| let is_linear_map = new_definition |
| `is_linear_map (zeroV:'v) (addV:'v->'v->'v) (smul:'r->'v->'v) |
| (zeroW:'w) (addW:'w->'w->'w) (smulW:'r->'w->'w) |
| (toFun:'v->'w) <=> |
| (!u v. addV u v = addV v u) /\ |
| (!u v w. addV (addV u v) w = addV u (addV v w)) /\ |
| (!u. addV u zeroV = u) /\ |
| (!a. smul a zeroV = zeroV) /\ |
| (!u v. addW u v = addW v u) /\ |
| (!u v w. addW (addW u v) w = addW u (addW v w)) /\ |
| (!u. addW u zeroW = u) /\ |
| (!a. smulW a zeroW = zeroW) /\ |
| (!u v. toFun (addV u v) = addW (toFun u) (toFun v)) /\ |
| (!a u. toFun (smul a u) = smulW a (toFun u))`;; |
|
|
| let ker = new_definition |
| `ker (toFun:'v->'w) (zeroW:'w) x <=> toFun x = zeroW`;; |
|
|
| let im = new_definition |
| `im (toFun:'v->'w) y <=> ?x. toFun x = y`;; |
|
|
| let ker_add = prove |
| (`!zeroV addV smul zeroW addW smulW (toFun:'v->'w) x y. |
| is_linear_map zeroV addV smul zeroW addW smulW toFun ==> |
| ker toFun zeroW x ==> ker toFun zeroW y ==> ker toFun zeroW (addV x y)`, |
| REWRITE_TAC[is_linear_map; ker] THEN MESON_TAC[]);; |
|
|
| let ker_smul = prove |
| (`!zeroV addV smul zeroW addW smulW (toFun:'v->'w) a x. |
| is_linear_map zeroV addV smul zeroW addW smulW toFun ==> |
| ker toFun zeroW x ==> ker toFun zeroW (smul a x)`, |
| REWRITE_TAC[is_linear_map; ker] THEN MESON_TAC[]);; |
|
|
| let im_add = prove |
| (`!zeroV addV smul zeroW addW smulW (toFun:'v->'w) y z. |
| is_linear_map zeroV addV smul zeroW addW smulW toFun ==> |
| im toFun y ==> im toFun z ==> im toFun (addW y z)`, |
| REWRITE_TAC[is_linear_map; im] THEN MESON_TAC[]);; |
|
|
| let im_smul = prove |
| (`!zeroV addV smul zeroW addW smulW (toFun:'v->'w) a y. |
| is_linear_map zeroV addV smul zeroW addW smulW toFun ==> |
| im toFun y ==> im toFun (smulW a y)`, |
| REWRITE_TAC[is_linear_map; im] THEN MESON_TAC[]);; |
|
|
|
|
