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let rec DISJ_TAC thm = DISJ_CASES_TAC thm THENL[ALL_TAC;TRY (POP_ASSUM DISJ_TAC)];;
let INTERPSIGNS_CONJ = prove_by_refinement(
`!P Q eqs l.
interpsigns eqs (\x. P x) l /\
interpsigns eqs (\x. Q x) l ==>
interpsigns eqs (\x. P x \/ Q x) l`,
(* {{{ Proof *)
[
STRIP_TAC THEN STRIP_TAC;
REPEAT LIST_INDUCT_TAC THEN REWRITE_TAC[interpsigns;ALL2;interpsign];
REPEAT (POP_ASSUM MP_TAC);
DISJ_TAC (ISPEC `h':sign` SIGN_CASES) THEN ASM_REWRITE_TAC[interpsign;interpsigns] THEN REPEAT STRIP_TAC THEN ASM_MESON_TAC[];
]);;
(* }}} *)
let INTERPMAT_TRIO = prove_by_refinement(
`!eqs x y l r t.
interpmat (CONS x (CONS y t)) eqs (CONS l (CONS l (CONS l r))) ==>
interpmat (CONS y t) eqs (CONS l r)`,
(* {{{ Proof *)
[
REWRITE_TAC[interpmat;partition_line;NOT_CONS_NIL;ALL2;HD;TL;APPEND];
REPEAT_N 6 STRIP_TAC;
DISJ_CASES_TAC (ISPEC `t:real list` list_CASES);
ASM_REWRITE_TAC[];
REPEAT STRIP_TAC;
MATCH_ACCEPT_TAC ROL_SING;
REWRITE_TAC[ALL2];
REWRITE_ASSUMS[TL];
STRIP_TAC;
MATCH_MP_TAC INTERPSIGNS_SUBSET;
EXISTS_TAC `\z. z < x \/ (z = x) \/ (x < z /\ z < y)`;
STRIP_TAC;
MATCH_MP_TAC INTERPSIGNS_CONJ;
ASM_REWRITE_TAC[];
MATCH_MP_TAC INTERPSIGNS_CONJ;
ASM_REWRITE_TAC[];
REWRITE_TAC[SUBSET;IN];
REAL_ARITH_TAC;
FIRST_ASSUM MATCH_ACCEPT_TAC;
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[NOT_CONS_NIL;TL];
REPEAT STRIP_TAC;
ASM_MESON_TAC[ROL_TAIL;TL;NOT_CONS_NIL;];
REWRITE_TAC[ALL2];
ASM_REWRITE_TAC[];
MATCH_MP_TAC INTERPSIGNS_SUBSET;
EXISTS_TAC `\z. z < x \/ (z = x) \/ (x < z /\ z < y)`;
STRIP_TAC;
MATCH_MP_TAC INTERPSIGNS_CONJ;
ASM_REWRITE_TAC[];
MATCH_MP_TAC INTERPSIGNS_CONJ;
ASM_REWRITE_TAC[];
REWRITE_TAC[SUBSET;IN];
REAL_ARITH_TAC;
]);;
(* }}} *)
let PARTITION_LINE_NOT_NIL = prove_by_refinement(
`!l. ~(partition_line l = [])`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[partition_line;NOT_CONS_NIL;];
REWRITE_TAC[partition_line];
COND_CASES_TAC;
REWRITE_TAC[NOT_CONS_NIL];
ASM_MESON_TAC[APPEND_EQ_NIL;NOT_CONS_NIL];
]);;
(* }}} *)
let ALL2_LENGTH = prove_by_refinement(
`!P l1 l2. ALL2 P l1 l2 ==> (LENGTH l1 = LENGTH l2)`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC THEN REWRITE_TAC[ALL2;LENGTH];
ASM_MESON_TAC[];
]);;
(* }}} *)
let LENGTH_TL = prove_by_refinement(
`!l:A list. ~(l = []) ==> (LENGTH (TL l) = PRE (LENGTH l))`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[NOT_CONS_NIL;TL;LENGTH;];
ARITH_TAC;
]);;
(* }}} *)
let PARTITION_LINE_LENGTH = prove_by_refinement(
`!l. LENGTH (partition_line l) = 2 * LENGTH l + 1`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[partition_line;LENGTH;];
ARITH_TAC;
REWRITE_TAC[partition_line;LENGTH;];
COND_CASES_TAC;
ASM_REWRITE_TAC[LENGTH;];
ARITH_TAC;
REWRITE_TAC[APPEND;LENGTH;];
ASM_SIMP_TAC[PARTITION_LINE_NOT_NIL;LENGTH_TL];
ARITH_TAC;
]);;
(* }}} *)
let PARTITION_LINE_LENGTH_TL = prove_by_refinement(
`!l. LENGTH (TL (partition_line l)) = 2 * LENGTH l`,
(* {{{ Proof *)
[
STRIP_TAC;
REWRITE_TAC[MATCH_MP LENGTH_TL (ISPEC `l:real list` PARTITION_LINE_NOT_NIL)];
REWRITE_TAC[PARTITION_LINE_LENGTH];
ARITH_TAC;
]);;
(* }}} *)
let PL_ALL2_LENGTH = prove_by_refinement(
