Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 3,613 Bytes
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let le_lem = prove_by_refinement(
`(!y. y <= Y ==> P y) ==>
(!y. y < Y ==> P y) /\
(!y. (y = Y) ==> P y)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
FIRST_ASSUM MATCH_MP_TAC;
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
]);;
(* }}} *)
let lt_int_lem = prove_by_refinement(
`(!y. y < Y ==> P y) ==> X < Y ==>
(!y. X < y /\ y < Y ==> P y)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_ASSUM MATCH_ACCEPT_TAC;
]);;
(* }}} *)
let ge_lem = prove_by_refinement(
`(!y. Y <= y ==> P y) ==>
(!y. Y < y ==> P y) /\
(!y. (y = Y) ==> P y)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
FIRST_ASSUM MATCH_MP_TAC;
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
]);;
(* }}} *)
let gt_int_lem = prove_by_refinement(
`(!y. Y < y ==> P y) ==> Y < X ==>
(!y. Y < y /\ y < X ==> P y)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
FIRST_ASSUM MATCH_ACCEPT_TAC;
]);;
(* }}} *)
let rest_lt_lem = prove_by_refinement(
`Y < X ==> (!x. x < X ==> P x) ==> (!x. x < Y ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS;real_gt];
]);;
(* }}} *)
let rest_gt_lem = prove_by_refinement(
`X < Y ==> (!x. X < x ==> P x) ==> (!x. Y < x ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS;real_gt];
]);;
(* }}} *)
let rest_eq_lt_lem = prove_by_refinement(
`Y < X ==> (!x. x < X ==> P x) ==> (!x. (x = Y) ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS];
]);;
(* }}} *)
let rest_eq_gt_lem = prove_by_refinement(
`X < Y ==> (!x. X < x ==> P x) ==> (!x. (x = Y) ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS];
]);;
(* }}} *)
let rest_int_lt_lem = prove_by_refinement(
`Y < X ==> (!x. x < X ==> P x) ==> (!x. Y < x /\ x < X ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS];
]);;
(* }}} *)
let rest_int_gt_lem = prove_by_refinement(
`X < Y ==> (!x. X < x ==> P x) ==> (!x. X < x /\ x < Y ==> P x)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC;
FIRST_ASSUM MATCH_MP_TAC;
ASM_MESON_TAC[REAL_LT_TRANS];
]);;
(* }}} *)
let INTERPSIGN_SUBSET = prove_by_refinement(
`!P Q p s. interpsign P p s /\ Q SUBSET P ==> interpsign Q p s`,
(* {{{ Proof *)
[
REWRITE_TAC[SUBSET;IN];
REPEAT_N 4 STRIP_TAC;
STRUCT_CASES_TAC (ISPEC `s:sign` SIGN_CASES) THEN
REWRITE_TAC[interpsign] THEN MESON_TAC[];
]);;
(* }}} *)
let INTERPSIGNS_SUBSET = prove_by_refinement(
`!P Q ps ss. interpsigns ps P ss /\ Q SUBSET P ==> interpsigns ps Q ss`,
(* {{{ Proof *)
[
REWRITE_TAC[SUBSET;IN];
REPEAT_N 2 STRIP_TAC;
LIST_INDUCT_TAC;
LIST_INDUCT_TAC;
REWRITE_TAC[ALL2;interpsigns;interpsign];
REWRITE_TAC[ALL2;interpsigns;interpsign];
LIST_INDUCT_TAC;
REWRITE_TAC[ALL2;interpsigns;interpsign];
REWRITE_TAC[ALL2;interpsigns;interpsign];
(* save *)
REPEAT STRIP_TAC;
MATCH_MP_TAC INTERPSIGN_SUBSET;
ASM_MESON_TAC[SUBSET;IN];
REWRITE_ASSUMS[ALL2;interpsigns;interpsign];
FIRST_ASSUM MATCH_MP_TAC;
ASM_REWRITE_TAC[];
]);;
(* }}} *)
let NOPOINT_LEM = prove_by_refinement(
`!pl sl. interpsigns pl (\x. T) sl ==>
(interpsigns pl (\x. x < &0) sl /\
interpsigns pl (\x. x = &0) sl /\
interpsigns pl (\x. &0 < x) sl)`,
(* {{{ Proof *)
[
REPEAT STRIP_TAC THEN MATCH_MP_TAC INTERPSIGNS_SUBSET THEN ASM_MESON_TAC[SUBSET;IN]
]);;
(* }}} *)
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