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inferpsign_thms.ml
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let EVEN_DIV_LEM = prove_by_refinement(
`!set p q c d a n.
(!x. a pow n * p x = c x * q x + d x) ==>
a <> &0 ==>
EVEN n ==>
((interpsign set q Zero) ==>
(interpsign set d Neg) ==>
(interpsign set p Neg)) /\
((interpsign set q Zero) ==>
(interpsign set d Pos) ==>
(interpsign set p Pos)) /\
((interpsign set q Zero) ==>
(interpsign set d Zero) ==>
(interpsign set p Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign];
REPEAT STRIP_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `&0 < a pow n`;
ASM_MESON_TAC[EVEN_ODD_POW;real_gt];
STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `&0 < a pow n`;
ASM_MESON_TAC[EVEN_ODD_POW;real_gt];
STRIP_TAC;
CLAIM `a pow n * p x > &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `&0 < a pow n`;
ASM_MESON_TAC[EVEN_ODD_POW;real_gt];
STRIP_TAC;
CLAIM `a pow n * p x = &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
ASM_MESON_TAC[REAL_ENTIRE;REAL_POS_NZ];
]);;
(* }}} *)
let GT_DIV_LEM = prove_by_refinement(
`!set p q c d a n.
(!x. a pow n * p x = c x * q x + d x) ==>
a > &0 ==>
((interpsign set q Zero) ==>
(interpsign set d Neg) ==>
(interpsign set p Neg)) /\
((interpsign set q Zero) ==>
(interpsign set d Pos) ==>
(interpsign set p Pos)) /\
((interpsign set q Zero) ==>
(interpsign set d Zero) ==>
(interpsign set p Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign];
REPEAT_N 9 STRIP_TAC;
CLAIM `a pow n > &0`;
ASM_MESON_TAC[REAL_POW_LT;real_gt;];
STRIP_TAC;
REPEAT STRIP_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
(* save *)
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x > &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x = &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
]);;
(* }}} *)
let NEG_ODD_LEM = prove_by_refinement(
`!set p q c d a n.
(!x. a pow n * p x = c x * q x + d x) ==>
a < &0 ==>
ODD n ==>
((interpsign set q Zero) ==>
(interpsign set (\x. -- d x) Neg) ==>
(interpsign set p Neg)) /\
((interpsign set q Zero) ==>
(interpsign set (\x. -- d x) Pos) ==>
(interpsign set p Pos)) /\
((interpsign set q Zero) ==>
(interpsign set (\x. -- d x) Zero) ==>
(interpsign set p Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign;POLY_NEG];
REPEAT_N 10 STRIP_TAC;
CLAIM `a pow n < &0`;
ASM_MESON_TAC[PARITY_POW_LT;real_gt;];
STRIP_TAC;
REAL_SIMP_TAC;
REPEAT STRIP_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x > &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
(* save *)
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `a pow n * p x = &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
]);;
(* }}} *)
let NEQ_ODD_LEM = prove_by_refinement(
`!set p q c d a n.
