src/HOL/Library/Old_SMT/old_smt_real.ML
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(*  Title:      HOL/Library/Old_SMT/old_smt_real.ML
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    Author:     Sascha Boehme, TU Muenchen
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SMT setup for reals.
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*)
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structure Old_SMT_Real: sig end =
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struct
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(* SMT-LIB logic *)
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fun smtlib_logic ts =
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  if exists (Term.exists_type (Term.exists_subtype (equal @{typ real}))) ts
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  then SOME "AUFLIRA"
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  else NONE
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(* SMT-LIB and Z3 built-ins *)
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local
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  fun real_num _ i = SOME (string_of_int i ^ ".0")
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  fun is_linear [t] = Old_SMT_Utils.is_number t
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    | is_linear [t, u] = Old_SMT_Utils.is_number t orelse Old_SMT_Utils.is_number u
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    | is_linear _ = false
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  fun mk_times ts = Term.list_comb (@{const times (real)}, ts)
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  fun times _ _ ts = if is_linear ts then SOME ("*", 2, ts, mk_times) else NONE
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in
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val setup_builtins =
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  Old_SMT_Builtin.add_builtin_typ Old_SMTLIB_Interface.smtlibC
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    (@{typ real}, K (SOME "Real"), real_num) #>
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  fold (Old_SMT_Builtin.add_builtin_fun' Old_SMTLIB_Interface.smtlibC) [
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    (@{const less (real)}, "<"),
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    (@{const less_eq (real)}, "<="),
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    (@{const uminus (real)}, "~"),
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    (@{const plus (real)}, "+"),
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    (@{const minus (real)}, "-") ] #>
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  Old_SMT_Builtin.add_builtin_fun Old_SMTLIB_Interface.smtlibC
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    (Term.dest_Const @{const times (real)}, times) #>
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  Old_SMT_Builtin.add_builtin_fun' Old_Z3_Interface.smtlib_z3C
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    (@{const times (real)}, "*") #>
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  Old_SMT_Builtin.add_builtin_fun' Old_Z3_Interface.smtlib_z3C
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    (@{const divide (real)}, "/")
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end
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(* Z3 constructors *)
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local
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  fun z3_mk_builtin_typ (Old_Z3_Interface.Sym ("Real", _)) = SOME @{typ real}
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    | z3_mk_builtin_typ (Old_Z3_Interface.Sym ("real", _)) = SOME @{typ real}
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        (*FIXME: delete*)
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    | z3_mk_builtin_typ _ = NONE
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  fun z3_mk_builtin_num _ i T =
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    if T = @{typ real} then SOME (Numeral.mk_cnumber @{ctyp real} i)
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    else NONE
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  fun mk_nary _ cu [] = cu
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    | mk_nary ct _ cts = uncurry (fold_rev (Thm.mk_binop ct)) (split_last cts)
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  val mk_uminus = Thm.apply (Thm.cterm_of @{context} @{const uminus (real)})
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  val add = Thm.cterm_of @{context} @{const plus (real)}
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  val real0 = Numeral.mk_cnumber @{ctyp real} 0
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  val mk_sub = Thm.mk_binop (Thm.cterm_of @{context} @{const minus (real)})
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  val mk_mul = Thm.mk_binop (Thm.cterm_of @{context} @{const times (real)})
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  val mk_div = Thm.mk_binop (Thm.cterm_of @{context} @{const divide (real)})
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  val mk_lt = Thm.mk_binop (Thm.cterm_of @{context} @{const less (real)})
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  val mk_le = Thm.mk_binop (Thm.cterm_of @{context} @{const less_eq (real)})
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  fun z3_mk_builtin_fun (Old_Z3_Interface.Sym ("-", _)) [ct] = SOME (mk_uminus ct)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("+", _)) cts =
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        SOME (mk_nary add real0 cts)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("-", _)) [ct, cu] =
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        SOME (mk_sub ct cu)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("*", _)) [ct, cu] =
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        SOME (mk_mul ct cu)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("/", _)) [ct, cu] =
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        SOME (mk_div ct cu)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("<", _)) [ct, cu] =
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        SOME (mk_lt ct cu)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym ("<=", _)) [ct, cu] =
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        SOME (mk_le ct cu)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym (">", _)) [ct, cu] =
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        SOME (mk_lt cu ct)
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    | z3_mk_builtin_fun (Old_Z3_Interface.Sym (">=", _)) [ct, cu] =
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        SOME (mk_le cu ct)
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    | z3_mk_builtin_fun _ _ = NONE
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in
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val z3_mk_builtins = {
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  mk_builtin_typ = z3_mk_builtin_typ,
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  mk_builtin_num = z3_mk_builtin_num,
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  mk_builtin_fun = (fn _ => fn sym => fn cts =>
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    (case try (Thm.typ_of_cterm o hd) cts of
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      SOME @{typ real} => z3_mk_builtin_fun sym cts
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    | _ => NONE)) }
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end
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(* Z3 proof reconstruction *)
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val real_rules = @{lemma
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  "0 + (x::real) = x"
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  "x + 0 = x"
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  "0 * x = 0"
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  "1 * x = x"
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  "x + y = y + x"
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  by auto}
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val real_linarith_proc =
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  Simplifier.make_simproc @{context} "fast_real_arith"
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   {lhss = [@{term "(m::real) < n"}, @{term "(m::real) \<le> n"}, @{term "(m::real) = n"}],
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    proc = K Lin_Arith.simproc}
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(* setup *)
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val _ =
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  Theory.setup
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   (Context.theory_map (
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      Old_SMTLIB_Interface.add_logic (10, smtlib_logic) #>
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      setup_builtins #>
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      Old_Z3_Interface.add_mk_builtins z3_mk_builtins #>
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      fold Old_Z3_Proof_Reconstruction.add_z3_rule real_rules #>
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      Old_Z3_Proof_Tools.add_simproc real_linarith_proc))
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bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
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end