6,A7,Th12; t is positive proof given a being expression of C, an_Adj C such that A10: t = (non_op C)term a; t = [non_op, the carrier of C]-tree <*a*> by A10,ABCMIZ_1:43; then t.{} = [non_op, the carrier of C] by TREES_4:def 4; then (t.{})`1 = non_op; then (t.{})`1 in {*, non_op} by TARSKI:def 2; hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3; end; hence thesis; end; suppose A11: (t.{})`1 = *; then (t.{})`1 in {*, non_op} by TARSKI:def 2; then A12: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3; consider a being expression of C, an_Adj C, q being expression of C, a_Type C such that A13: t = [*,3]-tree(a,q) by A11,Th18; t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6; then t.{} = [*, 3] by TREES_4:def 4; then (t.{})`1 = *; then t is expression of C, a_Type C & a_Type C = a_Type & a_Type = 0 & an_Adj C = an_Adj & an_Adj = 1 by Th13; hence thesis by A12,ABCMIZ_1:48; end; suppose A14: (t.{})`1 = non_op; then (t.{})`1 in {*, non_op} by TARSKI:def 2; then A15: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3; consider a being expression of C, an_Adj C such that A16: t = [non_op,3]-tree a by A14,Th19; t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5 .= [non_op, the carrier of C]-tree <*a*> by ABCMIZ_1:def 9,YELLOW11:1 .= (non_op C)term a by ABCMIZ_1:43; hence thesis by A15,ABCMIZ_1:def 37; end; end; theorem for C being initialized standardized ConstructorSignature for a being quasi-adjective of C holds a is negative iff (a .{})`1 = non_op proof let C be initialized standardized ConstructorSignature; let t be quasi-adjective of C; per cases; suppose A1: t is positive expression of C, an_Adj C; then (t.{})`1 in Constructors & non_op in {*, non_op} by Th24,TARSKI:def 2; hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3; end; suppose A2: t is negative expression of C, an_Adj C; then consider a being expression of C, an_Adj C such that A3: a is positive & t = (non_op C)term a by ABCMIZ_1:def 38; t = [non_op, the carrier of C]-tree <*a*> by A3,ABCMIZ_1:43; then t.{} = [non_op, the carrier of C] by TREES_4:def 4; hence thesis by A2; end; end; theorem for C being initialized standardized ConstructorSignature for t being expression of C holds t is pure expression of C, a_Type C iff (t.{})`1 in Constructors & (t.{})`1`1 = a_Type proof let C be initialized standardized ConstructorSignature; set X = MSVars C; set V = X (\/) ((the carrier of C) --> {0}); let t be expression of C; consider A being MSSubset of FreeMSA V such that A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1; the Sorts of Free(C, X) is MSSubset of FreeMSA V by A1,MSUALG_2:def 9; then the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then A2: (the Sorts of Free(C, MSVars C)).a_Type C c= (the Sorts of FreeMSA V).a_Type C; per cases by Th14; suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C; hence thesis by Th8,ABCMIZ_1:48,XBOOLE_0:3; end; suppose that A3: (t.{})`2 = the carrier of C and A4: (t.{})`1 in Constructors and A5: (t.{})`1 in the carrier' of C; reconsider o = (t.{})`1 as OperSymbol of C by A5; reconsider tt = t as Term of C, V by MSAFREE3:8; not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then o <> * & o <> non_op by TARSKI:def 2; then o is constructor; then A6: o`1 = the_result_sort_of o by Def1; A7: t.{} = [o, (t.{})`2]; then A8: the_sort_of tt = the_result_sort_of o by A3,MSATERM:17; thus now assume t is pure expression of C, a_Type C; then A9: t in (the Sorts of Free(C, MSVars C)).a_Type C by ABCMIZ_1:def 28; thus (t.{})`1 in Constructors by A4; assume (t.{})`1`1 <> a_Type; hence contradiction by A2,A9,A6,A8,MSAFREE3:7; end; assume (t.{})`1 in Constructors; assume (t.{})`1`1 = a_Type; then reconsider t as expression of C, a_Type C by A3,A7,Th12,A6; t is pure proof given a being expression of C, an_Adj C, q being expression of C, a_Type C such that A10: t = (ast C)term(a,q); t = [*, the carrier of C]-tree <*a,q*> by A10,ABCMIZ_1:46; then t.{} = [*, the carrier of C] by TREES_4:def 4; then (t.{})`1 = *; then (t.{})`1 in {*, non_op} by TARSKI:def 2; hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3; end; hence thesis; end; suppose A11: (t.{})`1 = *; then (t.{})`1 in {*, non_op} by TARSKI:def 2; then A12: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3; consider a being expression of C, an_Adj C, q being expression of C, a_Type C such that A13: t = [*,3]-tree(a,q) by A11,Th18; t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6 .= [*, the carrier of C]-tree <*a,q*> by ABCMIZ_1:def 9,YELLOW11:1 .= (ast C)term(a,q) by ABCMIZ_1:46; hence thesis by A12,ABCMIZ_1:def 41; end; suppose A14: (t.{})`1 = non_op; then (t.{})`1 in {*, non_op} by TARSKI:def 2; then A15: not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3; consider a being expression of C, an_Adj C such that A16: t = [non_op,3]-tree a by A14,Th19; t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5; then t.{} = [non_op,3] by TREES_4:def 4; then (t.