(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
Simprocs for the (integer) numerals.
*)
(*To quote from Provers/Arith/cancel_numeral_factor.ML:
Cancels common coefficients in balanced expressions:
u*#m ~~ u'*#m' == #n*u ~~ #n'*u'
where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
and d = gcd(m,m') and n=m/d and n'=m'/d.
*)
signature NUMERAL_SIMPROCS =
sig
val trans_tac: Proof.context -> thm option -> tactic
val assoc_fold: Simplifier.proc
val combine_numerals: Simplifier.proc
val eq_cancel_numerals: Simplifier.proc
val less_cancel_numerals: Simplifier.proc
val le_cancel_numerals: Simplifier.proc
val eq_cancel_factor: Simplifier.proc
val le_cancel_factor: Simplifier.proc
val less_cancel_factor: Simplifier.proc
val div_cancel_factor: Simplifier.proc
val mod_cancel_factor: Simplifier.proc
val dvd_cancel_factor: Simplifier.proc
val divide_cancel_factor: Simplifier.proc
val eq_cancel_numeral_factor: Simplifier.proc
val less_cancel_numeral_factor: Simplifier.proc
val le_cancel_numeral_factor: Simplifier.proc
val div_cancel_numeral_factor: Simplifier.proc
val divide_cancel_numeral_factor: Simplifier.proc
val field_combine_numerals: Simplifier.proc
val field_divide_cancel_numeral_factor: simproc
val num_ss: simpset
val field_comp_conv: Proof.context -> conv
end;
structure Numeral_Simprocs : NUMERAL_SIMPROCS =
struct
fun trans_tac _ NONE = all_tac
| trans_tac ctxt (SOME th) = ALLGOALS (resolve_tac ctxt [th RS trans]);
val mk_number = Arith_Data.mk_number;
val mk_sum = Arith_Data.mk_sum;
val long_mk_sum = Arith_Data.long_mk_sum;
val dest_sum = Arith_Data.dest_sum;
val mk_times = HOLogic.mk_binop \<^const_name>\<open>Groups.times\<close>;
fun one_of T = Const(\<^const_name>\<open>Groups.one\<close>, T);
(* build product with trailing 1 rather than Numeral 1 in order to avoid the
unnecessary restriction to type class number_ring
which is not required for cancellation of common factors in divisions.
UPDATE: this reasoning no longer applies (number_ring is gone)
*)
fun mk_prod T =
let val one = one_of T
fun mk [] = one
| mk [t] = t
| mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
in mk end;
(*This version ALWAYS includes a trailing one*)
fun long_mk_prod T [] = one_of T
| long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
val dest_times = HOLogic.dest_bin \<^const_name>\<open>Groups.times\<close> dummyT;
fun dest_prod t =
let val (t,u) = dest_times t
in dest_prod t @ dest_prod u end
handle TERM _ => [t];
fun find_first_numeral past (t::terms) =
((snd (HOLogic.dest_number t), rev past @ terms)
handle TERM _ => find_first_numeral (t::past) terms)
| find_first_numeral past [] = raise TERM("find_first_numeral", []);
(*DON'T do the obvious simplifications; that would create special cases*)
fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const (\<^const_name>\<open>Groups.uminus\<close>, _) $ t) = dest_coeff (~sign) t
| dest_coeff sign t =
let val ts = sort Term_Ord.term_ord (dest_prod t)
val (n, ts') = find_first_numeral [] ts
handle TERM _ => (1, ts)
in (sign*n, mk_prod (Term.fastype_of t) ts') end;
(*Find first coefficient-term THAT MATCHES u*)
fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
| find_first_coeff past u (t::terms) =
let val (n,u') = dest_coeff 1 t
in if u aconv u' then (n, rev past @ terms)
else find_first_coeff (t::past) u terms
end
handle TERM _ => find_first_coeff (t::past) u terms;
(*Fractions as pairs of ints. Can't use Rat.rat because the representation
needs to preserve negative values in the denominator.*)
fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
Fractions are reduced later by the cancel_numeral_factor simproc.*)
fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
val mk_divide = HOLogic.mk_binop \<^const_name>\<open>Rings.divide\<close>;
(*Build term (p / q) * t*)
fun mk_fcoeff ((p, q), t) =
