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(* Title : PRat.thy
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Author : Jacques D. Fleuriot
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Copyright : 1998 University of Cambridge
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Description : The positive rationals
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*)
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PRat = PNat + Equiv +
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constdefs
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ratrel :: "((pnat * pnat) * (pnat * pnat)) set"
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"ratrel == {p. ? x1 y1 x2 y2. p=((x1::pnat,y1),(x2,y2)) & x1*y2 = x2*y1}"
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typedef prat = "{x::(pnat*pnat).True}/ratrel" (Equiv.quotient_def)
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instance
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prat :: {ord,plus,times}
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constdefs
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prat_pnat :: pnat => prat ("$#_" [80] 80)
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"$# m == Abs_prat(ratrel^^{(m,Abs_pnat 1)})"
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qinv :: prat => prat
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"qinv(Q) == Abs_prat(UN p:Rep_prat(Q). split (%x y. ratrel^^{(y,x)}) p)"
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defs
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prat_add_def
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"P + Q == Abs_prat(UN p1:Rep_prat(P). UN p2:Rep_prat(Q).
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split(%x1 y1. split(%x2 y2. ratrel^^{(x1*y2 + x2*y1, y1*y2)}) p2) p1)"
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prat_mult_def
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"P * Q == Abs_prat(UN p1:Rep_prat(P). UN p2:Rep_prat(Q).
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split(%x1 y1. split(%x2 y2. ratrel^^{(x1*x2, y1*y2)}) p2) p1)"
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(*** Gleason p. 119 ***)
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prat_less_def
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"P < (Q::prat) == ? T. P + T = Q"
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prat_le_def
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"P <= (Q::prat) == ~(Q < P)"
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end
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