`!eqs pts sgns. ALL2 (interpsigns eqs) (partition_line pts) sgns ==>
(LENGTH sgns = 2 * LENGTH pts + 1)`,
(* {{{ Proof *)
[
REPEAT_N 3 STRIP_TAC;
DISJ_CASES_TAC (ISPEC `pts:real list` list_CASES);
ASM_REWRITE_TAC[interpmat;LENGTH;ROL_NIL;partition_line;];
ARITH_SIMP_TAC[];
DISJ_CASES_TAC (ISPEC `sgns:(sign list) list` list_CASES);
ASM_REWRITE_TAC[ALL2];
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ALL2];
DISJ_CASES_TAC (ISPEC `t:(sign list) list` list_CASES);
ASM_REWRITE_TAC[ALL2;LENGTH;ONE];
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ALL2];
(* save *)
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[interpmat;partition_line;];
COND_CASES_TAC;
ASM_REWRITE_TAC[ROL_SING;LENGTH;GSYM ONE];
ARITH_SIMP_TAC[];
STRIP_TAC;
CLAIM `LENGTH [\x. x < h; \x. x = h; \x. h < x] = LENGTH sgns`;
ASM_MESON_TAC[ALL2_LENGTH];
REWRITE_TAC[LENGTH];
ARITH_TAC;
REWRITE_ASSUMS[NOT_NIL];
POP_ASSUM MP_TAC THEN STRIP_TAC;
REWRITE_TAC[LENGTH];
STRIP_TAC;
CLAIM `LENGTH sgns = LENGTH (APPEND [\x. x < h; \x. x = h; \x. h < x /\ x < HD t] (TL (partition_line t)))`;
ASM_MESON_TAC[ ALL2_LENGTH];
DISCH_THEN SUBST1_TAC;
REWRITE_TAC[LENGTH_APPEND];
REWRITE_TAC[PARTITION_LINE_LENGTH_TL];
REWRITE_TAC[LENGTH];
ARITH_TAC;
]);;
(* }}} *)
let INTERPMAT_LENGTH = prove_by_refinement(
`!eqs pts sgns. interpmat pts eqs sgns ==>
(LENGTH sgns = 2 * LENGTH pts + 1)`,
(* {{{ Proof *)
[
REPEAT_N 3 STRIP_TAC;
DISJ_CASES_TAC (ISPEC `pts:real list` list_CASES);
ASM_REWRITE_TAC[interpmat;LENGTH;ROL_NIL;partition_line;];
ARITH_SIMP_TAC[];
DISJ_CASES_TAC (ISPEC `sgns:(sign list) list` list_CASES);
ASM_REWRITE_TAC[ALL2];
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ALL2];
DISJ_CASES_TAC (ISPEC `t:(sign list) list` list_CASES);
ASM_REWRITE_TAC[ALL2;LENGTH;ONE];
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ALL2];
(* save *)
POP_ASSUM MP_TAC THEN STRIP_TAC;
ASM_REWRITE_TAC[interpmat;partition_line;];
COND_CASES_TAC;
ASM_REWRITE_TAC[ROL_SING;LENGTH;GSYM ONE];
ARITH_SIMP_TAC[];
STRIP_TAC;
CLAIM `LENGTH [\x. x < h; \x. x = h; \x. h < x] = LENGTH sgns`;
ASM_MESON_TAC[ALL2_LENGTH];
REWRITE_TAC[LENGTH];
ARITH_TAC;
REWRITE_ASSUMS[NOT_NIL];
POP_ASSUM MP_TAC THEN STRIP_TAC;
REWRITE_TAC[LENGTH];
STRIP_TAC;
CLAIM `LENGTH sgns = LENGTH (APPEND [\x. x < h; \x. x = h; \x. h < x /\ x < HD t] (TL (partition_line t)))`;
ASM_MESON_TAC[ ALL2_LENGTH];
DISCH_THEN SUBST1_TAC;
REWRITE_TAC[LENGTH_APPEND];
REWRITE_TAC[PARTITION_LINE_LENGTH_TL];
REWRITE_TAC[LENGTH];
ARITH_TAC;
]);;
(* }}} *)
let ALL2_HD = prove_by_refinement(
`!b d a c. (LENGTH a = LENGTH c) ==>
ALL2 P (APPEND a b) (APPEND c d) ==> ALL2 P a c`,
(* {{{ Proof *)
[
REPEAT_N 2 STRIP_TAC;
LIST_INDUCT_TAC;
ONCE_REWRITE_TAC[prove(`(x = y) <=> (y = x)`,MESON_TAC[])];
REWRITE_TAC[LENGTH;LENGTH_EQ_NIL];
MESON_TAC[ALL2];
REWRITE_TAC[LENGTH;APPEND;];
LIST_INDUCT_TAC;
REWRITE_TAC[LENGTH];
ARITH_TAC;
REWRITE_TAC[LENGTH;APPEND;ALL2;SUC_INJ];
ASM_MESON_TAC[];
]);;
(* }}} *)
let ALL2_TL = prove_by_refinement(
`!b d a c. (LENGTH a = LENGTH c) ==>
ALL2 P (APPEND a b) (APPEND c d) ==> ALL2 P b d`,
(* {{{ Proof *)
[
REPEAT_N 2 STRIP_TAC;
LIST_INDUCT_TAC;
ONCE_REWRITE_TAC[prove(`(x = y) <=> (y = x)`,MESON_TAC[])];
