(* Title: ZF/arith_data.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
Arithmetic simplification: cancellation of common terms
*)
signature ARITH_DATA =
sig
(*the main outcome*)
val nat_cancel: simproc list
(*tools for use in similar applications*)
val gen_trans_tac: thm -> thm option -> tactic
val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option
val simplify_meta_eq: thm list -> simpset -> thm -> thm
(*debugging*)
structure EqCancelNumeralsData : CANCEL_NUMERALS_DATA
structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
end;
structure ArithData: ARITH_DATA =
struct
val iT = Ind_Syntax.iT;
val zero = Const("0", iT);
val succ = Const("succ", iT --> iT);
fun mk_succ t = succ $ t;
val one = mk_succ zero;
val mk_plus = FOLogic.mk_binop "Arith.add";
(*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
fun mk_sum [] = zero
| mk_sum [t,u] = mk_plus (t, u)
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
(*this version ALWAYS includes a trailing zero*)
fun long_mk_sum [] = zero
| long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
val dest_plus = FOLogic.dest_bin "Arith.add" iT;
(* dest_sum *)
fun dest_sum (Const("0",_)) = []
| dest_sum (Const("succ",_) $ t) = one :: dest_sum t
| dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
| dest_sum tm = [tm];
(*Apply the given rewrite (if present) just once*)
fun gen_trans_tac th2 NONE = all_tac
| gen_trans_tac th2 (SOME th) = ALLGOALS (rtac (th RS th2));
(*Use <-> or = depending on the type of t*)
fun mk_eq_iff(t,u) =
if fastype_of t = iT then FOLogic.mk_eq(t,u)
else FOLogic.mk_iff(t,u);
(*We remove equality assumptions because they confuse the simplifier and
because only type-checking assumptions are necessary.*)
fun is_eq_thm th =
can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));
fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
fun prove_conv name tacs ctxt prems (t,u) =
if t aconv u then NONE
else
let val prems' = List.filter (not o is_eq_thm) prems
val goal = Logic.list_implies (map (#prop o Thm.rep_thm) prems',
FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs)))
handle ERROR msg =>
(warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)
end;
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (the_context ()) name pats proc;
(*** Use CancelNumerals simproc without binary numerals,
just for cancellation ***)
val mk_times = FOLogic.mk_binop "Arith.mult";
fun mk_prod [] = one
| mk_prod [t] = t
| mk_prod (t :: ts) = if t = one then mk_prod ts
else mk_times (t, mk_prod ts);
val dest_times = FOLogic.dest_bin "Arith.mult" iT;
fun dest_prod t =
let val (t,u) = dest_times t
in dest_prod t @ dest_prod u end
handle TERM _ => [t];
(*Dummy version: the only arguments are 0 and 1*)
fun mk_coeff (0, t) = zero
| mk_coeff (1, t) = t
| mk_coeff _ = raise TERM("mk_coeff", []);
(*Dummy version: the "coefficient" is always 1.
In the result, the factors are sorted terms*)
fun dest_coeff t = (1, mk_prod (sort Term.term_ord (dest_prod t)));
(*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 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;
(*Simplify #1*n and n*#1 to n*)
val add_0s = [@{thm add_0_natify}, @{thm add_0_right_natify}];
val add_succs = [@{thm add_succ}, @{thm add_succ_right}];
val mult_1s = [@{thm mult_1_natify}, @{thm mult_1_right_natify}];
val tc_rules = [@{thm natify_in_nat}, @{thm add_type}, @{thm diff_type}, @{thm mult_type}];
val natifys = [@{thm natify_0}, @{thm natify_ident}, @{thm add_natify1}, @{thm add_natify2},
@{thm diff_natify1}, @{thm diff_natify2}];
(*Final simplification: cancel + and **)
fun simplify_meta_eq rules =
let val ss0 =
FOL_ss addeqcongs [@{thm eq_cong2}, @{thm iff_cong2}]
delsimps @{thms iff_simps} (*these could erase the whole rule!*)
addsimps rules
in fn ss => mk_meta_eq o simplify (Simplifier.inherit_context ss ss0) end;