(!x. a pow n * p x = c x * q x + d x) ==>
a <> &0 ==>
ODD n ==>
((interpsign set q Zero) ==>
(interpsign set (\x. a * d x) Neg) ==>
(interpsign set p Neg)) /\
((interpsign set q Zero) ==>
(interpsign set (\x. a * d x) Pos) ==>
(interpsign set p Pos)) /\
((interpsign set q Zero) ==>
(interpsign set (\x. a * d x) Zero) ==>
(interpsign set p Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign;POLY_CMUL];
REPEAT_N 10 STRIP_TAC;
CLAIM `a < &0 \/ a > &0 \/ (a = &0)`;
REAL_ARITH_TAC;
REWRITE_ASSUMS[NEQ];
ASM_REWRITE_TAC[];
LABEL_ALL_TAC;
STRIP_TAC;
(* save *)
CLAIM `a pow n < &0`;
ASM_MESON_TAC[PARITY_POW_LT];
STRIP_TAC;
REPEAT STRIP_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x > &0`;
POP_ASSUM MP_TAC;
ASM_REWRITE_TAC[real_gt;REAL_MUL_LT];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
POP_ASSUM MP_TAC;
POP_ASSUM MP_TAC;
REWRITE_TAC[REAL_MUL_LT];
REPEAT STRIP_TAC;
CLAIM `&0 < a pow n * p x`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_GT];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
(* save *)
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x < &0`;
POP_ASSUM MP_TAC;
REWRITE_TAC[REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x = &0`;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
STRIP_TAC;
CLAIM `a pow n * p x = &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
(* save *)
CLAIM `a pow n > &0`;
ASM_MESON_TAC[EVEN_ODD_POW;NEQ;real_gt];
STRIP_TAC;
REPEAT STRIP_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x < &0`;
POP_ASSUM MP_TAC;
ASM_REWRITE_TAC[real_gt;REAL_MUL_LT];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
POP_ASSUM MP_TAC;
POP_ASSUM MP_TAC;
REWRITE_TAC[REAL_MUL_LT];
REPEAT STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
(* save *)
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x > &0`;
POP_ASSUM MP_TAC;
REWRITE_TAC[REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
CLAIM `a pow n * p x < &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
STRIP_TAC;
CLAIM `a pow n * p x > &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
REWRITE_TAC[REAL_MUL_LT;REAL_MUL_GT;real_gt];
REPEAT STRIP_TAC;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
RULE_ASSUM_TAC (fun y -> try ISPEC `x:real` y with _ -> y);
POP_ASSUM (fun x -> REWRITE_ASSUMS[x]);
POP_ASSUM MP_TAC;
POP_ASSUM (fun x -> REWRITE_ASSUMS[x;REAL_MUL_RZERO;REAL_ADD_LID;]);
STRIP_TAC;
CLAIM `d x = &0`;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
STRIP_TAC;
CLAIM `a pow n * p x = &0`;
EVERY_ASSUM MP_TAC THEN REAL_ARITH_TAC;
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
]);;
(* }}} *)
let NEQ_MULT_LT_LEM = prove_by_refinement(
`!a q d d' set.
a < &0 ==>
((interpsign set d Neg) ==>
(interpsign set (\x. a * d x) Pos)) /\
((interpsign set d Pos) ==>
(interpsign set (\x. a * d x) Neg)) /\
((interpsign set d Zero) ==>
(interpsign set (\x. a * d x) Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign;POLY_NEG];
REPEAT STRIP_TAC;
ASM_MESON_TAC[REAL_MUL_GT;real_gt];
ASM_MESON_TAC[REAL_MUL_LT;real_gt];
ASM_MESON_TAC[REAL_ENTIRE;REAL_NOT_EQ;real_gt];
]);;
(* }}} *)
let NEQ_MULT_GT_LEM = prove_by_refinement(
`!a q d d' set.
a > &0 ==>
((interpsign set d Neg) ==>
(interpsign set (\x. a * d x) Neg)) /\
((interpsign set d Pos) ==>
(interpsign set (\x. a * d x) Pos)) /\
((interpsign set d Zero) ==>
(interpsign set (\x. a * d x) Zero))`,
(* {{{ Proof *)
[
REWRITE_TAC[interpsign;POLY_NEG] THEN
MESON_TAC[REAL_MUL_LT;REAL_ENTIRE;REAL_NOT_EQ;REAL_MUL_GT;real_gt];
]);;
(* }}} *)
let unknown_thm = prove(
`!set p. (interpsign set p Unknown)`,
MESON_TAC[interpsign]);;
let ips_gt_nz_thm = prove_by_refinement(
`!x. x > &0 ==> x <> &0`,
(* {{{ Proof *)
[
REWRITE_TAC[NEQ];
REAL_ARITH_TAC;
]);;
(* }}} *)
let ips_lt_nz_thm = prove_by_refinement(
`!x. x < &0 ==> x <> &0`,
(* {{{ Proof *)
[
REWRITE_TAC[NEQ];
REAL_ARITH_TAC;
]);;
(* }}} *)