REWRITE_TAC[LENGTH;LENGTH_EQ_NIL];
MESON_TAC[APPEND];
REWRITE_TAC[LENGTH;APPEND;];
LIST_INDUCT_TAC;
REWRITE_TAC[LENGTH];
ARITH_TAC;
REWRITE_TAC[ALL2;APPEND;LENGTH;SUC_INJ];
ASM_MESON_TAC[];
]);;
(* }}} *)
let ALL2_APPEND_LENGTH = prove_by_refinement(
`!P a c b d. (LENGTH a = LENGTH c) ==>
ALL2 P (APPEND a b) (APPEND c d) ==> ALL2 P a c /\ ALL2 P b d`,
(* {{{ Proof *)
[
ASM_MESON_TAC[ALL2_HD;ALL2_TL];
]);;
(* }}} *)
let ALL2_APPEND = prove_by_refinement(
`!a c b d. ALL2 P a c /\ ALL2 P b d ==>
ALL2 P (APPEND a b) (APPEND c d)`,
(* {{{ Proof *)
[
REPEAT LIST_INDUCT_TAC THEN REWRITE_ALL[APPEND;ALL2;LENGTH;ARITH_RULE `~(0 = SUC x)`;APPEND_NIL];
REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
FIRST_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[ALL2];
]);;
(* }}} *)
let ALL2_SPLIT = prove_by_refinement(
`!a c b d. (LENGTH a = LENGTH c) ==>
(ALL2 P (APPEND a b) (APPEND c d) <=> ALL2 P a c /\ ALL2 P b d)`,
(* {{{ Proof *)
[
ASM_MESON_TAC[ALL2_APPEND;ALL2_APPEND_LENGTH];
]);;
(* }}} *)
let BUTLAST_THM = prove_by_refinement(
`(BUTLAST [] = []) /\
(BUTLAST [x] = []) /\
(BUTLAST (CONS h1 (CONS h2 t)) = CONS h1 (BUTLAST (CONS h2 t)))`,
(* {{{ Proof *)
[
ASM_MESON_TAC[BUTLAST;NOT_CONS_NIL;];
]);;
(* }}} *)
let HD_BUTLAST = prove_by_refinement(
`!l. ~(l = []) ==> (!x. ~(l = [x])) ==> (HD (BUTLAST l) = HD l)`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[NOT_CONS_NIL;HD;BUTLAST];
COND_CASES_TAC;
ASM_REWRITE_TAC[];
ASM_MESON_TAC[];
REPEAT STRIP_TAC;
REWRITE_TAC[HD];
]);;
(* }}} *)
let SUBLIST = new_recursive_definition list_RECURSION
`(SUBLIST l [] <=> (l = [])) /\
(SUBLIST l (CONS h t) <=> (l = []) \/
SUBLIST l t \/
((HD l = h) /\ SUBLIST (TL l) t))`;;
let SUBLIST_NIL = prove_by_refinement(
`!l. SUBLIST [] l`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC THEN
ASM_MESON_TAC[SUBLIST];
]);;
(* }}} *)
let SUBLIST_CONS = prove_by_refinement(
`!l1 l2 h. SUBLIST l1 l2 ==> SUBLIST l1 (CONS h l2)`,
(* {{{ Proof *)
[
REPEAT LIST_INDUCT_TAC THEN ASM_MESON_TAC[SUBLIST];
]);;
(* }}} *)
let SUBLIST_TL = prove_by_refinement(
`!l1 l2. SUBLIST l1 l2 ==> ~(l1 = []) ==> SUBLIST (TL l1) l2`,
(* {{{ Proof *)
[
REPEAT LIST_INDUCT_TAC THEN ASM_MESON_TAC[SUBLIST;]
]);;
(* }}} *)
let SUBLIST_CONS2 = prove_by_refinement(
`!h t l. SUBLIST (CONS h t) l ==> SUBLIST t l`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC THEN ASM_MESON_TAC[SUBLIST;NOT_CONS_NIL;HD;TL];
]);;
(* }}} *)
let SUBLIST_ID = prove_by_refinement(
`!l. SUBLIST l l`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC THEN ASM_MESON_TAC[SUBLIST;SUBLIST_NIL;NOT_CONS_NIL;HD;TL];
]);;
(* }}} *)
let SUBLIST_CONS_CONS = prove_by_refinement(
`!h t1 t2. SUBLIST (CONS h t1) (CONS h t2) = SUBLIST t1 t2`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
ASM_MESON_TAC[SUBLIST;SUBLIST_NIL;SUBLIST_ID];
ASM_MESON_TAC[SUBLIST;SUBLIST_NIL;SUBLIST_ID];
REWRITE_TAC[SUBLIST;SUBLIST_NIL;SUBLIST_ID;NOT_CONS_NIL;HD;TL];
REWRITE_TAC[SUBLIST;SUBLIST_NIL;SUBLIST_ID;NOT_CONS_NIL;HD;TL;SUBLIST_CONS2;SUBLIST_CONS];
ASM_MESON_TAC[SUBLIST;SUBLIST_NIL;SUBLIST_ID;NOT_CONS_NIL;HD;TL];
]);;
(* }}} *)
let SUBLIST_NEQ = prove_by_refinement(