let val T = Term.fastype_of t
in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
(*Express t as a product of a fraction with other sorted terms*)
fun dest_fcoeff sign (Const (\<^const_name>\<open>Groups.uminus\<close>, _) $ t) = dest_fcoeff (~sign) t
| dest_fcoeff sign (Const (\<^const_name>\<open>Rings.divide\<close>, _) $ t $ u) =
let val (p, t') = dest_coeff sign t
val (q, u') = dest_coeff 1 u
in (mk_frac (p, q), mk_divide (t', u')) end
| dest_fcoeff sign t =
let val (p, t') = dest_coeff sign t
val T = Term.fastype_of t
in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
(** New term ordering so that AC-rewriting brings numerals to the front **)
(*Order integers by absolute value and then by sign. The standard integer
ordering is not well-founded.*)
fun num_ord (i,j) =
(case int_ord (abs i, abs j) of
EQUAL => int_ord (Int.sign i, Int.sign j)
| ord => ord);
(*This resembles Term_Ord.term_ord, but it puts binary numerals before other
non-atomic terms.*)
local open Term
in
fun numterm_ord (t, u) =
case (try HOLogic.dest_number t, try HOLogic.dest_number u) of
(SOME (_, i), SOME (_, j)) => num_ord (i, j)
| (SOME _, NONE) => LESS
| (NONE, SOME _) => GREATER
| _ => (
case (t, u) of
(Abs (_, T, t), Abs(_, U, u)) =>
(prod_ord numterm_ord Term_Ord.typ_ord ((t, T), (u, U)))
| _ => (
case int_ord (size_of_term t, size_of_term u) of
EQUAL =>
let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
(prod_ord Term_Ord.hd_ord numterms_ord ((f, ts), (g, us)))
end
| ord => ord))
and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
end;
val num_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> |> Simplifier.set_term_ord numterm_ord);
(*Maps 1 to Numeral1 so that arithmetic isn't complicated by the abstract 1.*)
val numeral_syms = @{thms numeral_One [symmetric]};
(*Simplify 0+n, n+0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
val add_0s = @{thms add_0_left add_0_right};
val mult_1s = @{thms mult_1s divide_numeral_1 mult_1_left mult_1_right mult_minus1 mult_minus1_right div_by_1};
(* For post-simplification of the rhs of simproc-generated rules *)
val post_simps =
@{thms numeral_One
add_0_left add_0_right
mult_zero_left mult_zero_right
mult_1_left mult_1_right
mult_minus1 mult_minus1_right}
val field_post_simps =
post_simps @ @{thms div_0 div_by_1}
(*Simplify inverse Numeral1*)
val inverse_1s = @{thms inverse_numeral_1}
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the Numeral_Simprocs, such as 3 * (5 * x). *)
val simps =
@{thms numeral_One [symmetric]
add_numeral_left
add_neg_numeral_left
mult_numeral_left
arith_simps rel_simps}
(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
during re-arrangement*)
val non_add_simps =
subtract Thm.eq_thm
@{thms add_numeral_left
add_neg_numeral_left
numeral_plus_numeral
add_neg_numeral_simps} simps;
(*To let us treat subtraction as addition*)
val diff_simps = @{thms diff_conv_add_uminus minus_add_distrib minus_minus};
(*To let us treat division as multiplication*)
val divide_simps = @{thms divide_inverse inverse_mult_distrib inverse_inverse_eq};
(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
@{thms minus_minus mult_minus_left mult_minus_right};
(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
@{thms mult.assoc minus_mult_right minus_mult_commute numeral_times_minus_swap};
val norm_ss1 =
simpset_of (put_simpset num_ss \<^context>
addsimps numeral_syms @ add_0s @ mult_1s @
diff_simps @ @{thms minus_zero ac_simps})
val norm_ss2 =
simpset_of (put_simpset num_ss \<^context>
addsimps non_add_simps @ mult_minus_simps)
val norm_ss3 =
simpset_of (put_simpset num_ss \<^context>
addsimps minus_from_mult_simps @ @{thms ac_simps minus_mult_commute})
structure CancelNumeralsCommon =
struct
val mk_sum = mk_sum
val dest_sum = dest_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val find_first_coeff = find_first_coeff []
