(* 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 prep_simproc: theory -> string * string list * (Proof.context -> term -> thm option)
-> simproc
val trans_tac: Proof.context -> thm option -> tactic
val assoc_fold: Proof.context -> cterm -> thm option
val combine_numerals: Proof.context -> cterm -> thm option
val eq_cancel_numerals: Proof.context -> cterm -> thm option
val less_cancel_numerals: Proof.context -> cterm -> thm option
val le_cancel_numerals: Proof.context -> cterm -> thm option
val eq_cancel_factor: Proof.context -> cterm -> thm option
val le_cancel_factor: Proof.context -> cterm -> thm option
val less_cancel_factor: Proof.context -> cterm -> thm option
val div_cancel_factor: Proof.context -> cterm -> thm option
val mod_cancel_factor: Proof.context -> cterm -> thm option
val dvd_cancel_factor: Proof.context -> cterm -> thm option
val divide_cancel_factor: Proof.context -> cterm -> thm option
val eq_cancel_numeral_factor: Proof.context -> cterm -> thm option
val less_cancel_numeral_factor: Proof.context -> cterm -> thm option
val le_cancel_numeral_factor: Proof.context -> cterm -> thm option
val div_cancel_numeral_factor: Proof.context -> cterm -> thm option
val divide_cancel_numeral_factor: Proof.context -> cterm -> thm option
val field_combine_numerals: Proof.context -> cterm -> thm option
val field_divide_cancel_numeral_factor: simproc list
val num_ss: simpset
val field_comp_conv: Proof.context -> conv
end;
structure Numeral_Simprocs : NUMERAL_SIMPROCS =
struct
fun prep_simproc thy (name, pats, proc) =
Simplifier.simproc_global thy name pats proc;
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 Groups.times};
fun one_of T = Const(@{const_name Groups.one}, 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 Groups.times} 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 Groups.uminus}, _) $ 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 Fields.divide};
(*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 Groups.uminus}, _) $ t) = dest_fcoeff (~sign) t
| dest_fcoeff sign (Const (@{const_name Fields.divide}, _) $ 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;
fun numtermless tu = (numterm_ord tu = LESS);
val num_ss =
simpset_of (put_simpset HOL_basic_ss @{context}
|> Simplifier.set_termless numtermless);
(*Maps 1 to Numeral1 so that arithmetic isn't complicated by the abstract 1.*)
val numeral_syms = [@{thm numeral_1_eq_1} RS sym];
(*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 divide_1};
(* For post-simplification of the rhs of simproc-generated rules *)
val post_simps =
[@{thm numeral_1_eq_1},
@{thm add_0_left}, @{thm add_0_right},
@{thm mult_zero_left}, @{thm mult_zero_right},
@{thm mult_1_left}, @{thm mult_1_right},
@{thm mult_minus1}, @{thm mult_minus1_right}]
val field_post_simps =
post_simps @ [@{thm divide_zero_left}, @{thm divide_1}]
(*Simplify inverse Numeral1*)
val inverse_1s = [@{thm inverse_numeral_1}];
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the Numeral_Simprocs, such as 3 * (5 * x). *)
val simps =
[@{thm numeral_1_eq_1} RS sym] @
@{thms add_numeral_left} @
@{thms add_neg_numeral_left} @
@{thms mult_numeral_left} @
@{thms arith_simps} @ @{thms 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} @
@{thms add_neg_numeral_left} @
@{thms numeral_plus_numeral} @
@{thms add_neg_numeral_simps}) simps;
(*To evaluate binary negations of coefficients*)
val minus_simps = [@{thm minus_zero}, @{thm minus_minus}];
(*To let us treat subtraction as addition*)
val diff_simps = [@{thm diff_conv_add_uminus}, @{thm minus_add_distrib}, @{thm minus_minus}];
(*To let us treat division as multiplication*)
val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
[@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
[@{thm mult.assoc}, @{thm minus_mult_right}, @{thm minus_mult_commute}];
val norm_ss1 =
simpset_of (put_simpset num_ss @{context}
addsimps numeral_syms @ add_0s @ mult_1s @
diff_simps @ minus_simps @ @{thms 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} @ @{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 add_0s @ 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 HOL.eq} 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 Orderings.less}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} 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 Orderings.less_eq}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} dummyT
