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|
(* ========================================================================= *)
(* Part 1: Background theories. *)
(* ========================================================================= *)
let EMPTY_IS_FINITE = prove
(`!s. (s = EMPTY) ==> FINITE s`,
SIMP_TAC[FINITE_RULES]);;
let SING_IS_FINITE = prove
(`!s a. (s = {a}) ==> FINITE s`,
SIMP_TAC[FINITE_INSERT; FINITE_RULES]);;
let UNION_NONZERO = prove
(`{a | ~(f a + g a = 0)} = {a | ~(f a = 0)} UNION {a | ~(g a = 0)}`,
REWRITE_TAC[ADD_EQ_0; EXTENSION; IN_UNION; IN_ELIM_THM; DE_MORGAN_THM]);;
(* ------------------------------------------------------------------------- *)
(* Definition of type of finite multisets with a few basic operations. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("mmember",(11,"right"));;
parse_as_infix("munion",(16,"right"));;
parse_as_infix("mdiff",(18,"left"));;
let multiset_tybij_th = prove
(`?f. FINITE {a:A | ~(f a = 0)}`,
EXISTS_TAC `\a:A. 0` THEN
SIMP_TAC[EMPTY_IS_FINITE; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY]);;
let multiset_tybij = new_type_definition
"multiset" ("multiset","multiplicity") multiset_tybij_th;;
let mempty = new_definition
`mempty = multiset (\b. 0)`;;
let mmember = new_definition
`a mmember M <=> ~(multiplicity M a = 0)`;;
let msing = new_definition
`msing a = multiset (\b. if b = a then 1 else 0)`;;
let munion = new_definition
`M munion N = multiset(\b. multiplicity M b + multiplicity N b)`;;
let mdiff = new_definition
`M mdiff N = multiset(\b. multiplicity M b - multiplicity N b)`;;
(* ------------------------------------------------------------------------- *)
(* Extensionality for multisets. *)
(* ------------------------------------------------------------------------- *)
let MEXTENSION = prove
(`(M = N) = !a. multiplicity M a = multiplicity N a`,
REWRITE_TAC[GSYM FUN_EQ_THM] THEN CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN
MESON_TAC[multiset_tybij]);;
(* ------------------------------------------------------------------------- *)
(* Basic properties of multisets. *)
(* ------------------------------------------------------------------------- *)
let MULTIPLICITY_MULTISET = prove
(`FINITE {a | ~(f a = 0)} /\ (f a = y) ==> (multiplicity(multiset f) a = y)`,
SIMP_TAC[multiset_tybij]);;
let MEMPTY = prove
(`multiplicity mempty a = 0`,
REWRITE_TAC[mempty] THEN MATCH_MP_TAC MULTIPLICITY_MULTISET THEN
SIMP_TAC[EMPTY_IS_FINITE; EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY]);;
let MSING = prove
(`multiplicity (msing (a:A)) b = if b = a then 1 else 0`,
REWRITE_TAC[msing] THEN MATCH_MP_TAC MULTIPLICITY_MULTISET THEN
REWRITE_TAC[] THEN MATCH_MP_TAC SING_IS_FINITE THEN EXISTS_TAC `a:A` THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INSERT; NOT_IN_EMPTY] THEN
GEN_TAC THEN COND_CASES_TAC THEN REWRITE_TAC[ARITH_EQ]);;
let MUNION = prove
(`multiplicity (M munion N) a = multiplicity M a + multiplicity N a`,
REWRITE_TAC[munion] THEN MATCH_MP_TAC MULTIPLICITY_MULTISET THEN
REWRITE_TAC[UNION_NONZERO; FINITE_UNION] THEN SIMP_TAC[multiset_tybij] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN REWRITE_TAC[multiset_tybij]);;
