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(* Title: FOL/ex/nat2.thy
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1991 University of Cambridge
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Theory for examples of simplification and induction on the natural numbers
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*)
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Nat2 = FOL +
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types nat 0
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arities nat :: term
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consts succ,pred :: "nat => nat"
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"0" :: "nat" ("0")
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"+" :: "[nat,nat] => nat" (infixr 90)
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"<","<=" :: "[nat,nat] => o" (infixr 70)
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rules
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pred_0 "pred(0) = 0"
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pred_succ "pred(succ(m)) = m"
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plus_0 "0+n = n"
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plus_succ "succ(m)+n = succ(m+n)"
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nat_distinct1 "~ 0=succ(n)"
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nat_distinct2 "~ succ(m)=0"
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succ_inject "succ(m)=succ(n) <-> m=n"
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leq_0 "0 <= n"
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leq_succ_succ "succ(m)<=succ(n) <-> m<=n"
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leq_succ_0 "~ succ(m) <= 0"
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lt_0_succ "0 < succ(n)"
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lt_succ_succ "succ(m)<succ(n) <-> m<n"
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lt_0 "~ m < 0"
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nat_ind "[| P(0); ALL n. P(n)-->P(succ(n)) |] ==> All(P)"
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end
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