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environ
environ
40 行: 60 行:
       XXREAL_2, CARD_1, FUNCT_2, RELSET_1, ZFMISC_1, FINSEQ_2, PRE_POLY,
       XXREAL_2, CARD_1, FUNCT_2, RELSET_1, ZFMISC_1, FINSEQ_2, PRE_POLY,
       XREAL_0, RVSUM_1;
       XREAL_0, RVSUM_1;
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requirements NUMERALS, SUBSET, ARITHM, REAL, BOOLE;
 
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definitions TARSKI, XBOOLE_0, INT_2, NAT_D, FINSEQ_1, VALUED_0,
 
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    PRE_POLY,FINSET_1,CARD_1;
 
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theorems ORDINAL1, NEWTON, NAT_1, XCMPLX_1, INT_1, CARD_4, XREAL_0, RVSUM_1,
 
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      INT_2, PEPIN, FUNCT_1, CARD_2, PREPOWER, FINSEQ_1, TARSKI, XBOOLE_1,
 
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      FUNCOP_1, WSIERP_1, XBOOLE_0, FINSEQ_2, FINSEQ_3, FINSEQ_4, RELAT_1,
 
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      FINSOP_1, FUNCT_2, XREAL_1, XXREAL_0, NAT_D, VALUED_0, XXREAL_2,
 
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      FINSET_1,PARTFUN1, PRE_POLY, CARD_1;
 
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schemes NAT_1, PRE_CIRC, FINSEQ_1, FINSEQ_2, PBOOLE, CLASSES1;
 
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begin
 
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now
 
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let
 
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  Humankind be finite set,
 
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  Tokyoite be Subset of  Humankind,
 
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  Numberofhair be  Function of Tokyoite,NAT ;
 
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assume  LM1:
 
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  card (Tokyoite) = 12*10|^6;
 
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assume  LM2:
 
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  for x be object
 
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    st x in Tokyoite
 
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  holds Numberofhair.x <= 10|^6;
 
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LM0:
 
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  10|^6 + 1 < 12*10|^6
 
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proof
 
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0 < 10|^6 by PREPOWER:6;
 
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then
 
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P2: 1*10|^6 < 11* 10|^6 by XREAL_1:68;
 
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P3: 1 <  10 & 2 <= 6;
 
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then
 
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10 < 10 |^6 by PREPOWER:13;
 
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then
 
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1 < 10 |^6 by XXREAL_0:2,P3;
 
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then
 
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1 < 11*10|^6 by P2,XXREAL_0:2;
 
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then
 
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P4: 1*10|^6 + 1 < 1*10|^6 + 11*10|^6 by XREAL_1:8;
 
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1*10|^6 + 11*10|^6 = (1+11)*10|^6  ;
 
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hence thesis by P4;
 
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end;
 
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LM3:
 
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  card (rng Numberofhair) <= 10|^6+1
 
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proof
 
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now let y be  object ;
 
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  assume
 
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  y in  rng Numberofhair;
 
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  then
 
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  consider  x be object
 
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    such that
 
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    A1: x in Tokyoite & y=Numberofhair.x  by FUNCT_2:11;
 
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  Numberofhair.x <= 10|^6 by A1,LM2;
 
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  then
 
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  Numberofhair.x < 10|^6+1 by NAT_1:16,XXREAL_0:2;
 
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  then
 
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  Numberofhair.x  in Segm (10|^6+1) by NAT_1:44,A1;
 
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  hence
 
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  y in Segm (10|^6+1) by A1;
 
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end;
 
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then
 
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A2: rng Numberofhair
 
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  c= Segm (10|^6+1) by TARSKI:def 3;
 
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then
 
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card rng Numberofhair <= card Segm (10|^6+1) by NAT_1:43;
 
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then
 
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card rng Numberofhair <= card (10|^6+1) by ORDINAL1:def 17;
 
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hence
 
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card rng Numberofhair <= (10|^6+1)  ;
 
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end;
 
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LM4:
 
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card (rng (Numberofhair))
 
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< card  (Tokyoite)
 
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proof
 
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reconsider N1= card (rng (Numberofhair))
 
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as Element of NAT ;
 
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reconsider N2= card (Tokyoite)
 