let MDIFF = prove
(`multiplicity (M mdiff N) (a:A) = multiplicity M a - multiplicity N a`,
REWRITE_TAC[mdiff] THEN MATCH_MP_TAC MULTIPLICITY_MULTISET THEN
REWRITE_TAC[] THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{a:A | ~(multiplicity M a = 0)}` THEN
SIMP_TAC[SUBSET; IN_ELIM_THM; multiset_tybij] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN REWRITE_TAC[multiset_tybij] THEN
ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Some trivial properties of multisets that we use later. *)
(* ------------------------------------------------------------------------- *)
let MUNION_MEMPTY = prove
(`~(M munion (msing(a:A)) = mempty)`,
REWRITE_TAC[MEXTENSION; MEMPTY; MSING; MUNION] THEN
DISCH_THEN(MP_TAC o SPEC `a:A`) THEN
REWRITE_TAC[ADD_EQ_0; ARITH_EQ]);;
let MMEMBER_MUNION = prove
(`x mmember (M munion N) <=> x mmember M \/ x mmember N`,
REWRITE_TAC[mmember; MUNION; ADD_EQ_0; DE_MORGAN_THM]);;
let MMEMBER_MSING = prove
(`x mmember (msing a) <=> (x = a)`,
REWRITE_TAC[mmember; MSING] THEN COND_CASES_TAC THEN REWRITE_TAC[ARITH_EQ]);;
let MUNION_EMPTY = prove
(`M munion mempty = M`,
REWRITE_TAC[MEXTENSION; MUNION; MEMPTY; ADD_CLAUSES]);;
let MUNION_ASSOC = prove
(`M1 munion (M2 munion M3) = (M1 munion M2) munion M3`,
REWRITE_TAC[MEXTENSION; MUNION; ADD_ASSOC]);;
let MUNION_AC = prove
(`(M1 munion M2 = M2 munion M1) /\
((M1 munion M2) munion M3 = M1 munion M2 munion M3) /\
(M1 munion M2 munion M3 = M2 munion M1 munion M3)`,
REWRITE_TAC[MEXTENSION; MUNION; ADD_AC]);;
let MUNION_11 = prove
(`(M1 munion N = M2 munion N) <=> (M1 = M2)`,
REWRITE_TAC[MEXTENSION; MUNION; EQ_ADD_RCANCEL]);;
let MUNION_INUNION = prove
(`a mmember (M munion (msing b)) /\ ~(b = a) ==> a mmember M`,
REWRITE_TAC[mmember; MUNION; MSING; ADD_EQ_0] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[ARITH_EQ]);;
let MMEMBER_MDIFF = prove
(`(a:A) mmember M ==> (M = (M mdiff (msing a)) munion (msing a))`,
REWRITE_TAC[mmember; MEXTENSION; MUNION; MDIFF; MSING] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(multiplicity M (a:A) = 0)` THEN ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Induction principle for multisets. *)
(* ------------------------------------------------------------------------- *)
let MULTISET_INDUCT_LEMMA1 = prove
(`(!M. ({a | ~(multiplicity M a = 0)} SUBSET s) ==> P M) /\
(!a:A M. P M ==> P (M munion (msing a)))
==> !n M. (multiplicity M a = n) /\
{a:A | ~(multiplicity M a = 0)} SUBSET (a INSERT s)
==> P M`,
STRIP_TAC THEN INDUCT_TAC THEN REPEAT STRIP_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
UNDISCH_TAC `{a:A | ~(multiplicity M a = 0)} SUBSET (a INSERT s)` THEN
REWRITE_TAC[SUBSET; IN_ELIM_THM; IN_INSERT] THEN ASM_MESON_TAC[];
SUBGOAL_THEN `M = (M mdiff (msing(a:A))) munion (msing a)` SUBST1_TAC THENL
[MATCH_MP_TAC MMEMBER_MDIFF THEN ASM_REWRITE_TAC[mmember; NOT_SUC];
ALL_TAC] THEN
MAP_EVERY (MATCH_MP_TAC o ASSUME)
[`!a:A M. P M ==> P (M munion msing a)`;
`!M. (multiplicity M a = n) /\
{a:A | ~(multiplicity M a = 0)} SUBSET (a INSERT s)
==> P M`] THEN
ASM_REWRITE_TAC[MDIFF; MSING; ARITH_RULE `SUC n - 1 = n`] THEN
MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `{a:A | ~(multiplicity M a = 0)}` THEN
