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43 lines
1.3 KiB
Plaintext
43 lines
1.3 KiB
Plaintext
--# -path=.:../prelude
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concrete ArithmEng of Arithm = LogicEng ** open LogicResEng in {
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lin
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Nat = {s = nomReg "number"} ;
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zero = ss "zero" ;
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succ = fun1 "successor" ;
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EqNat = adj2 ["equal to"] ;
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LtNat = adj2 ["smaller than"] ;
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Div = adj2 ["divisible by"] ;
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Even = adj1 "even" ;
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Odd = adj1 "odd" ;
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Prime = adj1 "prime" ;
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one = ss "one" ;
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two = ss "two" ;
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sum = fun2 "sum" ;
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prod = fun2 "product" ;
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evax1 = ss ["by the first axiom of evenness , zero is even"] ;
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evax2 n c = {s =
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c.s ++ [". By the second axiom of evenness , the successor of"] ++
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n.s ++ ["is odd"]} ;
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evax3 n c = {s =
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c.s ++ [". By the third axiom of evenness , the successor of"] ++
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n.s ++ ["is even"]} ;
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eqax1 = ss ["by the first axiom of equality , zero is equal to zero"] ;
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eqax2 m n c = {s =
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c.s ++ [". By the second axiom of equality , the successor of"] ++ m.s ++
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["is equal to the successor of"] ++ n.s} ;
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IndNat C d e = {s =
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["we proceed by induction . For the basis ,"] ++ d.s ++
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[". For the induction step, consider a number"] ++ C.$0 ++
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["and assume"] ++ C.s ++ "(" ++ e.$1 ++ ")" ++ "." ++ e.s ++
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["Hence, for all numbers"] ++ C.$0 ++ "," ++ C.s} ;
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ex1 = ss ["The first theorem and its proof ."] ;
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} ;
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