`!h1 h2 t1 t2. SUBLIST (CONS h1 t1) (CONS h2 t2) ==> ~(h1 = h2) ==>
SUBLIST (CONS h1 t1) t2`,
(* {{{ Proof *)
[
ASM_MESON_TAC[SUBLIST;NOT_CONS_NIL;HD;TL];
]);;
(* }}} *)
let SUBLIST_TRANS = prove_by_refinement(
`!l1 l2 l3. SUBLIST l1 l2 ==> SUBLIST l2 l3 ==> SUBLIST l1 l3`,
(* {{{ Proof *)
[
REPEAT LIST_INDUCT_TAC;
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
ASM_MESON_TAC[SUBLIST];
REPEAT STRIP_TAC;
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;HD;TL];
CASES_ON `h = h''`;
DISJ2_TAC;
ASM_REWRITE_TAC[];
POP_ASSUM (REWRITE_ALL o list);
CASES_ON `h' = h''`;
POP_ASSUM (REWRITE_ALL o list);
ASM_MESON_TAC[SUBLIST_CONS_CONS];
REWRITE_ASSUMS[IMP_AND_THM];
FIRST_ASSUM MATCH_MP_TAC;
EVERY_ASSUM (fun x -> try MP_TAC (MATCH_MP SUBLIST_CONS2 x) with _ -> ALL_TAC);
REPEAT STRIP_TAC;
ASM_MESON_TAC[SUBLIST;NOT_CONS_NIL;HD;TL;SUBLIST_CONS;SUBLIST_CONS2];
DISJ1_TAC;
CASES_ON `h' = h''`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_ASSUMS[SUBLIST_CONS_CONS];
CLAIM `SUBLIST (CONS h t) t'`;
ASM_MESON_TAC[SUBLIST_NEQ];
STRIP_TAC;
ASM_MESON_TAC[];
CASES_ON `h = h'`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_ASSUMS[SUBLIST_CONS_CONS];
ASM_MESON_TAC[SUBLIST_NEQ];
CLAIM `SUBLIST (CONS h t) t'`;
ASM_MESON_TAC[SUBLIST_NEQ];
STRIP_TAC;
CLAIM `SUBLIST (CONS h' t') t''`;
ASM_MESON_TAC[SUBLIST_NEQ];
STRIP_TAC;
CLAIM `SUBLIST t' t''`;
ASM_MESON_TAC[SUBLIST_CONS2];
STRIP_TAC;
ASM_MESON_TAC[];
]);;
(* }}} *)
let ROL_MEM = prove_by_refinement(
`!h t. real_ordered_list (CONS h t) ==> !x. MEM x t ==> h < x`,
(* {{{ Proof *)
[
STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[MEM];
REPEAT STRIP_TAC;
CASES_ON `x = h'`;
POP_ASSUM (REWRITE_ALL o list);
ASM_MESON_TAC[ROL_CONS_CONS];
CLAIM `real_ordered_list (CONS h t)`;
ASM_MESON_TAC[ROL_CONS_CONS_DELETE];
DISCH_THEN (REWRITE_ASSUMS o list);
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[MEM];
]);;
(* }}} *)
let SUBLIST_MEM = prove_by_refinement(
`!x l1 l2. SUBLIST l1 l2 ==> MEM x l1 ==> MEM x l2`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
REWRITE_TAC[MEM];
REWRITE_TAC[MEM];
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
REPEAT STRIP_TAC;
CASES_ON `h = h'`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_ASSUMS[SUBLIST_CONS_CONS];
CASES_ON `x = h'`;
ASM_MESON_TAC[MEM];
REWRITE_ASSUMS[IMP_AND_THM];
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[MEM;SUBLIST_CONS];
CASES_ON `x = h'`;
ASM_MESON_TAC[MEM];
ASM_MESON_TAC[SUBLIST_NEQ;SUBLIST;MEM];
]);;
(* }}} *)
let ROL_SUBLIST_LT = prove_by_refinement(
`!h t1 t2. real_ordered_list (CONS h t2) ==>
SUBLIST (CONS h t1) (CONS h t2) ==> !x. MEM x t1 ==> h < x`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
REWRITE_TAC[MEM];
REWRITE_TAC[MEM];
ASM_MESON_TAC[SUBLIST;NOT_CONS_NIL;HD;TL];
REPEAT STRIP_TAC;
REWRITE_ASSUMS[SUBLIST_CONS_CONS];
CLAIM `MEM x (CONS h'' t')`;
ASM_MESON_TAC[SUBLIST_MEM];
STRIP_TAC;
ASM_MESON_TAC[ROL_MEM];
]);;
(* }}} *)
let SUBLIST_DELETE = prove_by_refinement(
`!h1 h2 t l. SUBLIST (CONS h1 (CONS h2 t)) l ==>
SUBLIST (CONS h1 t) l`,
(* {{{ Proof *)
[
STRIP_TAC THEN STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