val trans_tac = trans_tac
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))
val numeral_simp_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps simps)
fun numeral_simp_tac ctxt =
ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
val simplify_meta_eq = Arith_Data.simplify_meta_eq post_simps
val prove_conv = Arith_Data.prove_conv
end;
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>HOL.eq\<close> dummyT
val bal_add1 = @{thm eq_add_iff1} RS trans
val bal_add2 = @{thm eq_add_iff2} RS trans
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> dummyT
val bal_add1 = @{thm less_add_iff1} RS trans
val bal_add2 = @{thm less_add_iff2} RS trans
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less_eq\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> dummyT
val bal_add1 = @{thm le_add_iff1} RS trans
val bal_add2 = @{thm le_add_iff2} RS trans
);
val eq_cancel_numerals = EqCancelNumerals.proc
val less_cancel_numerals = LessCancelNumerals.proc
val le_cancel_numerals = LeCancelNumerals.proc
structure CombineNumeralsData =
struct
type coeff = int
val iszero = (fn x => x = 0)
val add = op +
val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *)
val dest_sum = dest_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val left_distrib = @{thm combine_common_factor} RS trans
val prove_conv = Arith_Data.prove_conv_nohyps
val trans_tac = trans_tac
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))
val numeral_simp_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps simps)
fun numeral_simp_tac ctxt =
ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
val simplify_meta_eq = Arith_Data.simplify_meta_eq post_simps
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
(*Version for fields, where coefficients can be fractions*)
structure FieldCombineNumeralsData =
struct
type coeff = int * int
val iszero = (fn (p, _) => p = 0)
val add = add_frac
val mk_sum = long_mk_sum
val dest_sum = dest_sum
val mk_coeff = mk_fcoeff
val dest_coeff = dest_fcoeff 1
val left_distrib = @{thm combine_common_factor} RS trans
val prove_conv = Arith_Data.prove_conv_nohyps
val trans_tac = trans_tac
val norm_ss1a =
simpset_of (put_simpset norm_ss1 \<^context> addsimps (inverse_1s @ divide_simps))
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1a ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))
val numeral_simp_ss =
simpset_of (put_simpset HOL_basic_ss \<^context>
addsimps (simps @ @{thms add_frac_eq not_False_eq_True}))
fun numeral_simp_tac ctxt =
ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
val simplify_meta_eq = Arith_Data.simplify_meta_eq field_post_simps
end;
structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
val combine_numerals = CombineNumerals.proc
val field_combine_numerals = FieldCombineNumerals.proc
(** Constant folding for multiplication in semirings **)
(*We do not need folding for addition: combine_numerals does the same thing*)
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val assoc_ss = simpset_of (put_simpset HOL_basic_ss \<^context> addsimps @{thms ac_simps minus_mult_commute})
val eq_reflection = eq_reflection
val is_numeral = can HOLogic.dest_number
end;
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
fun assoc_fold ctxt ct = Semiring_Times_Assoc.proc ctxt (Thm.term_of ct)
structure CancelNumeralFactorCommon =
struct
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val trans_tac = trans_tac
val norm_ss1 =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps minus_from_mult_simps @ mult_1s)
val norm_ss2 =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps simps @ mult_minus_simps)
val norm_ss3 =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps @{thms ac_simps minus_mult_commute numeral_times_minus_swap})
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))