ASM_SIMP_TAC[SUBSET; IN_ELIM_THM; CONTRAPOS_THM; SUB_0]]);;
let MULTISET_INDUCT_LEMMA2 = prove
(`P mempty /\
(!a:A M. P M ==> P (M munion (msing a)))
==> !s. FINITE s ==> !M. {a:A | ~(multiplicity M a = 0)} SUBSET s ==> P M`,
STRIP_TAC THEN MATCH_MP_TAC FINITE_INDUCT THEN CONJ_TAC THENL
[REWRITE_TAC[SUBSET; IN_ELIM_THM; NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `M:(A)multiset = mempty` (fun th -> ASM_REWRITE_TAC[th]) THEN
ASM_REWRITE_TAC[MEXTENSION; MEMPTY]; X_GEN_TAC `a:A`] THEN
REPEAT STRIP_TAC THEN MP_TAC MULTISET_INDUCT_LEMMA1 THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[GSYM EXISTS_REFL]);;
let MULTISET_INDUCT = prove
(`P mempty /\
(!a:A M. P M ==> P (M munion (msing a)))
==> !M. P M`,
DISCH_THEN(MP_TAC o MATCH_MP MULTISET_INDUCT_LEMMA2) THEN
REWRITE_TAC[RIGHT_IMP_FORALL_THM] THEN
REWRITE_TAC[IMP_IMP] THEN
GEN_TAC THEN DISCH_THEN MATCH_MP_TAC THEN
EXISTS_TAC `{a:A | ~(multiplicity M a = 0)}` THEN
REWRITE_TAC[SUBSET_REFL; multiset_tybij] THEN
CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN REWRITE_TAC[multiset_tybij]);;
(* ========================================================================= *)
(* Part 2: Transcription of Tobias's paper. *)
(* ========================================================================= *)
parse_as_infix("<<",(12,"right"));;
(* ------------------------------------------------------------------------- *)
(* Wellfounded part of a relation. *)
(* ------------------------------------------------------------------------- *)
let WFP_RULES,WFP_INDUCT,WFP_CASES = new_inductive_definition
`!x. (!y. y << x ==> WFP(<<) y) ==> WFP(<<) x`;;
(* ------------------------------------------------------------------------- *)
(* Wellfounded part induction. *)
(* ------------------------------------------------------------------------- *)
let WFP_PART_INDUCT = prove
(`!P. (!x. x IN WFP(<<) /\ (!y. y << x ==> P(y)) ==> P(x))
==> !x:A. x IN WFP(<<) ==> P(x)`,
GEN_TAC THEN REWRITE_TAC[IN] THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `a ==> b <=> a ==> a /\ b`] THEN
MATCH_MP_TAC WFP_INDUCT THEN ASM_MESON_TAC[WFP_RULES]);;
(* ------------------------------------------------------------------------- *)
(* A relation is wellfounded iff WFP is the whole universe. *)
(* ------------------------------------------------------------------------- *)
let WFP_WF = prove
(`WF(<<) <=> (WFP(<<) = UNIV:A->bool)`,
EQ_TAC THENL
[REWRITE_TAC[WF_IND; EXTENSION; IN; UNIV] THEN MESON_TAC[WFP_RULES];
DISCH_TAC THEN MP_TAC WFP_PART_INDUCT THEN
ASM_REWRITE_TAC[IN; UNIV; WF_IND]]);;
(* ------------------------------------------------------------------------- *)
(* The multiset order. *)
(* ------------------------------------------------------------------------- *)
let morder = new_definition
`morder(<<) N M <=> ?M0 a K. (M = M0 munion (msing a)) /\
(N = M0 munion K) /\
(!b. b mmember K ==> b << a)`;;
(* ------------------------------------------------------------------------- *)
(* We separate off this part from the proof of LEMMA_2_1. *)
(* ------------------------------------------------------------------------- *)
let LEMMA_2_0 = prove