val final_rules = add_0s @ mult_1s @ [@{thm mult_0}, @{thm mult_0_right}];
structure CancelNumeralsCommon =
struct
val mk_sum = (fn T:typ => mk_sum)
val dest_sum = dest_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
val find_first_coeff = find_first_coeff []
val norm_ss1 = ZF_ss addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac}
val norm_ss2 = ZF_ss addsimps add_0s @ mult_1s @ @{thms add_ac} @
@{thms mult_ac} @ tc_rules @ natifys
fun norm_tac ss =
ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
val numeral_simp_ss = ZF_ss addsimps add_0s @ tc_rules @ natifys
fun numeral_simp_tac ss =
ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = simplify_meta_eq final_rules
end;
(** The functor argumnets are declared as separate structures
so that they can be exported to ease debugging. **)
structure EqCancelNumeralsData =
struct
open CancelNumeralsCommon
val prove_conv = prove_conv "nateq_cancel_numerals"
val mk_bal = FOLogic.mk_eq
val dest_bal = FOLogic.dest_eq
val bal_add1 = @{thm eq_add_iff} RS iff_trans
val bal_add2 = @{thm eq_add_iff} RS iff_trans
fun trans_tac _ = gen_trans_tac iff_trans
end;
structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
structure LessCancelNumeralsData =
struct
open CancelNumeralsCommon
val prove_conv = prove_conv "natless_cancel_numerals"
val mk_bal = FOLogic.mk_binrel "Ordinal.lt"
val dest_bal = FOLogic.dest_bin "Ordinal.lt" iT
val bal_add1 = @{thm less_add_iff} RS iff_trans
val bal_add2 = @{thm less_add_iff} RS iff_trans
fun trans_tac _ = gen_trans_tac iff_trans
end;
structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
structure DiffCancelNumeralsData =
struct
open CancelNumeralsCommon
val prove_conv = prove_conv "natdiff_cancel_numerals"
val mk_bal = FOLogic.mk_binop "Arith.diff"
val dest_bal = FOLogic.dest_bin "Arith.diff" iT
val bal_add1 = @{thm diff_add_eq} RS trans
val bal_add2 = @{thm diff_add_eq} RS trans
fun trans_tac _ = gen_trans_tac trans
end;
structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
val nat_cancel =
map prep_simproc
[("nateq_cancel_numerals",
["l #+ m = n", "l = m #+ n",
"l #* m = n", "l = m #* n",
"succ(m) = n", "m = succ(n)"],
(K EqCancelNumerals.proc)),
("natless_cancel_numerals",
["l #+ m < n", "l < m #+ n",
"l #* m < n", "l < m #* n",
"succ(m) < n", "m < succ(n)"],
(K LessCancelNumerals.proc)),
("natdiff_cancel_numerals",
["(l #+ m) #- n", "l #- (m #+ n)",
"(l #* m) #- n", "l #- (m #* n)",
"succ(m) #- n", "m #- succ(n)"],
(K DiffCancelNumerals.proc))];
end;
Addsimprocs ArithData.nat_cancel;
(*examples:
print_depth 22;
set timing;
set trace_simp;
fun test s = (Goal s; by (Asm_simp_tac 1));
test "x #+ y = x #+ z";
test "y #+ x = x #+ z";
test "x #+ y #+ z = x #+ z";
test "y #+ (z #+ x) = z #+ x";
test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
test "x #+ succ(y) = x #+ z";
test "x #+ succ(y) = succ(z #+ x)";
test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
test "(x #+ y) #- (x #+ z) = w";
test "(y #+ x) #- (x #+ z) = dd";
test "(x #+ y #+ z) #- (x #+ z) = dd";
test "(y #+ (z #+ x)) #- (z #+ x) = dd";
test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
(*BAD occurrence of natify*)
test "(x #+ succ(y)) #- (x #+ z) = dd";
test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
(*use of typing information*)
test "x : nat ==> x #+ y = x";
test "x : nat --> x #+ y = x";
test "x : nat ==> x #+ y < x";
test "x : nat ==> x < y#+x";
test "x : nat ==> x le succ(x)";
(*fails: no typing information isn't visible*)
test "x #+ y = x";
test "x #+ y < x #+ z";
test "y #+ x < x #+ z";
test "x #+ y #+ z < x #+ z";
test "y #+ z #+ x < x #+ z";
test "y #+ (z #+ x) < z #+ x";
test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
test "x #+ succ(y) < x #+ z";
test "x #+ succ(y) < succ(z #+ x)";
test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
test "x #+ succ(y) le succ(z #+ x)";
*)