(* simp_thms are necessary because some of the cancellation rules below
(e.g. mult_less_cancel_left) introduce various logical connectives *)
val numeral_simp_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps simps @ @{thms simp_thms})
fun numeral_simp_tac ctxt =
ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
val simplify_meta_eq = Arith_Data.simplify_meta_eq
(@{thms Nat.add_0 Nat.add_0_right} @ post_simps)
val prove_conv = Arith_Data.prove_conv
end
(*Version for semiring_div*)
structure DivCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val mk_bal = HOLogic.mk_binop \<^const_name>\<open>Rings.divide\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Rings.divide\<close> dummyT
val cancel = @{thm div_mult_mult1} RS trans
val neg_exchanges = false
)
(*Version for fields*)
structure DivideCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val mk_bal = HOLogic.mk_binop \<^const_name>\<open>Rings.divide\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Rings.divide\<close> dummyT
val cancel = @{thm mult_divide_mult_cancel_left} RS trans
val neg_exchanges = false
)
structure EqCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>HOL.eq\<close> dummyT
val cancel = @{thm mult_cancel_left} RS trans
val neg_exchanges = false
)
structure LessCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> dummyT
val cancel = @{thm mult_less_cancel_left} RS trans
val neg_exchanges = true
)
structure LeCancelNumeralFactor = CancelNumeralFactorFun
(
open CancelNumeralFactorCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less_eq\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> dummyT
val cancel = @{thm mult_le_cancel_left} RS trans
val neg_exchanges = true
)
val eq_cancel_numeral_factor = EqCancelNumeralFactor.proc
val less_cancel_numeral_factor = LessCancelNumeralFactor.proc
val le_cancel_numeral_factor = LeCancelNumeralFactor.proc
val div_cancel_numeral_factor = DivCancelNumeralFactor.proc
val divide_cancel_numeral_factor = DivideCancelNumeralFactor.proc
val field_divide_cancel_numeral_factor =
\<^simproc_setup>\<open>passive field_divide_cancel_numeral_factor
("((l::'a::field) * m) / n" | "(l::'a::field) / (m * n)" |
"((numeral v)::'a::field) / (numeral w)" |
"((numeral v)::'a::field) / (- numeral w)" |
"((- numeral v)::'a::field) / (numeral w)" |
"((- numeral v)::'a::field) / (- numeral w)") =
\<open>K DivideCancelNumeralFactor.proc\<close>\<close>;
val field_eq_cancel_numeral_factor =
\<^simproc_setup>\<open>passive field_eq_cancel_numeral_factor
("(l::'a::field) * m = n" | "(l::'a::field) = m * n") =
\<open>K EqCancelNumeralFactor.proc\<close>\<close>;
val field_cancel_numeral_factors =
[field_divide_cancel_numeral_factor, field_eq_cancel_numeral_factor]
(** Declarations for ExtractCommonTerm **)
(*Find first term that matches u*)
fun find_first_t past u [] = raise TERM ("find_first_t", [])
| find_first_t past u (t::terms) =
if u aconv t then (rev past @ terms)
else find_first_t (t::past) u terms
handle TERM _ => find_first_t (t::past) u terms;
(** Final simplification for the CancelFactor simprocs **)
val simplify_one = Arith_Data.simplify_meta_eq
[@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_by_1}, @{thm numeral_One}];
fun cancel_simplify_meta_eq ctxt cancel_th th =
simplify_one ctxt (([th, cancel_th]) MRS trans);
local
val Tp_Eq = Thm.reflexive (Thm.cterm_of \<^theory_context>\<open>HOL\<close> HOLogic.Trueprop)
fun Eq_True_elim Eq =
Thm.equal_elim (Thm.combination Tp_Eq (Thm.symmetric Eq)) @{thm TrueI}
in
fun sign_conv pos_th neg_th ctxt t =
let val T = fastype_of t;
val zero = Const(\<^const_name>\<open>Groups.zero\<close>, T);
val less = Const(\<^const_name>\<open>Orderings.less\<close>, [T,T] ---> HOLogic.boolT);
val pos = less $ zero $ t and neg = less $ t $ zero
fun prove p =