{})`1 = non_op; then t is expression of C, an_Adj C by Th13; hence thesis by A15,ABCMIZ_1:48; end; end; begin :: Expressions reserve i,j for Nat, x for variable, l for quasi-loci; set MC = MaxConstrSign; definition mode expression is expression of MaxConstrSign; mode valuation is valuation of MaxConstrSign; mode quasi-adjective is quasi-adjective of MaxConstrSign; func QuasiAdjs -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals QuasiAdjs MaxConstrSign; coherence; mode quasi-term is quasi-term of MaxConstrSign; func QuasiTerms -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals QuasiTerms MaxConstrSign; coherence; mode quasi-type is quasi-type of MaxConstrSign; func QuasiTypes -> set equals QuasiTypes MaxConstrSign; coherence; end; registration cluster QuasiAdjs -> non empty; coherence; cluster QuasiTerms -> non empty; coherence; cluster QuasiTypes -> non empty; coherence; end; definition redefine func Modes -> non empty Subset of Constructors; coherence proof Modes c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7; hence thesis by XBOOLE_1:1; end; redefine func Attrs -> non empty Subset of Constructors; coherence proof Attrs c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7; hence thesis by XBOOLE_1:1; end; redefine func Funcs -> non empty Subset of Constructors; coherence by XBOOLE_1:7; end; reserve C for initialized ConstructorSignature, c for constructor OperSymbol of C; definition func set-constr -> Element of Modes equals [a_Type,[{},0]]; coherence proof a_Type in {a_Type} & [{},0] in [:QuasiLoci,NAT:] by ABCMIZ_1:29,TARSKI:def 1,ZFMISC_1:def 2; hence thesis by ZFMISC_1:def 2; end; end; theorem kind_of set-constr = a_Type & loci_of set-constr = {} & index_of set-constr = 0; theorem Th28: Constructors = [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:] proof thus Constructors = [:{a_Type},[:QuasiLoci,NAT:]:] \/ [:{an_Adj},[:QuasiLoci,NAT:]:] \/ Funcs .= [:{a_Type} \/ {an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ZFMISC_1:97 .= [:{a_Type, an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ENUMSET1:1 .= [:{a_Type, an_Adj} \/ {a_Term}, [:QuasiLoci,NAT:]:] by ZFMISC_1:97 .= [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:] by ENUMSET1:3; end; theorem Th29: [rng l, i] in Vars & l^<*[rng l,i]*> is quasi-loci proof varcl rng l = rng l by ABCMIZ_1:33; hence [rng l, i] in Vars by ABCMIZ_1:17; then reconsider x = [rng l, i] as variable; rng l in {rng l, i} & {rng l, i} in x by TARSKI:def 2; then vars x = rng l & x nin rng l by XREGULAR:7; hence thesis by ABCMIZ_1:31; end; theorem Th30: ex l st len l = i proof defpred P[Nat] means ex l st len l = $1; <*>Vars is quasi-loci & len <*>Vars = 0 by ABCMIZ_1:29; then A1: P[ 0 ]; A2: P[j] implies P[j+1] proof given l such that A3: len l = j; reconsider l1 = l^<*[rng l, 1]*> as quasi-loci by Th29; take l1; thus thesis by A3,FINSEQ_2:16; end; P[j] from NAT_1:sch 2(A1,A2); hence thesis; end; theorem Th31: for X being finite Subset of Vars ex l st rng l = varcl X proof let X be finite Subset of Vars; reconsider Y = varcl X as finite Subset of Vars by ABCMIZ_1:24; defpred R[set, set] means $1 in $2`1; A1: for x,y being set st x in Y & y in Y & R[x,y] holds not R[y,x] proof let x,y be set such that A2: x in Y & y in Y & R[x,y] & R[y,x]; x in Vars by A2; then consider A being Subset of Vars, j being Element of NAT such that A3: x = [varcl A, j] & A is finite by ABCMIZ_1:18; y in Vars by A2; then consider B being Subset of Vars, k being Element of NAT such that A4: y = [varcl B, k] & B is finite by ABCMIZ_1:18; A5: y in varcl A & x in varcl B by A2,A3,A4; A6: varcl A in {varcl A} & varcl B in {varcl B} by TARSKI:def 1; {varcl A} in x & {varcl B} in y by A4,A3,TARSKI:def 2; hence thesis by A5,A6,XREGULAR:10; end; A7: for x,y,z being set st x in Y & y in Y & z in Y & R[x,y] & R[y,z] holds R[x,z] proof let x,y,z be set such that A8: x in Y & y in Y & z in Y & R[x,y] & R[y,z]; y in Vars by A8; then consider B being Subset of Vars, k being Element of NAT such that A9: y = [varcl B, k] & B is finite by ABCMIZ_1:18; z in Vars by A8; then consider C being Subset of Vars, j being Element of NAT such that A10: z = [varcl C, j] & C is finite by ABCMIZ_1:18; A11: z`1 = varcl C & y`1 = varcl B by A10,A9; then varcl B c= varcl C by A8,A9,ABCMIZ_1:def 1; hence R[x,z] by A11,A8; end; consider l being one-to-one FinSequence such that A12: rng l = Y and A13: for i,j st i in dom l & j in dom l & R[l.i, l.j] holds i < j from Numeration(A1,A7); reconsider l as one-to-one FinSequence of Vars by A12,FINSEQ_1:def 4; now let i be Nat, x be variable; assume A14: i in dom l & x = l.i; let y be variable; assume A15: y in vars x; x in Vars; then consider A being Subset of Vars, j being Element of NAT such that A16: x = [varcl A, j] & A is finite by ABCMIZ_1:18; x in rng l & vars x = varcl A by A14,A16,FUNCT_1:def 3; then vars x c= Y by A12,A16,ABCMIZ_1:def 1; then consider a being object such that A17: a in dom l & y = l.a by A12,A15,FUNCT_1:def 3; reconsider a as Nat by A17; take a; thus a in dom l & a < i & y = l.a by A13,A14,A15,A17; end; then reconsider l as quasi-loci by ABCMIZ_1:30; take l; thus rng l = varcl X by A12; end; theorem Th32: :: to mozna uogolnic na X zamkniety na poddrzewa :: (troche dodatkowych pojec i twierdzen) for S being non empty non void ManySortedSign for Y being non empty-yielding ManySortedSet of the carrier of S for X,o being set, p being DTree-yielding FinSequence st ex C st X = Union the Sorts of Free(S, Y) holds o-tree p in X implies p is FinSequence of X proof let S be non empty non void ManySortedSign; let Y be non empty-yielding ManySortedSet of the carrier of S; let X,o be set; let p be DTree-yielding FinSequence; given C such that A1: X = Union the Sorts of Free(S, Y); assume o-tree p in X; then reconsider e = o-tree p as Element of Free(S, Y) by A1; rng p c= X proof let z be object; assume z in rng p; then consider i being object such that A2: i in dom p & z = p.i by FUNCT_1:def 3; reconsider i as Nat by A2; reconsider ppi = p.i as DecoratedTree by