author | paulson |
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child 10797 | 028d22926a41 |
permissions | -rw-r--r-- |
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(* Title : HyperNat.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Description : Explicit construction of hypernaturals using |
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ultrafilters |
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*) |
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HyperNat = Star + |
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constdefs |
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hypnatrel :: "((nat=>nat)*(nat=>nat)) set" |
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"hypnatrel == {p. EX X Y. p = ((X::nat=>nat),Y) & |
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{n::nat. X(n) = Y(n)} : FreeUltrafilterNat}" |
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typedef hypnat = "UNIV//hypnatrel" (Equiv.quotient_def) |
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instance |
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hypnat :: {ord,zero,plus,times,minus} |
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consts |
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"1hn" :: hypnat ("1hn") |
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"whn" :: hypnat ("whn") |
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constdefs |
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(* embedding the naturals in the hypernaturals *) |
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hypnat_of_nat :: nat => hypnat |
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"hypnat_of_nat m == Abs_hypnat(hypnatrel^^{%n::nat. m})" |
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(* hypernaturals as members of the hyperreals; the set is defined as *) |
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(* the nonstandard extension of set of naturals embedded in the reals *) |
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HNat :: "hypreal set" |
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"HNat == *s* {n. EX no. n = real_of_nat no}" |
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(* the set of infinite hypernatural numbers *) |
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HNatInfinite :: "hypnat set" |
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"HNatInfinite == {n. n ~: SNat}" |
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(* explicit embedding of the hypernaturals in the hyperreals *) |
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hypreal_of_hypnat :: hypnat => hypreal |
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"hypreal_of_hypnat N == Abs_hypreal(UN X: Rep_hypnat(N). |
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hyprel^^{%n::nat. real_of_nat (X n)})" |
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defs |
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(** the overloaded constant "SNat" **) |
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(* set of naturals embedded in the hyperreals*) |
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SNat_def "SNat == {n. EX N. n = hypreal_of_nat N}" |
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(* set of naturals embedded in the hypernaturals*) |
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SHNat_def "SNat == {n. EX N. n = hypnat_of_nat N}" |
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(** hypernatural arithmetic **) |
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hypnat_zero_def "0 == Abs_hypnat(hypnatrel^^{%n::nat. 0})" |
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hypnat_one_def "1hn == Abs_hypnat(hypnatrel^^{%n::nat. 1})" |
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(* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *) |
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hypnat_omega_def "whn == Abs_hypnat(hypnatrel^^{%n::nat. n})" |
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hypnat_add_def |
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"P + Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q). |
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hypnatrel^^{%n::nat. X n + Y n})" |
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hypnat_mult_def |
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"P * Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q). |
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hypnatrel^^{%n::nat. X n * Y n})" |
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hypnat_minus_def |
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"P - Q == Abs_hypnat(UN X:Rep_hypnat(P). UN Y:Rep_hypnat(Q). |
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hypnatrel^^{%n::nat. X n - Y n})" |
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hypnat_less_def |
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"P < (Q::hypnat) == EX X Y. X: Rep_hypnat(P) & |
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Y: Rep_hypnat(Q) & |
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{n::nat. X n < Y n} : FreeUltrafilterNat" |
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hypnat_le_def |
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"P <= (Q::hypnat) == ~(Q < P)" |
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end |
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