val bal_add1 = @{thm le_add_iff1} RS trans
val bal_add2 = @{thm le_add_iff2} RS trans
);
fun eq_cancel_numerals ctxt ct = EqCancelNumerals.proc ctxt (term_of ct)
fun less_cancel_numerals ctxt ct = LessCancelNumerals.proc ctxt (term_of ct)
fun le_cancel_numerals ctxt ct = LeCancelNumerals.proc ctxt (term_of ct)
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 add_0s @ 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 add_0s @ simps @ [@{thm add_frac_eq}, @{thm 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);
fun combine_numerals ctxt ct = CombineNumerals.proc ctxt (term_of ct)
fun field_combine_numerals ctxt ct = FieldCombineNumerals.proc ctxt (term_of ct)
(** 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 (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})
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
([@{thm Nat.add_0}, @{thm 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 Divides.div}
val dest_bal = HOLogic.dest_bin @{const_name Divides.div} 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 Fields.divide}
val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} 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 HOL.eq} 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 Orderings.less}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} 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 Orderings.less_eq}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} dummyT
val cancel = @{thm mult_le_cancel_left} RS trans
val neg_exchanges = true
)
fun eq_cancel_numeral_factor ctxt ct = EqCancelNumeralFactor.proc ctxt (term_of ct)
fun less_cancel_numeral_factor ctxt ct = LessCancelNumeralFactor.proc ctxt (term_of ct)
fun le_cancel_numeral_factor ctxt ct = LeCancelNumeralFactor.proc ctxt (term_of ct)
fun div_cancel_numeral_factor ctxt ct = DivCancelNumeralFactor.proc ctxt (term_of ct)
fun divide_cancel_numeral_factor ctxt ct = DivideCancelNumeralFactor.proc ctxt (term_of ct)
val field_divide_cancel_numeral_factor =
[prep_simproc @{theory}
("field_divide_cancel_numeral_factor",
["((l::'a::field_inverse_zero) * m) / n",
"(l::'a::field_inverse_zero) / (m * n)",
"((numeral v)::'a::field_inverse_zero) / (numeral w)",
"((numeral v)::'a::field_inverse_zero) / (- numeral w)",
"((- numeral v)::'a::field_inverse_zero) / (numeral w)",
"((- numeral v)::'a::field_inverse_zero) / (- numeral w)"],
DivideCancelNumeralFactor.proc)];
val field_cancel_numeral_factors =
prep_simproc @{theory}
("field_eq_cancel_numeral_factor",
["(l::'a::field) * m = n",
"(l::'a::field) = m * n"],
EqCancelNumeralFactor.proc)
:: field_divide_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_1_eq_1}];
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 HOL} 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 Groups.zero}, T);
val less = Const(@{const_name Orderings.less}, [T,T] ---> HOLogic.boolT);
val pos = less $ zero $ t and neg = less $ t $ zero
val thy = Proof_Context.theory_of ctxt
fun prove p =
SOME (Eq_True_elim (Simplifier.asm_rewrite ctxt (Thm.cterm_of thy 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 HOL.eq} 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 Orderings.less_eq}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} 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 Orderings.less}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} 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 Divides.div}
val dest_bal = HOLogic.dest_bin @{const_name Divides.div} dummyT
fun simp_conv _ _ = SOME @{thm div_mult_mult1_if}
);
structure ModCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binop @{const_name Divides.mod}
val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} dummyT
fun simp_conv _ _ = SOME @{thm mod_mult_mult1}
);
(*for idoms*)
structure DvdCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val mk_bal = HOLogic.mk_binrel @{const_name Rings.dvd}
val dest_bal = HOLogic.dest_bin @{const_name Rings.dvd} 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 Fields.divide}
val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} dummyT
fun simp_conv _ _ = SOME @{thm mult_divide_mult_cancel_left_if}