A2,TREES_3:24; A3: 1 <= i & i <= len p by A2,FINSEQ_3:25; then A4: (i-'1)+1 = i by XREAL_1:235; then A5: i-'1 < len p by A3,NAT_1:13; A6: len doms p = len p by TREES_3:38; A7: (doms p).i = dom ppi by A2,FUNCT_6:22; A8: dom e = tree doms p by TREES_4:10; <*i-'1*>^<*>NAT = <*i-'1*> & <*>NAT in dom ppi by FINSEQ_1:34,TREES_1:22; then reconsider q = <*i-'1*> as Element of dom e by A4,A5,A6,A7,A8,TREES_3:def 15; e|q = z by A2,A4,A5,TREES_4:def 4; then z is Element of Free(S, Y) by MSAFREE3:33; hence thesis by A1; end; hence p is FinSequence of X by FINSEQ_1:def 4; end; definition let C; let e be expression of C; mode subexpression of e -> expression of C means it in Subtrees e; existence by TREES_9:11; func constrs e -> set equals (proj1 rng e)/\the set of all o where o is constructor OperSymbol of C; coherence; end; definition let S be non empty non void ManySortedSign; let X be non empty-yielding ManySortedSet of the carrier of S; let e be Element of Free(S,X); func main-constr e -> set equals: :: dobre dla zestandaryzowanych (nie ma def) Def9: (e.{})`1 if e is compound otherwise {}; :: x-term C = [x, a_Term]-tree {} :: (ast C)term(a,t) = [*, the carrier of C]-tree<*a,t*> :: (non_op C)term a = [non_op, the carrier of C]-tree<*a*> :: c-trm p = [c, the carrier of C]-tree p :: problem gdy '{}' moze byc 'c' correctness; func args e -> DTree-yielding FinSequence means: ARGS: e = (e.{})-tree it; existence proof consider v being set, p being DTree-yielding FinSequence such that A1: e = v-tree p by TREES_9:8; A2: v = e.{} by A1,TREES_4:def 4; thus thesis by A1,A2; end; uniqueness by TREES_4:15; end; definition let S be non empty non void ManySortedSign; let X be non empty-yielding ManySortedSet of the carrier of S; let e be Element of Free(S,X); redefine func args e -> FinSequence of Free(S, X); coherence proof A1: e = (e.{})-tree args e by ARGS; args e is FinSequence of Free(S, X) by A1,Th32; hence thesis; end; end; theorem Th33: for C for e being expression of C holds e is subexpression of e proof let C be initialized ConstructorSignature; let e be expression of C; thus e in Subtrees e by TREES_9:11; end; theorem main-constr (x -term C) = {} by Def9; theorem Th35: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c holds main-constr (c-trm p) = c proof let c be constructor OperSymbol of C; let p be FinSequence of QuasiTerms C; assume len p = len the_arity_of c; then c-trm p = [c, the carrier of C]-tree p by ABCMIZ_1:def 35; then (c-trm p).{} = [c, the carrier of C] by TREES_4:def 4; hence main-constr (c-trm p) = [c, the carrier of C]`1 by Def9 .= c; end; definition let C; let e be expression of C; attr e is constructor means: Def11: e is compound & main-constr e is constructor OperSymbol of C; end; registration let C; cluster constructor -> compound for expression of C; coherence; end; registration let C; cluster constructor for expression of C; existence proof consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type & the_arity_of m = {} & the_result_sort_of a = an_Adj & the_arity_of a = {} by ABCMIZ_1:def 12; the_arity_of ast C = <*an_Adj C, a_Type C*> & the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; then reconsider m as constructor OperSymbol of C by A1,ABCMIZ_1:def 11; set p = <*>QuasiTerms C; take e = m-trm p; thus e is compound; len p = len the_arity_of m by A1; hence thesis by Th35; end; end; registration let C; let e be constructor expression of C; cluster constructor for subexpression of e; existence proof e is subexpression of e by Th33; hence thesis; end; end; registration let S be non void Signature; let X be non empty-yielding ManySortedSet of S; let t be Element of Free(S,X); cluster rng t -> Relation-like; coherence proof set Z = (the carrier of S)-->{0}; set Y = X (\/) Z; t is Term of S,Y by MSAFREE3:8; then rng t c= the carrier of DTConMSA Y by RELAT_1:def 19; hence thesis; end; end; theorem for e being constructor expression of C holds main-constr e in constrs e proof let e be constructor expression of C; A1: main-constr e = (e.{})`1 by Def9; {} in dom e by TREES_1:22; then e.{} in rng e by FUNCT_1:def 3; then A2: main-constr e in proj1 rng e by A1,MCART_1:86; main-constr e is constructor OperSymbol of C by Def11; then main-constr e in the set of all c; hence main-constr e in constrs e by A2,XBOOLE_0:def 4; end; begin :: Arity reserve a,a9 for quasi-adjective, t,t1,t2 for quasi-term, T for quasi-type, c for Element of Constructors; definition let C be non void Signature; attr C is arity-rich means: Def12: for n being Nat, s being SortSymbol of C holds {o where o is OperSymbol of C: the_result_sort_of o = s & len the_arity_of o = n} is infinite; let o be OperSymbol of C; attr o is nullary means: Def13: the_arity_of o = {}; attr o is unary means: Def14: len the_arity_of o = 1; attr o is binary means: Def15: len the_arity_of o = 2; end; theorem Th37: for C being non void Signature for o being OperSymbol of C holds (o is nullary implies o is not unary) & (o is nullary implies o is not binary) & (o is unary implies o is not binary); registration let C be ConstructorSignature; cluster non_op C -> unary; coherence proof the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; hence len the_arity_of non_op C = 1 by FINSEQ_1:39; end; cluster ast C -> binary; coherence proof the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38; hence len the_arity_of ast C = 2 by FINSEQ_1:44; end; end; registration let C be ConstructorSignature; cluster nullary -> constructor for OperSymbol