Quelle  THF_Arith.thy

(*  Title:      HOL/TPTP/THF_Arith.thy
  Author: Jasmin Blanchette
  Copyright 2011, 2012
 
 Experimental setup for THF arithmetic. This is not connected with the TPTP
 parser yet.
*)

theory THF_Arith
imports Complex_Main
begin

consts
  is_int :: "'a ==> bool"
  is_rat :: "'a ==> bool"

overloading rat_is_int ≡ "is_int :: rat ==> bool"
begin
  definition "rat_is_int (q::rat) ⟷ (∃n::int. q = of_int n)"
end

overloading real_is_int ≡ "is_int :: real ==> bool"
begin
  definition "real_is_int (x::real) ⟷ x ∈ ℤ"
end

overloading real_is_rat ≡ "is_rat :: real ==> bool"
begin
  definition "real_is_rat (x::real) ⟷ x ∈ ℚ"
end

consts
  to_int :: "'a ==> int"
  to_rat :: "'a ==> rat"
  to_real :: "'a ==> real"

overloading rat_to_int ≡ "to_int :: rat ==> int"
begin
  definition "rat_to_int (q::rat) = ⌊q⌋"
end

overloading real_to_int ≡ "to_int :: real ==> int"
begin
  definition "real_to_int (x::real) = ⌊x⌋"
end

overloading int_to_rat ≡ "to_rat :: int ==> rat"
begin
  definition "int_to_rat (n::int) = (of_int n::rat)"
end

overloading real_to_rat ≡ "to_rat :: real ==> rat"
begin
  definition "real_to_rat (x::real) = (inv of_rat x::rat)"
end

overloading int_to_real ≡ "to_real :: int ==> real"
begin
  definition "int_to_real (n::int) = real_of_int n"
end

overloading rat_to_real ≡ "to_real :: rat ==> real"
begin
  definition "rat_to_real (x::rat) = (of_rat x::real)"
end

declare
  rat_is_int_def [simp]
  real_is_int_def [simp]
  real_is_rat_def [simp]
  rat_to_int_def [simp]
  real_to_int_def [simp]
  int_to_rat_def [simp]
  real_to_rat_def [simp]
  int_to_real_def [simp]
  rat_to_real_def [simp]

lemma to_rat_is_int [intro, simp]: "is_int (to_rat (n::int))"
by (metis int_to_rat_def rat_is_int_def)

lemma to_real_is_int [intro, simp]: "is_int (to_real (n::int))"
by (metis Ints_of_int int_to_real_def real_is_int_def)

lemma to_real_is_rat [intro, simp]: "is_rat (to_real (q::rat))"
by (metis Rats_of_rat rat_to_real_def real_is_rat_def)

lemma inj_of_rat [intro, simp]: "inj (of_rat::rat==>real)"
by (metis injI of_rat_eq_iff)

end