(`morder(<<) N (M0 munion (msing a))
==> (?M. morder(<<) M M0 /\ (N = M munion (msing a))) \/
(?K. (N = M0 munion K) /\ (!b:A. b mmember K ==> b << a))`,
GEN_REWRITE_TAC LAND_CONV [morder] THEN
DISCH_THEN(EVERY_TCL (map X_CHOOSE_THEN
[`M1:(A)multiset`; `b:A`; `K:(A)multiset`]) STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `b:A = a` THENL
[DISJ2_TAC THEN UNDISCH_THEN `b:A = a` SUBST_ALL_TAC THEN
EXISTS_TAC `K:(A)multiset` THEN ASM_MESON_TAC[MUNION_11]; DISJ1_TAC] THEN
SUBGOAL_THEN `?M2. M1 = M2 munion (msing(a:A))` STRIP_ASSUME_TAC THENL
[EXISTS_TAC `M1 mdiff (msing(a:A))` THEN
MAP_EVERY MATCH_MP_TAC [MMEMBER_MDIFF; MUNION_INUNION] THEN
UNDISCH_TAC `M0 munion (msing a) = M1 munion (msing(b:A))` THEN
ASM_REWRITE_TAC[MEXTENSION; MUNION; MSING; mmember] THEN
DISCH_THEN(MP_TAC o SPEC `a:A`) THEN ASM_REWRITE_TAC[] THEN
ARITH_TAC; ALL_TAC] THEN
EXISTS_TAC `M2 munion K:(A)multiset` THEN ASM_REWRITE_TAC[MUNION_AC] THEN
REWRITE_TAC[morder] THEN
MAP_EVERY EXISTS_TAC [`M2:(A)multiset`; `b:A`; `K:(A)multiset`] THEN
UNDISCH_TAC `M0 munion msing (a:A) = M1 munion msing b` THEN
ASM_REWRITE_TAC[MUNION_AC] THEN MESON_TAC[MUNION_AC; MUNION_11]);;
(* ------------------------------------------------------------------------- *)
(* The sequence of lemmas from Tobias's paper. *)
(* ------------------------------------------------------------------------- *)
let LEMMA_2_1 = prove
(`(!M b:A. b << a /\ M IN WFP(morder(<<))
==> (M munion (msing b)) IN WFP(morder(<<))) /\
M0 IN WFP(morder(<<)) /\
(!M. morder(<<) M M0 ==> (M munion (msing a)) IN WFP(morder(<<)))
==> (M0 munion (msing a)) IN WFP(morder(<<))`,
STRIP_TAC THEN REWRITE_TAC[IN] THEN MATCH_MP_TAC WFP_RULES THEN
X_GEN_TAC `N:(A)multiset` THEN
DISCH_THEN(DISJ_CASES_THEN MP_TAC o MATCH_MP LEMMA_2_0) THENL
[ASM_MESON_TAC[IN]; REWRITE_TAC[LEFT_IMP_EXISTS_THM]] THEN
SPEC_TAC(`N:(A)multiset`,`N:(A)multiset`) THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
MATCH_MP_TAC MULTISET_INDUCT THEN REPEAT STRIP_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[MUNION_ASSOC; MMEMBER_MUNION; MMEMBER_MSING]) THEN
ASM_MESON_TAC[IN; MUNION_EMPTY]);;
let LEMMA_2_2 = prove
(`(!M b. b << a /\ M IN WFP(morder(<<))
==> (M munion (msing b)) IN WFP(morder(<<)))
==> !M. M IN WFP(morder(<<)) ==> (M munion (msing a)) IN WFP(morder(<<))`,
STRIP_TAC THEN MATCH_MP_TAC WFP_PART_INDUCT THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC LEMMA_2_1 THEN ASM_REWRITE_TAC[]);;
let LEMMA_2_3 = prove
(`WF(<<)
==> !a M. M IN WFP(morder(<<)) ==> (M munion (msing a)) IN WFP(morder(<<))`,
REWRITE_TAC[WF_IND] THEN DISCH_THEN MATCH_MP_TAC THEN MESON_TAC[LEMMA_2_2]);;
let LEMMA_2_4 = prove
(`WF(<<) ==> !M. M IN WFP(morder(<<))`,
DISCH_TAC THEN MATCH_MP_TAC MULTISET_INDUCT THEN CONJ_TAC THENL
[REWRITE_TAC[IN] THEN MATCH_MP_TAC WFP_RULES THEN
REWRITE_TAC[morder; MUNION_MEMPTY];
ASM_SIMP_TAC[LEMMA_2_3]]);;
(* ------------------------------------------------------------------------- *)
(* Hence the final result. *)
(* ------------------------------------------------------------------------- *)
let MORDER_WF = prove
(`WF(<<) ==> WF(morder(<<))`,
SIMP_TAC[WFP_WF; EXTENSION; IN_UNIV; LEMMA_2_4]);;
|