CASES_ON `h1 = h`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;HD;TL;SUBLIST_NIL];
STRIP_TAC;
CLAIM `SUBLIST [h1; h2] t`;
ASM_MESON_TAC[SUBLIST_NEQ];
DISCH_THEN (REWRITE_ASSUMS o list);
ASM_MESON_TAC[SUBLIST_CONS];
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
CASES_ON `h1 = h'`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[SUBLIST_CONS_CONS];
MESON_TAC[SUBLIST_CONS2];
STRIP_TAC;
CLAIM `SUBLIST (CONS h1 (CONS h2 (CONS h t))) t'`;
ASM_MESON_TAC[SUBLIST_NEQ];
DISCH_THEN (REWRITE_ASSUMS o list);
ASM_MESON_TAC[SUBLIST_CONS];
]);;
(* }}} *)
let SUBLIST_MATCH = prove_by_refinement(
`!h t l. SUBLIST (CONS h t) l ==>
?(l1:A list) l2. (l = APPEND l1 (CONS h l2)) /\ SUBLIST t l2`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
CASES_ON `h = h'`;
POP_ASSUM (REWRITE_ALL o list);
STRIP_TAC;
EXISTS_TAC `[]`;
REWRITE_TAC[APPEND;SUBLIST_NIL];
ASM_MESON_TAC[];
REWRITE_TAC[SUBLIST_NIL];
STRIP_TAC;
CLAIM `SUBLIST [h] t`;
ASM_MESON_TAC[SUBLIST_NEQ];
DISCH_THEN (REWRITE_ASSUMS o list);
REPEAT (POP_ASSUM MP_TAC);
REPEAT STRIP_TAC;
EXISTS_TAC `CONS h' l1`;
EXISTS_TAC `l2`;
REWRITE_TAC[APPEND];
AP_TERM_TAC;
ASM_MESON_TAC[];
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
(* save *)
CASES_ON `h = h''`;
POP_ASSUM (REWRITE_ALL o list);
STRIP_TAC;
REWRITE_ASSUMS[SUBLIST_CONS_CONS];
EXISTS_TAC `[]:A list`;
EXISTS_TAC `t'`;
ASM_MESON_TAC[APPEND];
(* save *)
STRIP_TAC;
CLAIM `SUBLIST (CONS h (CONS h' t)) t'`;
ASM_MESON_TAC[SUBLIST_NEQ];
DISCH_THEN (REWRITE_ASSUMS o list);
REPEAT (POP_ASSUM MP_TAC) THEN REPEAT STRIP_TAC;
ASM_REWRITE_TAC[];
EXISTS_TAC `CONS h'' l1`;
EXISTS_TAC `l2`;
ASM_REWRITE_TAC[APPEND];
]);;
(* }}} *)
let ROL_SUBLIST = prove_by_refinement(
`!l1 l2. real_ordered_list l2 ==> SUBLIST l1 l2 ==> real_ordered_list l1`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[ROL_NIL];
REWRITE_TAC[real_ordered_list];
REPEAT STRIP_TAC;
REWRITE_ASSUMS[IMP_AND_THM];
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[SUBLIST_CONS2];
CASES_ON `t = []`;
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[];
REWRITE_ASSUMS[NOT_NIL];
POP_ASSUM MP_TAC THEN STRIP_TAC;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[HD];
DISJ_CASES_TAC (ISPEC `l2:real list` list_CASES);
ASM_MESON_TAC[SUBLIST;NOT_CONS_NIL];
POP_ASSUM MP_TAC THEN STRIP_TAC;
POP_ASSUM (REWRITE_ALL o list);
CASES_ON `h = h''`;
POP_ASSUM (REWRITE_ALL o list);
ASM_MESON_TAC[ROL_SUBLIST_LT;MEM];
FIRST_ASSUM (fun x -> MP_TAC (MATCH_MP SUBLIST_MATCH x));
STRIP_TAC;
CLAIM `real_ordered_list (CONS h l2')`;
ASM_MESON_TAC[ROL_APPEND];
STRIP_TAC;
CLAIM `MEM h' l2'`;
ASM_MESON_TAC[SUBLIST_MEM;MEM];
STRIP_TAC;
ASM_MESON_TAC[ROL_MEM];
]);;
(* }}} *)
let SUBLIST_BUTLAST = prove_by_refinement(
`!l. SUBLIST (BUTLAST l) l`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[BUTLAST;SUBLIST_NIL];
REWRITE_TAC[BUTLAST;SUBLIST_NIL;SUBLIST];
REPEAT COND_CASES_TAC;
REWRITE_TAC[SUBLIST_NIL];
ASM_REWRITE_TAC[HD;TL;NOT_CONS_NIL;];
]);;
(* }}} *)
let SUBLIST_APPEND_HD = prove_by_refinement(
`!l2 l3 l1. SUBLIST (APPEND l1 l2) (APPEND l1 l3) = SUBLIST l2 l3`,
(* {{{ Proof *)
[
REPEAT_N 2 STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[APPEND];