SOME (Eq_True_elim (Simplifier.asm_rewrite ctxt (Thm.cterm_of ctxt p)))
handle THM _ => NONE
in case prove pos of
SOME th => SOME(th RS pos_th)
| NONE => (case prove neg of
SOME th => SOME(th RS neg_th)
| NONE => NONE)
end;
end
structure CancelFactorCommon =
struct
val mk_sum = long_mk_prod
val dest_sum = dest_prod
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
val find_first = find_first_t []
val trans_tac = trans_tac
val norm_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps mult_1s @ @{thms ac_simps minus_mult_commute})
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
val simplify_meta_eq = cancel_simplify_meta_eq
fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
end;
(*mult_cancel_left requires a ring with no zero divisors.*)
structure EqCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>HOL.eq\<close> dummyT
fun simp_conv _ _ = SOME @{thm mult_cancel_left}
);
(*for ordered rings*)
structure LeCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less_eq\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> dummyT
val simp_conv = sign_conv
@{thm mult_le_cancel_left_pos} @{thm mult_le_cancel_left_neg}
);
(*for ordered rings*)
structure LessCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> dummyT
val simp_conv = sign_conv
@{thm mult_less_cancel_left_pos} @{thm mult_less_cancel_left_neg}
);
(*for semirings with division*)
structure DivCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binop \<^const_name>\<open>Rings.divide\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Rings.divide\<close> dummyT
fun simp_conv _ _ = SOME @{thm div_mult_mult1_if}
);
structure ModCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binop \<^const_name>\<open>modulo\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>modulo\<close> dummyT
fun simp_conv _ _ = SOME @{thm mod_mult_mult1}
);
(*for idoms*)
structure DvdCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Rings.dvd\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Rings.dvd\<close> dummyT
fun simp_conv _ _ = SOME @{thm dvd_mult_cancel_left}
);
(*Version for all fields, including unordered ones (type complex).*)
structure DivideCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binop \<^const_name>\<open>Rings.divide\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Rings.divide\<close> dummyT
fun simp_conv _ _ = SOME @{thm mult_divide_mult_cancel_left_if}
);
fun eq_cancel_factor ctxt ct = EqCancelFactor.proc ctxt (Thm.term_of ct)
fun le_cancel_factor ctxt ct = LeCancelFactor.proc ctxt (Thm.term_of ct)
fun less_cancel_factor ctxt ct = LessCancelFactor.proc ctxt (Thm.term_of ct)
fun div_cancel_factor ctxt ct = DivCancelFactor.proc ctxt (Thm.term_of ct)
fun mod_cancel_factor ctxt ct = ModCancelFactor.proc ctxt (Thm.term_of ct)
fun dvd_cancel_factor ctxt ct = DvdCancelFactor.proc ctxt (Thm.term_of ct)
fun divide_cancel_factor ctxt ct = DivideCancelFactor.proc ctxt (Thm.term_of ct)
local
val cterm_of = Thm.cterm_of \<^context>;
fun tvar S = (("'a", 0), S);
val zero_tvar = tvar \<^sort>\<open>zero\<close>;
val zero = cterm_of (Const (\<^const_name>\<open>zero_class.zero\<close>, TVar zero_tvar));
val type_tvar = tvar \<^sort>\<open>type\<close>;
val geq = cterm_of (Const (\<^const_name>\<open>HOL.eq\<close>, TVar type_tvar --> TVar type_tvar --> \<^typ>\<open>bool\<close>));
val add_frac_eq = mk_meta_eq @{thm add_frac_eq}
val add_frac_num = mk_meta_eq @{thm add_frac_num}
val add_num_frac = mk_meta_eq @{thm add_num_frac}
fun prove_nz ctxt T t =
let
val z = Thm.instantiate_cterm (TVars.make1 (zero_tvar, T), Vars.empty) zero
val eq = Thm.instantiate_cterm (TVars.make1 (type_tvar, T), Vars.empty) geq
val th =
Simplifier.rewrite (ctxt addsimps @{thms simp_thms})
(Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close>
(Thm.apply (Thm.apply eq t) z)))
in Thm.equal_elim (Thm.symmetric th) TrueI end