of C; coherence proof let o be OperSymbol of C such that A1: the_arity_of o = {}; the_arity_of ast C = <*an_Adj C, a_Type C*> & the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; hence o <> * & o <> non_op by A1; end; end; theorem Th38: for C being ConstructorSignature holds C is initialized iff ex m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C st m is nullary & a is nullary proof let C be ConstructorSignature; hereby assume C is initialized; then consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type & the_arity_of m = {} & the_result_sort_of a = an_Adj & the_arity_of a = {}; reconsider m as OperSymbol of a_Type C by A1,ABCMIZ_1:def 32; reconsider a as OperSymbol of an_Adj C by A1,ABCMIZ_1:def 32; take m, a; thus m is nullary by A1; thus a is nullary by A1; end; given m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C such that A2: m is nullary & a is nullary; take m,a; the_result_sort_of non_op C = an_Adj C & the_result_sort_of ast C = a_Type C by ABCMIZ_1:38; hence thesis by A2,ABCMIZ_1:def 32; end; registration let C be initialized ConstructorSignature; cluster nullary constructor for OperSymbol of a_Type C; existence proof consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C such that A1: m is nullary & a is nullary by Th38; take m; thus thesis by A1; end; cluster nullary constructor for OperSymbol of an_Adj C; existence proof consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C such that A2: m is nullary & a is nullary by Th38; take a; thus thesis by A2; end; end; registration let C be initialized ConstructorSignature; cluster nullary constructor for OperSymbol of C; existence proof set o = the nullary constructor OperSymbol of a_Type C; take o; thus thesis; end; end; registration cluster arity-rich -> with_an_operation_for_each_sort for non void Signature; coherence proof let S be non void Signature such that A1: for n being Nat, s being SortSymbol of S holds {o where o is OperSymbol of S: the_result_sort_of o = s & len the_arity_of o = n} is infinite; let v be object; assume v in the carrier of S; then reconsider v as SortSymbol of S; set A = {o where o is OperSymbol of S: the_result_sort_of o = v & len the_arity_of o = 0}; reconsider A as infinite set by A1; set a = the Element of A; a in A; then consider o being OperSymbol of S such that A2: a = o & the_result_sort_of o = v & len the_arity_of o = 0; thus thesis by A2,FUNCT_2:4; end; cluster arity-rich -> initialized for ConstructorSignature; coherence proof let C be ConstructorSignature such that A3: C is arity-rich; set Xt = {o where o is OperSymbol of C: the_result_sort_of o = a_Type C & len the_arity_of o = 0}; set x = the Element of Xt; Xt is infinite set by A3; then x in Xt; then consider m being OperSymbol of C such that A4: x = m & the_result_sort_of m = a_Type C & len the_arity_of m = 0; set Xa = {o where o is OperSymbol of C: the_result_sort_of o = an_Adj C & len the_arity_of o = 0}; set x = the Element of Xa; Xa is infinite set by A3; then x in Xa; then consider a being OperSymbol of C such that A5: x = a & the_result_sort_of a = an_Adj C & len the_arity_of a = 0; take m, a; thus thesis by A4,A5; end; end; registration cluster MaxConstrSign -> arity-rich; coherence proof set C = MaxConstrSign; let n be Nat, s be SortSymbol of C; A1: the carrier of C = {0,1,2} by ABCMIZ_1:def 9; set X = {o where o is OperSymbol of C: the_result_sort_of o = s & len the_arity_of o = n}; consider l being quasi-loci such that A2: len l = n by Th30; deffunc F(object) = [s,[l,$1]]; consider f being Function such that A3: dom f = NAT & for i being object st i in NAT holds f.i = F(i) from FUNCT_1:sch 3; f is one-to-one proof let i,j be object; assume i in dom f & j in dom f; then reconsider i,j as Element of NAT by A3; f.i = F(i) & f.j = F(j) by A3; then f.i = f.j implies [l,i] = [l,j] by XTUPLE_0:1; hence thesis by XTUPLE_0:1; end; then A4: rng f is infinite by A3,CARD_1:59; rng f c= X proof let y be object; assume y in rng f; then consider x being object such that A5: x in dom f & y = f.x by FUNCT_1:def 3; reconsider x as Element of NAT by A3,A5; A6: [l,x] in [:QuasiLoci, NAT:] & y = F(x) by A3,A5; then y in Constructors by A1,Th28,ZFMISC_1:def 2; then y in {*,non_op}\/Constructors by XBOOLE_0:def 3; then reconsider y as OperSymbol of C by ABCMIZ_1:def 24; A7: y is constructor by A6; then A8: the_result_sort_of y = y`1 by ABCMIZ_1:def 24 .= s by A6; len the_arity_of y = card y`2`1 by A7,ABCMIZ_1:def 24 .= card [l,x]`1 by A6 .= len l; hence thesis by A2,A8; end; hence X is infinite by A4; end; end; registration cluster arity-rich initialized for ConstructorSignature; existence proof take MaxConstrSign; thus thesis; end; end; registration let C be arity-rich ConstructorSignature; let s be SortSymbol of C; cluster nullary constructor for OperSymbol of s; existence proof set X = {o where o is OperSymbol of C: the_result_sort_of o = s & len the_arity_of o = 0}; X is infinite by Def12; then consider m1,m2 being object such that A1: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10; consider o1 being OperSymbol of C such that A2: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 0 by A1; reconsider m1 as OperSymbol of s by A2,ABCMIZ_1:def 32; the_arity_of m1 = {} by A2; then m1 is nullary; hence thesis; end; cluster unary constructor for OperSymbol of s; existence proof set X = {o where o is OperSymbol of C: the_result_sort_of o = s & len the_arity_of o = 1}; X is infinite by Def12; then consider m1,m2 being