);
fun eq_cancel_factor ctxt ct = EqCancelFactor.proc ctxt (term_of ct)
fun le_cancel_factor ctxt ct = LeCancelFactor.proc ctxt (term_of ct)
fun less_cancel_factor ctxt ct = LessCancelFactor.proc ctxt (term_of ct)
fun div_cancel_factor ctxt ct = DivCancelFactor.proc ctxt (term_of ct)
fun mod_cancel_factor ctxt ct = ModCancelFactor.proc ctxt (term_of ct)
fun dvd_cancel_factor ctxt ct = DvdCancelFactor.proc ctxt (term_of ct)
fun divide_cancel_factor ctxt ct = DivideCancelFactor.proc ctxt (term_of ct)
local
val zr = @{cpat "0"}
val zT = ctyp_of_term zr
val geq = @{cpat HOL.eq}
val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
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 ([(zT,T)],[]) zr
val eq = Thm.instantiate_cterm ([(eqT,T)],[]) geq
val th = Simplifier.rewrite (ctxt addsimps @{thms simp_thms})
(Thm.apply @{cterm "Trueprop"} (Thm.apply @{cterm "Not"}
(Thm.apply (Thm.apply eq t) z)))
in Thm.equal_elim (Thm.symmetric th) TrueI
end
fun proc phi 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 term_of) [x,y,z,w]
val T = ctyp_of_term x
val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
val th = 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 proc2 phi ctxt ct =
let
val (l,r) = Thm.dest_binop ct
val T = ctyp_of_term l
in (case (term_of l, term_of r) of
(Const(@{const_name Fields.divide},_)$_$_, _) =>
let val (x,y) = Thm.dest_binop l val z = r
val _ = map (HOLogic.dest_number o term_of) [x,y,z]
val ynz = prove_nz ctxt T y
in SOME (Thm.implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
end
| (_, Const (@{const_name Fields.divide},_)$_$_) =>
let val (x,y) = Thm.dest_binop r val z = l
val _ = map (HOLogic.dest_number o term_of) [x,y,z]
val ynz = prove_nz ctxt T y
in SOME (Thm.implies_elim (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 Fields.divide},_)$a$b) = is_number a andalso is_number b
| is_number t = can HOLogic.dest_number t
val is_number = is_number o term_of
fun proc3 phi ctxt ct =
(case term_of ct of
Const(@{const_name Orderings.less},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
in SOME (mk_meta_eq th) end
| Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
in SOME (mk_meta_eq th) end
| Const(@{const_name HOL.eq},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
let
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
in SOME (mk_meta_eq th) end
| Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
in SOME (mk_meta_eq th) end
| Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
in SOME (mk_meta_eq th) end
| Const(@{const_name HOL.eq},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
let
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
val _ = map is_number [a,b,c]
val T = ctyp_of_term c
val th = 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 =
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
name = "add_frac_frac_simproc",
proc = proc, identifier = []}
val add_frac_num_simproc =
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
name = "add_frac_num_simproc",
proc = proc2, identifier = []}
val ord_frac_simproc =
make_simproc
{lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
@{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
@{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
@{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
@{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
@{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
name = "ord_frac_simproc", proc = proc3, identifier = []}
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
@{thm "divide_numeral_1"},
@{thm "divide_zero"}, @{thm divide_zero_left},
@{thm "divide_divide_eq_left"},
@{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
@{thm "times_divide_times_eq"},
@{thm "divide_divide_eq_right"},
@{thm diff_conv_add_uminus}, @{thm "minus_divide_left"},
@{thm "add_divide_distrib"} RS sym,
@{thm Fields.field_divide_inverse} RS sym, @{thm inverse_divide},
Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult.commute}))))
(@{thm Fields.field_divide_inverse} RS sym)]
val field_comp_ss =
simpset_of
(put_simpset HOL_basic_ss @{context}
addsimps @{thms "semiring_norm"}
addsimps ths addsimps @{thms 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 [@{thm numeral_1_eq_1}])
end
end;