ASM_REWRITE_TAC[APPEND;SUBLIST_CONS_CONS];
]);;
(* }}} *)
let SUBLIST_ID_CONS = prove_by_refinement(
`!h l. ~(SUBLIST (CONS h l) l)`,
(* {{{ Proof *)
[
STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[SUBLIST;NOT_CONS_NIL;];
ASM_REWRITE_TAC[SUBLIST;NOT_CONS_NIL;HD;TL];
ASM_MESON_TAC[SUBLIST_DELETE];
]);;
(* }}} *)
let SUBLIST_ID_APPEND = prove_by_refinement(
`!m l. ~(l = []) ==> ~(SUBLIST (APPEND l m) m)`,
(* {{{ Proof *)
[
STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[APPEND;];
DISCH_THEN (fun x -> ALL_TAC);
CASES_ON `t = []`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[APPEND;SUBLIST_ID_CONS];
POP_ASSUM (fun x -> REWRITE_ASSUMS[x] THEN ASSUME_TAC x);
ASM_MESON_TAC[SUBLIST_CONS2];
]);;
(* }}} *)
let SUBLIST_APPEND_TL = prove_by_refinement(
`!l3 l1 l2. SUBLIST (APPEND l1 l3) (APPEND l2 l3) = SUBLIST l1 l2`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC;
REWRITE_TAC[APPEND;APPEND_NIL;SUBLIST;SUBLIST_ID];
REWRITE_ALL[SUBLIST_NIL;APPEND;APPEND_NIL;SUBLIST;SUBLIST_ID];
ASM_REWRITE_TAC[];
REWRITE_ALL[SUBLIST_NIL;SUBLIST;SUBLIST_ID;NOT_CONS_NIL;];
ASM_MESON_TAC[SUBLIST_ID_APPEND;APPEND;NOT_CONS_NIL;];
REWRITE_TAC[APPEND];
CASES_ON `h = h'`;
POP_ASSUM (REWRITE_ALL o list);
EQ_TAC;
REWRITE_TAC[SUBLIST;APPEND;HD;TL;NOT_CONS_NIL;];
STRIP_TAC;
ASM_MESON_TAC[APPEND;];
ASM_MESON_TAC[APPEND;];
REWRITE_TAC[SUBLIST_CONS_CONS];
ASM_MESON_TAC[];
EQ_TAC;
STRIP_TAC;
MATCH_MP_TAC SUBLIST_CONS;
CLAIM `SUBLIST (CONS h (APPEND t l3)) (APPEND t' l3)`;
ASM_MESON_TAC[SUBLIST_NEQ];
STRIP_TAC;
ASM_MESON_TAC[APPEND;];
ASM_REWRITE_TAC[NOT_CONS_NIL;SUBLIST;HD;TL];
STRIP_TAC;
ASM_MESON_TAC[APPEND;];
]);;
(* }}} *)
let SUBLIST_TRANS2 = REWRITE_RULE[IMP_AND_THM] SUBLIST_TRANS;;
let APPEND_CONS = prove_by_refinement(
`!h l1 l2. APPEND l1 (CONS h l2) = APPEND (APPEND l1 [h]) l2`,
(* {{{ Proof *)
[
STRIP_TAC;
REPEAT LIST_INDUCT_TAC THEN REWRITE_TAC[APPEND_NIL;APPEND];
ASM_MESON_TAC[];
]);;
(* }}} *)
let SUBLIST_APPEND = prove_by_refinement(
`!l1 l2 m1 m2. SUBLIST l1 l2 ==> SUBLIST m1 m2 ==>
SUBLIST (APPEND l1 m1) (APPEND l2 m2)`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[SUBLIST_NIL;APPEND];
LIST_INDUCT_TAC;
REWRITE_TAC[APPEND];
REPEAT STRIP_TAC;
POP_ASSUM (fun x -> FIRST_ASSUM (fun y -> ASSUME_TAC (MATCH_MP y x) THEN ASSUME_TAC x));
REWRITE_TAC[APPEND];
ASM_MESON_TAC[SUBLIST_CONS];
LIST_INDUCT_TAC;
MESON_TAC[SUBLIST;NOT_CONS_NIL];
CASES_ON `h = h'`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[SUBLIST_CONS_CONS];
REWRITE_TAC[SUBLIST_CONS_CONS;APPEND;];
ASM_MESON_TAC[];
REPEAT STRIP_TAC;
CLAIM `SUBLIST (CONS h t) t'`;
ASM_MESON_TAC[SUBLIST_NEQ];
DISCH_THEN (fun x -> FIRST_ASSUM (fun y -> MP_TAC (MATCH_MP y x)));
POP_ASSUM (fun x -> DISCH_THEN (fun y -> MP_TAC (MATCH_MP y x)));
ASM_MESON_TAC[APPEND;SUBLIST_CONS];
]);;
(* }}} *)
let SUBLIST_APPEND2 = REWRITE_RULE[IMP_AND_THM] SUBLIST_APPEND;;
let ROL_APPEND2 = prove_by_refinement(
`!l2 l1. real_ordered_list (APPEND l1 l2) ==>
real_ordered_list (APPEND l1 (BUTLAST l2))`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
MATCH_MP_TAC (REWRITE_RULE[IMP_AND_THM] ROL_SUBLIST);
EXISTS_TAC `APPEND l1 l2`;