object such that A3: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10; consider o1 being OperSymbol of C such that A4: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 1 by A3; consider o2 being OperSymbol of C such that A5: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 1 by A3; reconsider m1,m2 as OperSymbol of s by A4,A5,ABCMIZ_1:def 32; A6: m1 is unary & m2 is unary by A4,A5; then A7: m1 <> ast C & m2 <> ast C by Th37; m1 <> non_op or m2 <> non_op by A3; then m1 is constructor or m2 is constructor by A7; hence thesis by A6; end; cluster binary constructor for OperSymbol of s; existence proof set X = {o where o is OperSymbol of C: the_result_sort_of o = s & len the_arity_of o = 2}; X is infinite by Def12; then consider m1,m2 being object such that A8: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10; consider o1 being OperSymbol of C such that A9: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 2 by A8; consider o2 being OperSymbol of C such that A10: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 2 by A8; reconsider m1,m2 as OperSymbol of s by A9,A10,ABCMIZ_1:def 32; A11: m1 is binary & m2 is binary by A9,A10; then A12: m1 <> non_op C & m2 <> non_op C by Th37; m1 <> * or m2 <> * by A8; then m1 is constructor or m2 is constructor by A12; hence thesis by A11; end; end; registration let C be arity-rich ConstructorSignature; cluster unary constructor for OperSymbol of C; existence proof set o = the unary constructor OperSymbol of a_Type C; take o; thus thesis; end; cluster binary constructor for OperSymbol of C; existence proof set o = the binary constructor OperSymbol of a_Type C; take o; thus thesis; end; end; theorem Th39: for o being nullary OperSymbol of C holds [o, the carrier of C]-tree {} is expression of C, the_result_sort_of o proof let o be nullary OperSymbol of C; set X = MSVars C; set Z = (the carrier of C)-->{0}; set Y = X (\/) Z; A1: the_arity_of o = {} by Def13; A2: the Sorts of Free(C, X) = C-Terms(X, Y) by MSAFREE3:24; for i being Nat st i in dom {} ex t being Term of C,Y st t = {}.i & the_sort_of t = (the_arity_of o).i; then reconsider p = {} as ArgumentSeq of Sym(o, Y) by A1,MSATERM:24; A3: variables_in (Sym(o, Y)-tree p) c= X proof let s be object; assume s in the carrier of C; then reconsider s9 = s as SortSymbol of C; let x be object; assume x in (variables_in (Sym(o, Y)-tree p)).s; then ex t being DecoratedTree st t in rng p & x in (C variables_in t).s9 by MSAFREE3:11; hence thesis; end; set s9 = the_result_sort_of o; A4: the_sort_of (Sym(o, Y)-tree p) = the_result_sort_of o by MSATERM:20; (the Sorts of Free(C, X)).s9 = {t where t is Term of C,Y: the_sort_of t = s9 & variables_in t c= X} by A2,MSAFREE3:def 5; then [o, the carrier of C]-tree {} in (the Sorts of Free(C, X)).s9 by A3,A4; hence thesis by ABCMIZ_1:41; end; definition let C be initialized ConstructorSignature; let m be nullary constructor OperSymbol of a_Type C; redefine func m term -> pure expression of C, a_Type C; coherence proof set T = m term; the_arity_of m = 0 by Def13; then len the_arity_of m = 0; then A1: T = [m, the carrier of C]-tree {} by ABCMIZ_1:def 29; ex m, a being OperSymbol of C st the_result_sort_of m = a_Type & the_arity_of m = {} & the_result_sort_of a = an_Adj & the_arity_of a = {} by ABCMIZ_1:def 12; then the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then reconsider T as expression of C, a_Type C by A1,Th39; T is pure proof given a being expression of C, an_Adj C, t being expression of C, a_Type C such that A2: T = (ast C)term(a,t); T = [ *, the carrier of C]-tree <*a,t*> by A2,ABCMIZ_1:46; hence thesis by A1,TREES_4:15; end; hence thesis; end; end; definition let c be Element of Constructors; func @c -> constructor OperSymbol of MaxConstrSign equals c; coherence proof * in {*, non_op} & non_op in {*, non_op} & the carrier' of MC = {*, non_op} \/ Constructors by ABCMIZ_1:def 24,TARSKI:def 2; then c <> * & c <> non_op & c in the carrier' of MC by ABCMIZ_1:39,XBOOLE_0:3,def 3; hence thesis by ABCMIZ_1:def 11; end; end; definition let m be Element of Modes; redefine func @m -> constructor OperSymbol of a_Type MaxConstrSign; coherence proof A1: m`1 in {a_Type} by MCART_1:10; the_result_sort_of @m = m`1 by ABCMIZ_1:def 24 .= a_Type by A1,TARSKI:def 1; hence thesis by ABCMIZ_1:def 32; end; end; registration cluster @set-constr -> nullary; coherence proof len the_arity_of @set-constr = card set-constr`2`1 by ABCMIZ_1:def 24 .= card [{},0]`1 .= card 0; hence the_arity_of @set-constr = {}; end; end; theorem the_arity_of @set-constr = {} by Def13; definition func set-type -> quasi-type equals ({}QuasiAdjs MaxConstrSign) ast ((@set-constr) term); coherence; end; theorem adjs set-type = {} & the_base_of set-type = (@set-constr) term; definition let l be FinSequence of Vars; func args l -> FinSequence of QuasiTerms MaxConstrSign means: Def18: len it = len l & for i st i in dom l holds it.i = (l/.i)-term MaxConstrSign; existence proof deffunc F(Nat) = (l/.