fun add_frac_frac_proc ctxt ct =
let
val ((x,y),(w,z)) =
(Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
val _ = map (HOLogic.dest_number o Thm.term_of) [x,y,z,w]
val T = Thm.ctyp_of_cterm x
val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
val th = Thm.instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
in SOME (Thm.implies_elim (Thm.implies_elim th y_nz) z_nz) end
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
fun add_frac_num_proc ctxt ct =
let
val (l,r) = Thm.dest_binop ct
val T = Thm.ctyp_of_cterm l
in (case (Thm.term_of l, Thm.term_of r) of
(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_, _) =>
let val (x,y) = Thm.dest_binop l val z = r
val _ = map (HOLogic.dest_number o Thm.term_of) [x,y,z]
val ynz = prove_nz ctxt T y
in SOME (Thm.implies_elim (Thm.instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
end
| (_, Const (\<^const_name>\<open>Rings.divide\<close>,_)$_$_) =>
let val (x,y) = Thm.dest_binop r val z = l
val _ = map (HOLogic.dest_number o Thm.term_of) [x,y,z]
val ynz = prove_nz ctxt T y
in SOME (Thm.implies_elim (Thm.instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
end
| _ => NONE)
end
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
fun is_number (Const(\<^const_name>\<open>Rings.divide\<close>,_)$a$b) = is_number a andalso is_number b
| is_number t = can HOLogic.dest_number t
val is_number = is_number o Thm.term_of
fun ord_frac_proc ct =
(case Thm.term_of ct of
Const(\<^const_name>\<open>Orderings.less\<close>,_)$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
in SOME (mk_meta_eq th) end
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
in SOME (mk_meta_eq th) end
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
in SOME (mk_meta_eq th) end
| Const(\<^const_name>\<open>Orderings.less\<close>,_)$_$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
in SOME (mk_meta_eq th) end
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$_$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
in SOME (mk_meta_eq th) end
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$_$(Const(\<^const_name>\<open>Rings.divide\<close>,_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = Thm.ctyp_of_cterm c
val th = Thm.instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
in SOME (mk_meta_eq th) end
| _ => NONE) handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
val add_frac_frac_simproc =
\<^simproc_setup>\<open>passive add_frac_frac ("(x::'a::field) / y + (w::'a::field) / z") =
\<open>K add_frac_frac_proc\<close>\<close>
val add_frac_num_simproc =
\<^simproc_setup>\<open>passive add_frac_num ("(x::'a::field) / y + z" | "z + (x::'a::field) / y") =
\<open>K add_frac_num_proc\<close>\<close>
val ord_frac_simproc =
\<^simproc_setup>\<open>passive ord_frac
("(a::'a::{field,ord}) / b < c" |
"(a::'a::{field,ord}) / b \<le> c" |
"c < (a::'a::{field,ord}) / b" |
"c \<le> (a::'a::{field,ord}) / b" |
"c = (a::'a::{field,ord}) / b" |
"(a::'a::{field, ord}) / b = c") =
\<open>K (K ord_frac_proc)\<close>\<close>
val field_comp_ss =
simpset_of
(put_simpset HOL_basic_ss \<^context>
addsimps @{thms semiring_norm
mult_numeral_1
mult_numeral_1_right
divide_numeral_1
div_by_0
div_0
divide_divide_eq_left
times_divide_eq_left
times_divide_eq_right
times_divide_times_eq
divide_divide_eq_right
diff_conv_add_uminus
minus_divide_left
add_divide_distrib [symmetric]
Fields.field_divide_inverse [symmetric]
inverse_divide
divide_inverse_commute [symmetric]
simp_thms}
addsimprocs field_cancel_numeral_factors
addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, ord_frac_simproc]
|> Simplifier.add_cong @{thm if_weak_cong})
in
fun field_comp_conv ctxt =
Simplifier.rewrite (put_simpset field_comp_ss ctxt)
then_conv
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms numeral_One})
end
end;