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as Element of NAT ;
 
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A1: N1<=(10|^6+1) & N2=12*10|^6 by LM1,LM3;
 
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then 
 
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N1 < N2 by A1,XXREAL_0:2,LM0;
 
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hence thesis ;
 
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end;
 
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EX:
 
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  ex x,y be object 
 
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    st x in Tokyoite
 
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      & y in Tokyoite
 
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      & x <> y 
 
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      & Numberofhair.x = Numberofhair.y
 
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proof
 
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assume
 
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A1:
 
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  not
 
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    (  ex x,y be object
 
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    st x in Tokyoite
 
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      & y in Tokyoite
 
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      & x <> y 
 
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      & Numberofhair.x = Numberofhair.y ) ;
 
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then
 
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A2:  for x,y be object
 
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    st x in Tokyoite
 
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      & y in Tokyoite
 
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      & x <> y 
 
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    holds 
 
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      Numberofhair.x <> Numberofhair.y  ;
 
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A3: dom Numberofhair = Tokyoite by FUNCT_2:def 1;
 
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then
 
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for x,y be object st x in dom Numberofhair
 
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            & y in dom Numberofhair
 
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            & Numberofhair.x = Numberofhair.y
 
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      holds x = y by A2;
 
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then
 
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Numberofhair is one-to-one by FUNCT_1:def 4;
 
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then
 
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card  (dom Numberofhair) = card (rng Numberofhair)
 
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  by CARD_1:70;
 
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then
 
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card  (Tokyoite) = card (rng (Numberofhair))
 
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  by A3;
 
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hence contradiction by LM4;
 
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end;
 
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end;
 
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2020年12月22日 (火) 11:06 時点における最新版

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表組みの例

Food complements
オレンジ りんご
パン パイ
バター アイスクリーム

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environ

 vocabularies NUMBERS, REAL_1, FINSEQ_1, VALUED_0, XBOOLE_0, NEWTON, ARYTM_3,
      RELAT_1, NAT_1, XXREAL_0, ARYTM_1, SUBSET_1, CARD_1, CARD_3, ORDINAL4,
      TARSKI, INT_2, FUNCT_1, FINSEQ_2, PRE_POLY, PBOOLE, FINSET_1, XCMPLX_0,
      UPROOTS, FUNCT_2, BINOP_2, SETWISEO, INT_1, FUNCOP_1, NAT_3, XREAL_0;
 notations TARSKI, XBOOLE_0, SUBSET_1, FINSET_1, ORDINAL1, CARD_1, NUMBERS,
      XCMPLX_0, XXREAL_0, XREAL_0, REAL_1, NAT_D, INT_2, RELAT_1, FUNCT_1,
      FUNCT_2, FINSEQ_1, FINSEQ_2, VALUED_0, PBOOLE, RVSUM_1, NEWTON, WSIERP_1,
      TREES_4, BINOP_2, FUNCOP_1, XXREAL_2, SETWOP_2, PRE_POLY;
 constructors BINOP_1, SETWISEO, NAT_D, FINSEQOP, FINSOP_1, NEWTON, WSIERP_1,
      BINOP_2, XXREAL_2, RELSET_1, PRE_POLY, REAL_1,CARD_1;
 registrations XBOOLE_0, RELAT_1, FUNCT_1, FINSET_1, NUMBERS, XCMPLX_0,
      XXREAL_0, NAT_1, INT_1, BINOP_2, MEMBERED, NEWTON, VALUED_0, FINSEQ_1,
      XXREAL_2, CARD_1, FUNCT_2, RELSET_1, ZFMISC_1, FINSEQ_2, PRE_POLY,
      XREAL_0, RVSUM_1;
::---------------------------------------
:: Combined Circuit Structure of STC_TYPE0_Inter_Inter.

definition
  let x1,x2,x3,x5,x6,x7 be set;
  func STC0IIStr(x1,x2,x3,x5,x6,x7) ->
    unsplit gate`1=arity gate`2isBoolean
    non void strict non empty ManySortedSign
  equals
:: WALLACE1:def 1
    BitGFA0Str(x1,x2,x3) +* BitGFA0Str(x5,x6,x7);
end;

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