Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 1,469 Bytes
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(* ------------------------------------------------------------------------- *)
(* Condense subdivision by removing points with no relevant zeros. *)
(* ------------------------------------------------------------------------- *)
let real_cases = prove(`!x y. x < y \/ (x = y) \/ y < x`,REAL_ARITH_TAC);;
let gt_aux = prove(
`!x. (x1 < x2 /\ x2 < x3) /\ ((x1 < x /\ x < x2) \/ (x = x2) \/ (x2 < x /\ x < x3)) ==> x1 < x /\ x < x3`,
REAL_ARITH_TAC);;
let gen_thm = prove_by_refinement(
`!P x1 x2 x3.
(x1 < x3) ==>
(!x. x1 < x /\ x < x2 ==> P x) ==>
(!x. (x = x2) ==> P x) ==>
(!x. x2 < x /\ x < x3 ==> P x) ==>
(!x. x1 < x /\ x < x3 ==> P x)`,
(* {{{ Proof *)
[
MESON_TAC[real_cases;gt_aux;DE_MORGAN_THM;REAL_NOT_LT;REAL_LE_LT];
]);;
(* }}} *)
let gen_thm_noleft = prove(
`!P x2 x3.
(x2 < x3) ==>
(!x. x < x2 ==> P x) ==>
(!x. (x = x2) ==> P x) ==>
(!x. x2 < x /\ x < x3 ==> P x) ==>
(!x. x < x3 ==> P x)`,
MESON_TAC[real_cases;gt_aux]);;
let gen_thm_noright = prove(
`!P x1 x2.
(x1 < x2) ==>
(!x. x1 < x /\ x < x2 ==> P x) ==>
(!x. (x = x2) ==> P x) ==>
(!x. x2 < x ==> P x) ==>
(!x. x1 < x ==> P x)`,
MESON_TAC[real_cases;gt_aux]);;
let gen_thm_noboth = prove(
`!P Q x2.
Q ==>
(!x. x < x2 ==> P x) ==>
(!x. (x = x2) ==> P x) ==>
(!x. x2 < x ==> P x) ==>
(!x. T ==> P x)`,
MESON_TAC[real_cases;gt_aux]);;
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