src/HOL/Induct/Multiset.ML

field_injects [iffs];

(*  Title:      HOL/Induct/Multiset.ML
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   1998 TUM
*)

Addsimps [Abs_multiset_inverse,Rep_multiset_inverse,Rep_multiset];

(** Preservation of representing set `multiset' **)

Goalw [multiset_def] "(%a. 0) : multiset";
by(Simp_tac 1);
qed "const0_in_multiset";
Addsimps [const0_in_multiset];

Goalw [multiset_def] "(%b. if b=a then 1 else 0) : multiset";
by(Simp_tac 1);
qed "only1_in_multiset";
Addsimps [only1_in_multiset];

Goalw [multiset_def]
 "[| M : multiset; N : multiset |] ==> (%a. M a + N a) : multiset";
by(Asm_full_simp_tac 1);
bd finite_UnI 1;
ba 1;
by(asm_full_simp_tac (simpset() delsimps [finite_Un]addsimps [Un_def]) 1);
qed "union_preserves_multiset";
Addsimps [union_preserves_multiset];

Goalw [multiset_def]
 "[| M : multiset |] ==> (%a. M a - N a) : multiset";
by(Asm_full_simp_tac 1);
be (rotate_prems 1 finite_subset) 1;
by(Auto_tac);
qed "diff_preserves_multiset";
Addsimps [diff_preserves_multiset];

(** Injectivity of Rep_multiset **)

Goal "(M = N) = (Rep_multiset M = Rep_multiset N)";
br iffI 1;
 by(Asm_simp_tac 1);
by(dres_inst_tac [("f","Abs_multiset")] arg_cong 1);
by(Asm_full_simp_tac 1);
qed "multiset_eq_conv_Rep_eq";
Addsimps [multiset_eq_conv_Rep_eq];
Addsimps [expand_fun_eq];
(*
Goal
 "[| f : multiset; g : multiset |] ==> \
\ (Abs_multiset f = Abs_multiset g) = (!x. f x = g x)";
br iffI 1;
 by(dres_inst_tac [("f","Rep_multiset")] arg_cong 1);
 by(Asm_full_simp_tac 1);
by(subgoal_tac "f = g" 1);
 by(Asm_simp_tac 1);
br ext 1;
by(Blast_tac 1);
qed "Abs_multiset_eq";
Addsimps [Abs_multiset_eq];
*)

(** Equations **)

(* union *)

Goalw [union_def,empty_def]
 "M + {#} = M & {#} + M = M";
by(Simp_tac 1);
qed "union_empty";
Addsimps [union_empty];

Goalw [union_def]
 "(M::'a multiset) + N = N + M";
by(simp_tac (simpset() addsimps add_ac) 1);
qed "union_comm";

Goalw [union_def]
 "((M::'a multiset)+N)+K = M+(N+K)";
by(simp_tac (simpset() addsimps add_ac) 1);
qed "union_assoc";

qed_goal "union_lcomm" thy "M+(N+K) = N+((M+K)::'a multiset)"
 (fn _ => [rtac (union_comm RS trans) 1, rtac (union_assoc RS trans) 1,
           rtac (union_comm RS arg_cong) 1]);

val union_ac = [union_assoc, union_comm, union_lcomm];

(* diff *)

Goalw [empty_def,diff_def]
 "M-{#} = M & {#}-M = {#}";
by(Simp_tac 1);
qed "diff_empty";
Addsimps [diff_empty];

Goalw [union_def,diff_def]
 "M+{#a#}-{#a#} = M";
by(Simp_tac 1);
qed "diff_union_inverse2";
Addsimps [diff_union_inverse2];

(* count *)

Goalw [count_def,empty_def]
 "count {#} a = 0";
by(Simp_tac 1);
qed "count_empty";
Addsimps [count_empty];

Goalw [count_def,single_def]
 "count {#b#} a = (if b=a then 1 else 0)";
by(Simp_tac 1);
qed "count_single";
Addsimps [count_single];

Goalw [count_def,union_def]
 "count (M+N) a = count M a + count N a";
by(Simp_tac 1);
qed "count_union";
Addsimps [count_union];

(* set_of *)

