(* Title: ZF/arith_data.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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: Proof.context -> thm -> thm option -> tactic
val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option
val simplify_meta_eq: thm list -> Proof.context -> 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 zero = \<^Const>\<open>zero\<close>;
val succ = \<^Const>\<open>succ\<close>;
fun mk_succ t = succ $ t;
val one = mk_succ zero;
fun mk_plus (t, u) = \<^Const>\<open>Arith.add for t u\<close>;
(*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);
(* dest_sum *)
fun dest_sum \<^Const_>\<open>zero\<close> = []
| dest_sum \<^Const_>\<open>succ for t\<close> = one :: dest_sum t
| dest_sum \<^Const_>\<open>Arith.add for t u\<close> = dest_sum t @ dest_sum u
| dest_sum tm = [tm];
(*Apply the given rewrite (if present) just once*)
fun gen_trans_tac _ _ NONE = all_tac
| gen_trans_tac ctxt th2 (SOME th) = ALLGOALS (resolve_tac ctxt [th RS th2]);
(*Use <-> or = depending on the type of t*)
fun mk_eq_iff(t,u) =
if fastype_of t = \<^Type>\<open>i\<close>
then \<^Const>\<open>IFOL.eq \<^Type>\<open>i\<close> for t u\<close>
else \<^Const>\<open>IFOL.iff for t u\<close>;
(*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 (\<^dest_judgment> (Thm.prop_of th));
fun prove_conv name tacs ctxt prems (t,u) =
if t aconv u then NONE
else
let val prems' = filter_out is_eq_thm prems
val goal = Logic.list_implies (map Thm.prop_of prems', \<^make_judgment> (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;
(*** Use CancelNumerals simproc without binary numerals,
just for cancellation ***)
fun mk_times (t, u) = \<^Const>\<open>Arith.mult for t u\<close>;
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);
fun dest_prod tm =
let val (t,u) = \<^Const_fn>\<open>Arith.mult for t u => \<open>(t, u)\<close>\<close> tm
in dest_prod t @ dest_prod u end
handle TERM _ => [tm];
(*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_Ord.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 ctxt =
let val ctxt' =
put_simpset FOL_ss ctxt
delsimps @{thms iff_simps} (*these could erase the whole rule!*)
addsimps rules
|> fold Simplifier.add_eqcong [@{thm eq_cong2}, @{thm iff_cong2}]
in mk_meta_eq o simplify ctxt' 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 =
simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac})
val norm_ss2 =
simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ mult_1s @ @{thms add_ac} @
@{thms mult_ac} @ tc_rules @ natifys)
fun norm_tac ctxt =
ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt))
val numeral_simp_ss =
simpset_of (put_simpset ZF_ss \<^context> addsimps add_0s @ tc_rules @ natifys)
fun numeral_simp_tac ctxt =
ALLGOALS (asm_simp_tac (put_simpset numeral_simp_ss ctxt))
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 [THEN iff_trans]}
val bal_add2 = @{thm eq_add_iff [THEN iff_trans]}
fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans}
end;
structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
structure LessCancelNumeralsData =
struct
open CancelNumeralsCommon
val prove_conv = prove_conv "natless_cancel_numerals"
fun mk_bal (t, u) = \<^Const>\<open>Ordinal.lt for t u\<close>
val dest_bal = \<^Const_fn>\<open>Ordinal.lt for t u => \<open>(t, u)\<close>\<close>
val bal_add1 = @{thm less_add_iff [THEN iff_trans]}
val bal_add2 = @{thm less_add_iff [THEN iff_trans]}
fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans}
end;
structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
structure DiffCancelNumeralsData =
struct
open CancelNumeralsCommon
val prove_conv = prove_conv "natdiff_cancel_numerals"
fun mk_bal (t, u) = \<^Const>\<open>Arith.diff for t u\<close>
val dest_bal = \<^Const_fn>\<open>Arith.diff for t u => \<open>(t, u)\<close>\<close>
val bal_add1 = @{thm diff_add_eq [THEN trans]}
val bal_add2 = @{thm diff_add_eq [THEN trans]}
fun trans_tac ctxt = gen_trans_tac ctxt @{thm trans}
end;
structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
val nat_cancel =
[Simplifier.make_simproc \<^context> "nateq_cancel_numerals"
{lhss =
[\<^term>\<open>l #+ m = n\<close>, \<^term>\<open>l = m #+ n\<close>,
\<^term>\<open>l #* m = n\<close>, \<^term>\<open>l = m #* n\<close>,
\<^term>\<open>succ(m) = n\<close>, \<^term>\<open>m = succ(n)\<close>],
proc = K EqCancelNumerals.proc},
Simplifier.make_simproc \<^context> "natless_cancel_numerals"
{lhss =
[\<^term>\<open>l #+ m < n\<close>, \<^term>\<open>l < m #+ n\<close>,
\<^term>\<open>l #* m < n\<close>, \<^term>\<open>l < m #* n\<close>,
\<^term>\<open>succ(m) < n\<close>, \<^term>\<open>m < succ(n)\<close>],
proc = K LessCancelNumerals.proc},
Simplifier.make_simproc \<^context> "natdiff_cancel_numerals"
{lhss =
[\<^term>\<open>(l #+ m) #- n\<close>, \<^term>\<open>l #- (m #+ n)\<close>,
\<^term>\<open>(l #* m) #- n\<close>, \<^term>\<open>l #- (m #* n)\<close>,
\<^term>\<open>succ(m) #- n\<close>, \<^term>\<open>m #- succ(n)\<close>],
proc = K DiffCancelNumerals.proc}];
end;
val _ =
Theory.setup (Simplifier.map_theory_simpset (fn ctxt =>
ctxt addsimprocs ArithData.nat_cancel));
(*examples:
print_depth 22;
set timing;
set simp_trace;
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)";
*)