$1)-term MaxConstrSign; consider g being FinSequence such that A1: len g = len l and A2: for i st i in dom g holds g.i = F(i) from FINSEQ_1:sch 2; A3: dom g = dom l by A1,FINSEQ_3:29; rng g c= QuasiTerms MaxConstrSign proof let j be object; assume j in rng g; then consider z being object such that A4: z in dom g & j = g.z by FUNCT_1:def 3; reconsider z as Nat by A4; j = F(z) by A2,A4; hence thesis by ABCMIZ_1:49; end; then reconsider g as FinSequence of QuasiTerms MaxConstrSign by FINSEQ_1:def 4; take g; thus thesis by A1,A2,A3; end; uniqueness proof let a1,a2 be FinSequence of QuasiTerms MaxConstrSign such that A5: len a1 = len l and A6: for i st i in dom l holds a1.i = (l/.i)-term MaxConstrSign and A7: len a2 = len l and A8: for i st i in dom l holds a2.i = (l/.i)-term MaxConstrSign; A9: dom a1 = dom l & dom a2 = dom l by A5,A7,FINSEQ_3:29; now let i; assume i in dom a1; then a1.i = (l/.i)-term MaxConstrSign & a2.i = (l/.i)-term MaxConstrSign by A6,A8,A9; hence a1.i = a2.i; end; hence thesis by A9; end; end; definition let c; func base_exp_of c -> expression equals (@c)-trm args loci_of c; coherence; end; theorem Th42: for o being OperSymbol of MaxConstrSign holds o is constructor iff o in Constructors proof let o be OperSymbol of MaxConstrSign; A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors by ABCMIZ_1:def 24; o is constructor iff o nin {*,non_op} by TARSKI:def 2; hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3,def 3; end; theorem for m being nullary OperSymbol of MaxConstrSign holds main-constr (m term) = m proof set C = MaxConstrSign; let m be nullary OperSymbol of C; the_arity_of m = 0 by Def13; then len the_arity_of m = 0 & len {} = 0; then A1: m term = [m, the carrier of C]-tree {} & m-trm(<*>QuasiTerms C) = [m, the carrier of C]-tree {} by ABCMIZ_1:def 29,def 35; hence main-constr (m term) = ((m term).{})`1 by Def9 .= [m, the carrier of C]`1 by A1,TREES_4:def 4 .= m; end; theorem for m being unary constructor OperSymbol of MaxConstrSign for t holds main-constr (m term t) = m proof set C = MaxConstrSign; let m be unary constructor OperSymbol of C; let t; reconsider w = t as Element of QuasiTerms C by ABCMIZ_1:49; reconsider p = <*w*> as FinSequence of QuasiTerms C; A1: len the_arity_of m = 1 by Def14; then the_arity_of m = 1 |-> a_Term by ABCMIZ_1:37 .= <*a_Term*> by FINSEQ_2:59; then len p = 1 & (the_arity_of m).1 = a_Term C by FINSEQ_1:40; then A2: m term t = [m, the carrier of C]-tree <*t*> & m-trm p = [m, the carrier of C]-tree <*t*> by A1,ABCMIZ_1:def 30,def 35; hence main-constr (m term t) = ((m term t).{})`1 by Def9 .= [m, the carrier of C]`1 by A2,TREES_4:def 4 .= m; end; theorem for a holds main-constr ((non_op MaxConstrSign)term a) = non_op proof set C = MaxConstrSign; set m = non_op C; let a; A1: len the_arity_of m = 1 by Def14; the_arity_of m = <*an_Adj*> by ABCMIZ_1:38; then (the_arity_of m).1 = an_Adj C by FINSEQ_1:40; then A2: m term a = [m, the carrier of C]-tree <*a*> by A1,ABCMIZ_1:def 30; thus main-constr (m term a) = ((m term a).{})`1 by Def9 .= [m, the carrier of C]`1 by A2,TREES_4:def 4 .= non_op; end; theorem for m being binary constructor OperSymbol of MaxConstrSign for t1,t2 holds main-constr (m term(t1,t2)) = m proof set C = MaxConstrSign; let m be binary constructor OperSymbol of C; let t1,t2; reconsider w1 = t1, w2 = t2 as Element of QuasiTerms C by ABCMIZ_1:49; reconsider p = <*w1,w2*> as FinSequence of QuasiTerms C; A1: len the_arity_of m = 2 by Def15; then the_arity_of m = 2 |-> a_Term by ABCMIZ_1:37 .= <*a_Term,a_Term*> by FINSEQ_2:61; then (the_arity_of m).1 = a_Term C & (the_arity_of m).2 = a_Term C & len p = 2 by FINSEQ_1:44; then A2: m term(t1,t2) = [m, the carrier of C]-tree <*t1,t2*> & m-trm p = [m, the carrier of C]-tree p by A1,ABCMIZ_1:def 31,def 35; hence main-constr (m term(t1,t2)) = ((m term(t1,t2)).{})`1 by Def9 .= [m, the carrier of C]`1 by A2,TREES_4:def 4 .= m; end; theorem for q being expression of MaxConstrSign, a_Type MaxConstrSign for a holds main-constr ((ast MaxConstrSign)term(a,q)) = * proof set C = MaxConstrSign; set m = ast C; let q be expression of MaxConstrSign, a_Type MaxConstrSign; let a; A1: len the_arity_of m = 2 by Def15; the_arity_of m = <*an_Adj C,a_Type C*> by ABCMIZ_1:38; then (the_arity_of m).1 = an_Adj C & (the_arity_of m).2 = a_Type C by FINSEQ_1:44; then A2: m term(a,q) = [m, the carrier of C]-tree <*a,q*> by A1,ABCMIZ_1:def 31; thus main-constr (m term(a,q)) = ((m term(a,q)).{})`1 by Def9 .= [m, the carrier of C]`1 by A2,TREES_4:def 4 .= *; end; definition let T be quasi-type; func constrs T -> set equals (constrs the_base_of T) \/ union {constrs a: a in adjs T}; coherence; end; theorem for q being pure expression of MaxConstrSign, a_Type MaxConstrSign for A being finite Subset of QuasiAdjs MaxConstrSign holds constrs(A ast q) = (constrs q) \/ union {constrs a: a in A}; theorem constrs(a ast T) = (constrs a) \/ (constrs T) proof set A = {constrs a9: a9 in {a} \/ adjs T}; set B = {constrs a9: a9 in adjs T}; A1: A = B \/{constrs a} proof thus A c= B \/{constrs a} proof let z be object; assume z in A; then consider a9 such that A2: z = constrs a9 & a9 in {a} \/ adjs T; a9 in {a} or a9 in adjs T by A2,XBOOLE_0:def 3; then a9 = a or a9 in adjs T by TARSKI:def 1; then z in {constrs a} or z in B by A2,TARSKI:def 1; hence thesis by XBOOLE_0:def 3; end; let z be object; assume A3: z in B\/{constrs a}; A4: a in {a} by TARSKI:def 1; per cases by A3,XBOOLE_0:def 3; suppose z in B; then consider a9 such that A5: z = constrs a9 & a9 in adjs T; a9 in {a} \/ adjs T by A5,XBOOLE_0:def 3; hence thesis by A5; end; suppose z in {constrs a}; then z = constrs a & a in {a} \/ adjs T by A4,TARSKI:def 1,XBOOLE_0:def 3; hence thesis; end; end; thus constrs(a ast T) = (constrs the_base_of (a ast T)) \/ union A .= (constrs the_base_of T) \/ union A .= (constrs the_base_of T) \/ ((union {constrs a}) \/ union B) by A1,ZFMISC_1:78 .= (constrs the_base_of T) \/ ((constrs a) \/ union B) by ZFMISC_1:25 .