Goalw [set_of_def] "set_of {#} = {}";
by(Simp_tac 1);
qed "set_of_empty";
Addsimps [set_of_empty];

Goalw [set_of_def]
 "set_of {#b#} = {b}";
by(Simp_tac 1);
qed "set_of_single";
Addsimps [set_of_single];

Goalw [set_of_def]
 "set_of(M+N) = set_of M Un set_of N";
by(Auto_tac);
qed "set_of_union";
Addsimps [set_of_union];

(* size *)

Goalw [size_def] "size {#} = 0";
by(Simp_tac 1);
qed "size_empty";
Addsimps [size_empty];

Goalw [size_def]
 "size {#b#} = 1";
by(Simp_tac 1);
qed "size_single";
Addsimps [size_single];

(* Some other day...
Goalw [size_def]
 "size (M+N::'a multiset) = size M + size N";
*)

(* equalities *)

Goalw [single_def,empty_def] "{#a#} ~= {#}  &  {#} ~= {#a#}";
by(Simp_tac 1);
qed "single_not_empty";
Addsimps [single_not_empty];

Goalw [single_def] "({#a#}={#b#}) = (a=b)";
by(Auto_tac);
qed "single_eq_single";
Addsimps [single_eq_single];

Goalw [union_def,empty_def]
 "(M+N = {#}) = (M = {#} & N = {#})";
by(Simp_tac 1);
by(Blast_tac 1);
qed "union_eq_empty";
AddIffs [union_eq_empty];

Goalw [union_def,empty_def]
 "({#} = M+N) = (M = {#} & N = {#})";
by(Simp_tac 1);
by(Blast_tac 1);
qed "empty_eq_union";
AddIffs [empty_eq_union];

Goalw [union_def]
 "(M+K = N+K) = (M=(N::'a multiset))";
by(Simp_tac 1);
qed "union_right_cancel";
Addsimps [union_right_cancel];

Goalw [union_def]
 "(K+M = K+N) = (M=(N::'a multiset))";
by(Simp_tac 1);
qed "union_left_cancel";
Addsimps [union_left_cancel];

Goalw [empty_def,single_def,union_def]
 "(M+N = {#a#}) = (M={#a#} & N={#} | M={#} & N={#a#})";
by(simp_tac (simpset() addsimps [add_is_1]) 1);
by(Blast_tac 1);
qed "union_is_single";

Goalw [empty_def,single_def,union_def]
 "({#a#} = M+N) = ({#a#}=M & N={#} | M={#} & {#a#}=N)";
by(simp_tac (simpset() addsimps [one_is_add]) 1);
by(blast_tac (claset() addDs [sym]) 1);
qed "single_is_union";

Goalw [single_def,union_def,diff_def]
 "(M+{#a#} = N+{#b#}) = (M=N & a=b | M = N-{#a#}+{#b#} & N = M-{#b#}+{#a#})";
by(Simp_tac 1);
br conjI 1;
 by(Force_tac 1);
by(Clarify_tac 1);
br conjI 1;
 by(Blast_tac 1);
by(Clarify_tac 1);
br iffI 1;
 br conjI 1;
 by(Clarify_tac 1);
  br conjI 1;
   by(asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1);
(* PROOF FAILED:
by(Blast_tac 1);
*)
  by(Fast_tac 1);
 by(Asm_simp_tac 1);
by(Force_tac 1);
qed "add_eq_conv_diff";

(* FIXME
val prems = Goal
 "[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
by(res_inst_tac [("a","F"),("f","%A. if finite A then card A else 0")]
     measure_induct 1);
by(Clarify_tac 1);
brs prems 1;
 ba 1;
by(Clarify_tac 1);
by(subgoal_tac "finite G" 1);
 by(fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
be allE 1;
be impE 1;
 by(Blast_tac 2);
by(asm_simp_tac (simpset() addsimps [psubset_card]) 1);
val lemma = result();

val prems = Goal
 "[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
br (lemma RS mp) 1;
by (REPEAT(ares_tac prems 1));
qed "finite_psubset_induct";

Better: use wf_finite_psubset in WF_Rel
*)

(** Towards the induction rule **)