ASM_REWRITE_TAC[SUBLIST_APPEND_HD;SUBLIST_BUTLAST];
]);;
(* }}} *)
let PL_LEM = prove_by_refinement(
`!l. ~(l = []) ==> ~(TL (partition_line l) = [])`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
STRIP_TAC;
REWRITE_TAC[partition_line];
ASM_MESON_TAC[NOT_CONS_NIL;APPEND;TL];
]);;
(* }}} *)
let HD_APPEND2 = prove_by_refinement(
`!l m. ~(l = []) ==> (HD (APPEND l m) = HD l)`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
REPEAT STRIP_TAC;
REWRITE_TAC[APPEND;HD];
]);;
(* }}} *)
let BUTLAST_TL = prove_by_refinement(
`!l. LENGTH l > 1 ==> (BUTLAST (TL l) = TL (BUTLAST l))`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[LENGTH] THEN ARITH_TAC;
REWRITE_TAC[LENGTH];
STRIP_TAC;
REWRITE_TAC[TL;BUTLAST];
COND_CASES_TAC;
REWRITE_ASSUMS [GSYM LENGTH_0];
REPEAT (POP_ASSUM MP_TAC) THEN ARITH_TAC;
REWRITE_ASSUMS[NOT_NIL];
POP_ASSUM MP_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[];
REWRITE_TAC[TL;BUTLAST];
]);;
(* }}} *)
let APPEND_TL = prove_by_refinement(
`!m l. ~(l = []) ==> (APPEND (TL l) m = TL (APPEND l m))`,
(* {{{ Proof *)
[
STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[APPEND;TL];
]);;
(* }}} *)
let APPEND_HD = prove_by_refinement(
`!m l. ~(l = []) ==> (HD (APPEND l m) = HD l)`,
(* {{{ Proof *)
[
STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[];
STRIP_TAC;
REWRITE_TAC[APPEND;HD];
]);;
(* }}} *)
let PL_LEM2 = prove_by_refinement(
`!l. ~(l = []) ==> LENGTH (partition_line l) > 1`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC THEN REWRITE_TAC[];
REWRITE_TAC[partition_line];
STRIP_TAC;
COND_CASES_TAC;
REWRITE_TAC[LENGTH] THEN ARITH_TAC;
REWRITE_TAC[APPEND;LENGTH] THEN ARITH_TAC;
]);;
(* }}} *)
let BUTLAST_APPEND = prove_by_refinement(
`!l m. ~(m = []) ==>
(BUTLAST (APPEND l m) = APPEND l (BUTLAST m))`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[APPEND];
REPEAT STRIP_TAC;
REWRITE_TAC[APPEND;BUTLAST];
ASM_MESON_TAC[APPEND_EQ_NIL];
]);;
(* }}} *)
let LENGTH_TL1 = prove_by_refinement(
`!l. LENGTH l > 1 ==> ~(TL l = [])`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[LENGTH] THEN ARITH_TAC;
REWRITE_TAC[LENGTH;TL];
REPEAT STRIP_TAC;
POP_ASSUM (REWRITE_ASSUMS o list);
REWRITE_ASSUMS[LENGTH];
POP_ASSUM MP_TAC THEN ARITH_TAC;
]);;
(* }}} *)
let PL_BUTLAST = prove_by_refinement(
`!l. ~(l = []) ==> ~(BUTLAST (partition_line l) = [])`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[partition_line];
COND_CASES_TAC;
(* XXX REWRITE_TAC works here, but not MESON_TAC *)
REWRITE_TAC[APPEND;NOT_CONS_NIL;BUTLAST];
REWRITE_TAC[APPEND;NOT_CONS_NIL;BUTLAST];
]);;
(* }}} *)
let PARTITION_LINE_APPEND = prove_by_refinement(
`!h t l. ~(l = []) ==>
(partition_line (APPEND l (CONS h t)) =
APPEND (BUTLAST (partition_line l))
(CONS (\x. LAST l < x /\ x < h)
(TL (partition_line (CONS h t)))))`,
(* {{{ Proof *)
[
STRIP_TAC THEN STRIP_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[];
DISCH_THEN (fun x -> ALL_TAC);
CASES_ON `t' = []`;
POP_ASSUM (REWRITE_ALL o list);
REWRITE_TAC[HD;APPEND;partition_line;BUTLAST;LAST;TL;NOT_CONS_NIL;];
POP_ASSUM (fun x -> REWRITE_ASSUMS [x] THEN ASSUME_TAC x);
REWRITE_TAC[APPEND];
CONV_TAC (LAND_CONV (REWRITE_CONV[partition_line]));
