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(* Title : Real.thy
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Author : Jacques D. Fleuriot
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Copyright : 1998 University of Cambridge
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Description : The reals
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*)
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Real = PReal +
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constdefs
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realrel :: "((preal * preal) * (preal * preal)) set"
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"realrel == {p. ? x1 y1 x2 y2. p=((x1::preal,y1),(x2,y2)) & x1+y2 = x2+y1}"
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typedef real = "{x::(preal*preal).True}/realrel" (Equiv.quotient_def)
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instance
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real :: {ord,plus,times}
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consts
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"0r" :: real ("0r")
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"1r" :: real ("1r")
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defs
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real_zero_def "0r == Abs_real(realrel^^{(@#($#1p),@#($#1p))})"
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real_one_def "1r == Abs_real(realrel^^{(@#($#1p) + @#($#1p),@#($#1p))})"
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constdefs
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real_preal :: preal => real ("%#_" [80] 80)
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"%# m == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
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real_minus :: real => real ("%~ _" [80] 80)
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"%~ R == Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
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rinv :: real => real
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"rinv(R) == (@S. R ~= 0r & S*R = 1r)"
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real_nat :: nat => real ("%%# _" [80] 80)
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"%%# n == %#(@#($#(*# n)))"
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defs
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real_add_def
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"P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
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split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
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real_mult_def
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"P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
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split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
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real_less_def
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"P < (Q::real) == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 &
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(x1,y1::preal):Rep_real(P) &
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(x2,y2):Rep_real(Q)"
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real_le_def
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"P <= (Q::real) == ~(Q < P)"
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end
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