Goal "finite F ==> (setsum f F = 0) = (!a:F. f a = 0)";
be finite_induct 1;
by(Auto_tac);
qed "setsum_0";
Addsimps [setsum_0];

Goal "finite F ==> setsum f F = Suc n --> (? a:F. 0 < f a)";
be finite_induct 1;
by(Auto_tac);
by(asm_full_simp_tac
    (simpset() delsimps [setsum_0] addsimps [setsum_0 RS sym]) 1);
val lemma = result();

Goal "[| setsum f F = Suc n; finite F |] ==> ? a:F. 0 < f a";
bd lemma 1;
by(Fast_tac 1);
qed "setsum_SucD";

Goal "[| finite F; 0 < f a |] ==> \
\     setsum (f(a:=f(a)-1)) F = (if a:F then setsum f F - 1 else setsum f F)";
be finite_induct 1;
by(Auto_tac);
 by(asm_simp_tac (simpset() addsimps add_ac) 1);
by(dres_inst_tac [("a","a")] mk_disjoint_insert 1);
by(Auto_tac);
qed "setsum_decr";

val prems = Goalw [multiset_def]
 "[| P(%a.0); \
\    !!f b. [| f : multiset; P(f) |] ==> P(f(b:=f(b)+1)) |] \
\  ==> !f. f : multiset --> setsum f {x. 0 < f x} = n --> P(f)";
by(induct_tac "n" 1);
 by(Asm_simp_tac 1);
 by(Clarify_tac 1);
 by(subgoal_tac "f = (%a.0)" 1);
  by(Asm_simp_tac 1);
  brs prems 1;
 br ext 1;
 by(Force_tac 1);
by(Clarify_tac 1);
by(forward_tac [setsum_SucD] 1);
 ba 1;
by(Clarify_tac 1);
by(rename_tac "a" 1);
by(subgoal_tac "finite{x. 0 < (f(a:=f(a)-1)) x}" 1);
 be (rotate_prems 1 finite_subset) 2;
 by(Simp_tac 2);
 by(Blast_tac 2);
by(subgoal_tac
   "f = (f(a:=f(a)-1))(a:=(f(a:=f(a)-1))a+1)" 1);
 br ext 2;
 by(Asm_simp_tac 2);
by(EVERY1[etac ssubst, resolve_tac prems]);
 by(Blast_tac 1);
by(EVERY[etac allE 1, etac impE 1, etac mp 2]);
 by(Blast_tac 1);
by(asm_simp_tac (simpset() addsimps [setsum_decr] delsimps [fun_upd_apply]) 1);
by(subgoal_tac "{x. x ~= a --> 0 < f x} = {x. 0 < f x}" 1);
 by(Blast_tac 2);
by(subgoal_tac "{x. x ~= a & 0 < f x} = {x. 0 < f x} - {a}" 1);
 by(Blast_tac 2);
by(asm_simp_tac (simpset() addsimps [le_imp_diff_is_add,setsum_diff1]
                           addcongs [conj_cong]
                           addSolver cut_trans_tac) 1);
val lemma = result();

val major::prems = Goal
 "[| f : multiset; \
\    P(%a.0); \
\    !!f b. [| f : multiset; P(f) |] ==> P(f(b:=f(b)+1)) |] ==> P(f)";
by(rtac (major RSN (3, lemma RS spec RS mp RS mp)) 1);
by(REPEAT(ares_tac (refl::prems) 1));
qed "Rep_multiset_induct";

val [prem1,prem2] = Goalw [union_def,single_def,empty_def]
 "[| P({#}); !!M x. P(M) ==> P(M + {#x#}) |] ==> P(M)";
by (rtac (Rep_multiset_inverse RS subst) 1);
by (rtac (Rep_multiset RS Rep_multiset_induct) 1);
 by(rtac prem1 1);
by(Clarify_tac 1);
by(subgoal_tac
    "f(b := f b + 1) = (%a. f a + (if a = b then 1 else 0))" 1);
 by(Simp_tac 2);
be ssubst 1;
by(etac (Abs_multiset_inverse RS subst) 1);
by(etac(simplify (simpset()) prem2)1);
qed "multiset_induct";