COND_CASES_TAC;
ASM_MESON_TAC[NOT_CONS_NIL;APPEND_EQ_NIL];
POP_ASSUM (fun x -> ALL_TAC);
ASM_REWRITE_TAC[];
ASM_REWRITE_TAC[LAST];
ASM_SIMP_TAC[APPEND_HD];
CONV_TAC (RAND_CONV (LAND_CONV (RAND_CONV (REWRITE_CONV[partition_line]))));
ASM_REWRITE_TAC[];
REWRITE_TAC[APPEND;BUTLAST;NOT_CONS_NIL;];
REPEAT AP_TERM_TAC;
COND_CASES_TAC;
ASM_MESON_TAC[PL_LEM2;LENGTH_TL1];
REWRITE_TAC[APPEND];
AP_TERM_TAC;
MP_TAC (ISPEC `t':real list` PL_LEM2);
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_SIMP_TAC[BUTLAST_TL];
MP_TAC (ISPEC `t':real list` PL_BUTLAST);
ASM_REWRITE_TAC[];
STRIP_TAC;
ASM_SIMP_TAC[APPEND_TL];
]);;
(* }}} *)
let HD_TL = prove_by_refinement(
`!l. ~(l = []) ==> (l = CONS (HD l) (TL l))`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC;
REWRITE_TAC[];
REWRITE_TAC[HD;TL];
]);;
(* }}} *)
let HD_LEM = prove_by_refinement(
`!l1 l2. (TL l1 = l2) <=> (CONS (HD l1) (TL l1) = CONS (HD l1) l2)`,
(* {{{ Proof *)
[
MESON_TAC[CONS_11];
]);;
(* }}} *)
let rec LENGTH_N n ty =
let zero = `0` in
let neg = `(~)` in
let imp_thm = TAUT `(a ==> b) ==> (b ==> a) ==> (a <=> b)` in
match n with
0 -> CONJUNCT1 LENGTH
| 1 -> LENGTH_SING
| n ->
let len_tm = mk_const ("LENGTH",[ty,aty]) in
let tl_tm = mk_const ("TL",[ty,aty]) in
let hd_tm = mk_const ("HD",[ty,aty]) in
let t = mk_var("t",mk_type("list",[ty])) in
let n_tm = mk_small_numeral n in
let pren_tm = mk_small_numeral (n - 1) in
let len_thm = ASSUME (mk_eq(mk_comb(len_tm,t),n_tm)) in
let pre_thm = LENGTH_N (n - 1) ty in
let n_nz = prove(mk_neg(mk_eq(n_tm,zero)),ARITH_TAC) in
let not_nil_thm = EQ_MP (REWRITE_RULE[len_thm] (AP_TERM neg (ISPEC t LENGTH_0))) n_nz in
let n_suc = prove(mk_eq(n_tm,mk_comb(`SUC`,pren_tm)),ARITH_TAC) in
let len_tl = REWRITE_RULE[n_suc;PRE;ISPEC (mk_comb(tl_tm,t)) pre_thm;len_thm] (MATCH_MP LENGTH_TL not_nil_thm) in
let cons_thm = MATCH_MP (ISPEC t HD_TL) not_nil_thm in
let hd_thm = ONCE_REWRITE_RULE[HD_LEM] len_tl in
let thm = REWRITE_RULE[GSYM cons_thm] hd_thm in
let x0 = mk_var("x" ^ string_of_int n,ty) in
let hdt = mk_comb(hd_tm,t) in
let ex_thm = EXISTS (mk_exists(x0,subst[x0,hdt] (concl thm)),mk_comb(hd_tm,t)) thm in
let left = DISCH (concl len_thm) ex_thm in
let right = prove(mk_imp(concl ex_thm,concl len_thm),REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[LENGTH] THEN ARITH_TAC) in
GEN_ALL(MATCH_MPL[imp_thm;left;right]);;
let BUTLAST_LENGTH = prove_by_refinement(
`!l. ~(l = []) ==> (LENGTH (BUTLAST l) = PRE (LENGTH l))`,
(* {{{ Proof *)
[
LIST_INDUCT_TAC THEN REWRITE_TAC[];
REWRITE_TAC[BUTLAST;LENGTH];
COND_CASES_TAC;
ASM_REWRITE_TAC[NOT_CONS_NIL;LENGTH;];
ARITH_TAC;
ASM_REWRITE_TAC[NOT_CONS_NIL;LENGTH;];
ASM_SIMP_TAC[];
MATCH_MP_TAC (ARITH_RULE `~(n = 0) ==> (SUC(PRE n) = PRE(SUC n))`);
ASM_MESON_TAC[LENGTH_0];
]);;
(* }}} *)
let ALL2_LEM = prove_by_refinement(
`!a b x y s eqs pts sgns.
ALL2 (interpsigns eqs) (partition_line
(APPEND pts [x; y])) (APPEND sgns [a; b; s; s; s]) ==>
ALL2 (interpsigns eqs) (partition_line
(APPEND pts [x])) (APPEND sgns [a; b; s])`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
DISJ_CASES_TAC (ISPEC `pts:real list` list_CASES);
POP_ASSUM (REWRITE_ALL o list);