= (constrs the_base_of T) \/ (constrs a) \/ union B by XBOOLE_1:4 .= (constrs a) \/ ((constrs the_base_of T) \/ union B) by XBOOLE_1:4 .= (constrs a) \/ (constrs T); end; begin :: Unification definition let C be initialized ConstructorSignature; let t,p be expression of C; pred t matches_with p means ex f being valuation of C st t = p at f; reflexivity proof let t be expression of C; take f = the empty valuation of C; thus t at f = t by ABCMIZ_1:139; end; end; theorem for t1,t2,t3 being expression of C st t1 matches_with t2 & t2 matches_with t3 holds t1 matches_with t3 proof let t1,t2,t3 be expression of C; given f1 being valuation of C such that A1: t1 = t2 at f1; given f2 being valuation of C such that A2: t2 = t3 at f2; take f2 at f1; thus thesis by A1,A2,ABCMIZ_1:149; end; definition let C be initialized ConstructorSignature; let A,B be Subset of QuasiAdjs C; pred A matches_with B means ex f being valuation of C st B at f c= A; reflexivity proof let t be Subset of QuasiAdjs C; take f = the empty valuation of C; let x be object; assume x in t at f; then ex a being quasi-adjective of C st x = a at f & a in t; hence x in t by ABCMIZ_1:139; end; end; theorem for A1,A2,A3 being Subset of QuasiAdjs C st A1 matches_with A2 & A2 matches_with A3 holds A1 matches_with A3 proof let t1,t2,t3 be Subset of QuasiAdjs C; given f1 being valuation of C such that A1: t2 at f1 c= t1; given f2 being valuation of C such that A2: t3 at f2 c= t2; take f2 at f1; (t3 at f2) at f1 c= t2 at f1 by A2,ABCMIZ_1:146; then (t3 at f2) at f1 c= t1 by A1; hence thesis by ABCMIZ_1:150; end; definition let C be initialized ConstructorSignature; let T,P be quasi-type of C; pred T matches_with P means ex f being valuation of C st (adjs P) at f c= adjs T & (the_base_of P) at f = the_base_of T; reflexivity proof let t be quasi-type of C; take f = the empty valuation of C; thus (adjs t) at f c= adjs t proof let x be object; assume x in (adjs t) at f; then ex a being quasi-adjective of C st x = a at f & a in adjs t; hence x in adjs t by ABCMIZ_1:139; end; thus thesis by ABCMIZ_1:139; end; end; theorem for T1,T2,T3 being quasi-type of C st T1 matches_with T2 & T2 matches_with T3 holds T1 matches_with T3 proof let t1,t2,t3 be quasi-type of C; given f1 being valuation of C such that A1: (adjs t2) at f1 c= adjs t1 & (the_base_of t2) at f1 = the_base_of t1; given f2 being valuation of C such that A2: (adjs t3) at f2 c= adjs t2 & (the_base_of t3) at f2 = the_base_of t2; take f2 at f1; ((adjs t3) at f2) at f1 c= (adjs t2) at f1 by A2,ABCMIZ_1:146; then ((adjs t3) at f2) at f1 c= adjs t1 by A1; hence thesis by A1,A2,ABCMIZ_1:149,150; end; definition let C be initialized ConstructorSignature; let t1,t2 be expression of C; let f be valuation of C; ::$N Unification of Mizar terms pred f unifies t1,t2 means t1 at f = t2 at f; end; theorem for t1,t2 being expression of C for f being valuation of C st f unifies t1,t2 holds f unifies t2,t1; definition let C be initialized ConstructorSignature; let t1,t2 be expression of C; pred t1,t2 are_unifiable means ex f being valuation of C st f unifies t1,t2; reflexivity proof let t be expression of C; set f = the valuation of C; take f; thus t at f = t at f; end; symmetry proof let t1,t2 be expression of C; given f being valuation of C such that A1: f unifies t1,t2; take f; thus t2 at f = t1 at f by A1; end; end; definition let C be initialized ConstructorSignature; let t1,t2 be expression of C; pred t1,t2 are_weakly-unifiable means ex g being irrelevant one-to-one valuation of C st variables_in t2 c= dom g & t1,t2 at g are_unifiable; reflexivity proof let t be expression of C; take C idval variables_in t; thus thesis by ABCMIZ_1:131,137; end; :: symmetry; end; :: theorem :: for t1,t2 being expression of C st t1 matches_with t2 :: holds t1,t2 are_weakly-unifiable; theorem for t1,t2 being expression of C st t1,t2 are_unifiable holds t1,t2 are_weakly-unifiable proof let t1,t2 be expression of C; given f being valuation of C such that A1: f unifies t1,t2; take g = C idval variables_in t2; thus variables_in t2 c= dom g by ABCMIZ_1:131; take f; thus f unifies t1,t2 at g by A1,ABCMIZ_1:137; end; definition let C be initialized ConstructorSignature; let t,t1,t2 be expression of C; pred t is_a_unification_of t1,t2 means ex f being valuation of C st f unifies t1,t2 & t = t1 at f; end; theorem for t1,t2,t being expression of C st t is_a_unification_of t1,t2 holds t is_a_unification_of t2,t1 proof let t1,t2,t be expression of C; given f being valuation of C such that A1: f unifies t1,t2 & t = t1 at f; take f; thus f unifies t2,t1 & t = t2 at f by A1; end; theorem for t1,t2,t being expression of C st t is_a_unification_of t1,t2 holds t matches_with t1 & t matches_with t2 proof let t1,t2,t be expression of C; given f being valuation of C such that A1: f unifies t1,t2 & t = t1 at f; thus ex f being valuation of C st t = t1 at f by A1; take f; thus t = t2 at f by A1; end; definition let C be initialized ConstructorSignature; let t,t1,t2 be expression of C; pred t is_a_general-unification_of t1,t2 means t is_a_unification_of t1,t2 & for u being expression of C st u is_a_unification_of t1,t2 holds u matches_with t; end; :: theorem :: for t1,t2 being expression of C st t1,t2 are_unifiable :: ex t being expression of C st t is_a_general-unification_of t1,t2; begin :: Type distribution theorem Th57: for n being Nat for s being SortSymbol of MaxConstrSign ex m being constructor OperSymbol of s st len the_arity_of m = n proof set C = MaxConstrSign; let n be