Delsimps [multiset_eq_conv_Rep_eq, expand_fun_eq];
Delsimps [Abs_multiset_inverse,Rep_multiset_inverse,Rep_multiset];

Goal
 "(M+{#a#} = N+{#b#}) = (M = N & a = b | (? K. M = K+{#b#} & N = K+{#a#}))";
by(simp_tac (simpset() addsimps [add_eq_conv_diff]) 1);
by(Auto_tac);
qed "add_eq_conv_ex";

(** order **)

Goalw [mult1_def] "(M, {#}) ~: mult1(r)";
by(Simp_tac 1);
qed "not_less_empty";
AddIffs [not_less_empty];

Goalw [mult1_def]
 "(N,M0 + {#a#}) : mult1(r) = \
\ ((? M. (M,M0) : mult1(r) & N = M + {#a#}) | \
\  (? K. (!b. elem K b --> (b,a) : r) & N = M0 + K))";
br iffI 1;
 by(asm_full_simp_tac (simpset() addsimps [add_eq_conv_ex]) 1);
 by(Clarify_tac 1);
 be disjE 1;
  by(Blast_tac 1);
 by(Clarify_tac 1);
 by(res_inst_tac [("x","Ka+K")] exI 1);
 by(simp_tac (simpset() addsimps union_ac) 1);
 by(Blast_tac 1);
be disjE 1;
 by(Clarify_tac 1);
 by(res_inst_tac [("x","aa")] exI 1);
 by(res_inst_tac [("x","M0+{#a#}")] exI 1);
 by(res_inst_tac [("x","K")] exI 1);
 by(simp_tac (simpset() addsimps union_ac) 1);
by(Blast_tac 1);
qed "less_add_conv";

Open_locale "MSOrd";

val W_def = thm "W_def";

Goalw [W_def]
 "[| !b. (b,a) : r --> (!M : W. M+{#b#} : W); M0 : W; \
\    !M. (M,M0) : mult1(r) --> M+{#a#} : W |] \
\ ==> M0+{#a#} : W";
br accI 1;
by(rename_tac "N" 1);
by(full_simp_tac (simpset() addsimps [less_add_conv]) 1);
be disjE 1;
 by(Blast_tac 1);
by(Clarify_tac 1);
by(rotate_tac ~1 1);
be rev_mp 1;
by(res_inst_tac [("M","K")] multiset_induct 1);
 by(Asm_simp_tac 1);
by(simp_tac (simpset() addsimps [union_assoc RS sym]) 1);
by(Blast_tac 1);
qed "lemma1";

Goalw [W_def]
 "[| !b. (b,a) : r --> (!M : W. M+{#b#} : W); M : W |] ==> M+{#a#} : W";
be acc_induct 1;
by(blast_tac (claset() addIs [export lemma1]) 1);
qed "lemma2";

Goalw [W_def]
 "wf(r) ==> !M:W. M+{#a#} : W";
by(eres_inst_tac [("a","a")] wf_induct 1);
by(blast_tac (claset() addIs [export lemma2]) 1);
qed "lemma3";

Goalw [W_def] "wf(r) ==> M : W";
by(res_inst_tac [("M","M")] multiset_induct 1);
 br accI 1;
 by(Asm_full_simp_tac 1);
by(blast_tac (claset() addDs [export lemma3]) 1);
qed "all_accessible";

Close_locale();

Goal "wf(r) ==> wf(mult1 r)";
by(blast_tac (claset() addIs [acc_wfI, export all_accessible]) 1);
qed "wf_mult1";

Goalw [mult_def] "wf(r) ==> wf(mult r)";
by(blast_tac (claset() addIs [wf_trancl,wf_mult1]) 1);
qed "wf_mult";

Goalw [mult_def,set_of_def]
 "(M,N) : mult r = (? I J K. N = I+J & M = I+K & \
\                            (!k : set_of K. ? j : set_of J. (k,j) : r))";
br iffI 1;

be trancl_induct 1;
by(asm_full_simp_tac (simpset() addsimps [mult1_def]) 1);

by(Clarify_tac 1);
by(res_inst_tac [("x","M0")] exI 1);
by(Simp_tac 1);