Nat; let s be SortSymbol of C; deffunc F(Nat) = [{},$1]; consider l being FinSequence such that A1: len l = n and A2: for i st i in dom l holds l.i = F(i) from FINSEQ_1:sch 2; A3: l is one-to-one proof let i,j be object such that A4: i in dom l & j in dom l & l.i = l.j; reconsider i,j as Nat by A4; l.i = F(i) & l.i = F(j) by A2,A4; then i = F(j)`2; hence thesis; end; rng l c= Vars proof let z be object; assume z in rng l; then consider a being object such that A5: a in dom l & z = l.a by FUNCT_1:def 3; reconsider a as Nat by A5; z = F(a) by A2,A5; hence thesis by ABCMIZ_1:25; end; then reconsider l as one-to-one FinSequence of Vars by A3,FINSEQ_1:def 4; 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 proof let i,x; assume i in dom l & x = l.i; then x = F(i) by A2; hence thesis; end; then reconsider l as quasi-loci by ABCMIZ_1:30; set m = [s,[l,0]]; the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then A6: m in Constructors by Th28; then m in {*,non_op}\/Constructors by XBOOLE_0:def 3; then reconsider m as constructor OperSymbol of C by A6,Th42,ABCMIZ_1:def 24; the_result_sort_of m = m`1 by ABCMIZ_1:def 24 .= s; then reconsider m as constructor OperSymbol of s by ABCMIZ_1:def 32; take m; thus len the_arity_of m = card m`2`1 by ABCMIZ_1:def 24 .= card [l,0]`1 .= card l .= n by A1; end; theorem Th58: for l for s being SortSymbol of MaxConstrSign for m being constructor OperSymbol of s st len the_arity_of m = len l holds variables_in (m-trm args l) = rng l proof let l; set X = rng l; set n = len l; set C = MaxConstrSign; let s be SortSymbol of C; let m be constructor OperSymbol of s such that A1: len the_arity_of m = n; set p = args l; set Y = {variables_in t where t is quasi-term of C: t in rng p}; A2: len p = len the_arity_of m by A1,Def18; then A3: variables_in (m-trm p) = union Y by ABCMIZ_1:90; A4: dom p = dom l by A1,A2,FINSEQ_3:29; thus variables_in (m-trm p) c= X proof let s be object; assume s in variables_in (m-trm p); then consider A being set such that A5: s in A & A in Y by A3,TARSKI:def 4; consider t being quasi-term of C such that A6: A = variables_in t & t in rng p by A5; consider z being object such that A7: z in dom p & t = p.z by A6,FUNCT_1:def 3; reconsider z as Element of NAT by A7; l.z = l/.z by A4,A7,PARTFUN1:def 6; then A8: l/.z in X by A4,A7,FUNCT_1:def 3; p.z = (l/.z)-term C by A4,A7,Def18; then A = {l/.z} by A6,A7,ABCMIZ_1:86; hence thesis by A5,A8,TARSKI:def 1; end; let s be object; assume s in X; then consider z being object such that A9: z in dom l & s = l.z by FUNCT_1:def 3; reconsider z as Element of NAT by A9; set t = (l/.z)-term C; p.z = t & l.z = l/.z by A9,Def18,PARTFUN1:def 6; then variables_in t = {s} & t in rng p by A4,A9,ABCMIZ_1:86,FUNCT_1:def 3; then s in {s} & {s} in Y by TARSKI:def 1; hence thesis by A3,TARSKI:def 4; end; theorem Th59: for X being finite Subset of Vars st varcl X = X for s being SortSymbol of MaxConstrSign ex m being constructor OperSymbol of s st ::a_Type MaxConstrSign ex p being FinSequence of QuasiTerms MaxConstrSign st len p = len the_arity_of m & vars (m-trm p) = X proof let X be finite Subset of Vars; assume A1: varcl X = X; then consider l such that A2: rng l = X by Th31; set n = len l; set C = MaxConstrSign; let s be SortSymbol of C; consider m being constructor OperSymbol of s such that A3: len the_arity_of m = n by Th57; take m; set p = args l; take p; thus len p = len the_arity_of m by A3,Def18; thus thesis by A1,A2,A3,Th58; end; definition let d be PartFunc of Vars, QuasiTypes; attr d is even means for x,T st x in dom d & T = d.x holds vars T = vars x; end; definition let l be quasi-loci; mode type-distribution of l -> PartFunc of Vars, QuasiTypes means: Def30: dom it = rng l & it is even; existence proof defpred P[object,object] means ex x,T st $1 = x & $2 = T & vars T = vars x; A1: for z being object st z in rng l ex y being object st P[z,y] proof set C = MaxConstrSign; let z be object; assume z in rng l; then reconsider x = z as variable; varcl vars x = vars x by Th2; then consider m being constructor OperSymbol of a_Type C, p being FinSequence of QuasiTerms C such that A2: len p = len the_arity_of m & vars (m-trm p) = vars x by Th59; a_Type C in the carrier of C & the carrier of C c= rng the ResultSort of C by ABCMIZ_1:def 54; then consider o being object such that A3: o in dom the ResultSort of C & a_Type C = (the ResultSort of C).o by FUNCT_1:def 3; reconsider o as OperSymbol of C by A3; the_result_sort_of o = a_Type C by A3; then the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then reconsider q = m-trm p as pure expression of C, a_Type C by A2,ABCMIZ_1:75; set B = {} QuasiAdjs C; reconsider T = B ast q as quasi-type; take T, x, T; thus thesis by A2,ABCMIZ_1:106; end; consider f being Function such that A4: dom f = rng l and A5: for z being object st z in rng l holds P[z,f.z] from CLASSES1:sch 1(A1); rng f c= QuasiTypes proof let y be object; assume y in rng f; then consider z being object such that A6: z in dom f & y = f.z by FUNCT_1:def 3; P[z,y] by A4,A5,A6; hence thesis by ABCMIZ_1:def 43; end; then reconsider f as PartFunc of Vars, QuasiTypes by A4,RELSET_1:4; take f; thus dom f = rng l by A4; let x,T; assume x in dom f & T = f.x; then P[x,T] by A4,A5; hence thesis; end; end; theorem for l being empty quasi-loci holds {} is type-distribution of l proof let l be empty quasi-loci; reconsider d = {} as PartFunc of Vars, QuasiTypes by RELSET_1:12; dom d = rng l & d is even; hence thesis by Def30; end;