|
| Property name | Formalization |
|
| COMPUTE_FMATRIX_MUL_EQ | ∣- !fm1 fm2 x. compute_fmatrix fm1 x ** compute_fmatrix fm2 x = compute_fmatrix (fm1 ** fm2) x |
| FMATRIX_ADD_INDEX | ∣- !fm1 fm2 i j. i < dimindex (:'m) ∧ j < dimindex (:'n) ==>
((fm1 + fm2) ' i ' j = (∖x. fm1 ' i ' j x + fm2 ' i ' j x)) |
| FMATRIX_SUB_INDEX | ∣- !fm1 fm2 i j. i < dimindex (:'m) ∧ j < dimindex (:'n) ==>
((fm1 − fm2) ' i ' j = (∖x. fm1 ' i ' j x – fm2 ' i ' j x)) |
| FMATRIX_ROW_ADD | ∣- !fm1 fm2 i. i < dimindex (:'m) ==>
(fun_row fm1 i + fun_row fm2 i = fun_row (fm1 + fm2) i) |
| FMATRIX_COLUMN_ADD | ∣- !fm1 fm2 i. i < dimindex (:'n) ==>
(fun_column fm1 i + fun_column fm2 i = fun_column (fm1 + fm2)
i) |
| FMATRIX_ROW_SUB | ∣- !fm1 fm2 i. i < dimindex (:'m) ==>
(fun_row fm1 i − fun_row fm2 i = fun_row (fm1 − fm2) i) |
| FMATRIX_COLUMN_SUB | ∣- !fm1 fm2 i. i < dimindex (:'n) ==>
(fun_column fm1 i − fun_column fm2 i = fun_column (fm1 − fm2) i) |
| FMATRIX_NEG | ∣- !fm. ~fm = −1 ** fm |
| FMATRIX_NEG_NEG | ∣- !fm. ~~fm = fm |
| FMATRIX_MUL_K_EQ | ∣- !fm k. fm ** k = k ** fm |
| FMATRIX_MUL_KX_EQ | ∣- !fm f. fm ** f = f ** fm |
| FMATRIX_ADD_COMM | ∣- !fm1 fm1. fm1 + fm2 = fm2 + fm1 |
| FMATRIX_ADD_ASSOC | ∣- !fm1 fm2 fm3. fm1 + (fm2 + fm3) = fm1 + fm2 + fm3 |
| FMATRIX_ADD_MUL_LK | ∣- !fm1 fm2 k. k ** (fm1 + fm2) = k ** fm1 + k ** fm2 |
| FMATRIX_ADD_MUL_RK | ∣- !fm1 fm2 k. (fm1 + fm2) ** k = fm1 ** k + fm2 ** k |
| FMATRIX_ADD_MUL_LKX | ∣- !fm1 fm2 kx. kx ** (fm1 + fm2) = kx ** fm1 + kx ** fm2 |
| FMATRIX_ADD_MUL_RKX | ∣- !fm1 fm2 kx. (fm1 + fm2) ** kx = fm1 ** kx + fm2 ** kx |
| FMATRIX_ADD_MUL_LFVEC | ∣- !fm1 fm2 fv. fv ** (fm1 + fm2) = fv ** fm1 + fv ** fm2 |
| FMATRIX_ADD_MUL_RFVEC | ∣- !fm1 fm2 fv. (fm1 + fm2) ** fv = fm1 ** fv + fm2 ** fv |
| FMATRIX_SUB_MUL_LFVEC | ∣- !fm1 fm2 fv. fv ** (fm1 − fm2) = fv ** fm1 − fv ** fm2 |
| FMATRIX_SUB_MUL_RFVEC | ∣- !fm1 fm2 fv. (fm1 − fm2) ** fv = fm1 ** fv − fm2 ** fv |
| FMATRIX_MUL_LRADD | ∣- !fm k l. (k + l) ** fm = k ** fm + l ** fm |
| FMATRIX_MUL_RRADD | ∣- !fm k l. fm ** (k + l) = fm ** k + fm ** l |
| FMATRIX_MUL_LFADD | ∣- !fm f g. (∖x. f x + g x) ** fm = f ** fm + g ** fm |
| FMATRIX_MUL_RFADD | ∣- !fm f g. fm ** (∖x. f x + g x) = fm ** f + fm ** g |
| FMATRIX_MUL_RFVADD | ∣- !fm fv1 fv2. fm ** (fv1 + fv2) = fm ** fv1 + fm ** fv2 |
| FMATRIX_MUL_LFVADD | ∣- !fm fv1 fv2. (fv1 + fv2) ** fm = fv1 ** fm + fv2 ** fm |
| FMATRIX_ADD_LDISTRIB | ∣- !fm1 fm2 fm3. fm1 ** (fm2 + fm3) = fm1 ** fm2 + fm1 ** fm3 |
| FMATRIX_ADD_RDISTRIB | ∣- !fm1 fm2 fm3. (fm1 + fm2) ** fm3 = fm1 ** fm3 + fm2 ** fm3 |
| FMATRIX_MUL_LMUL_K | ∣- !fm1 fm2 k. k ** fm1 ** fm2 = (k ** fm1) ** fm2 |
| FMATRIX_MUL_RMUL_K | ∣- !fm1 fm2 k. k ** fm1 ** fm2 = fm1 ** k ** fm2 |
| FMATRIX_MUL_NEG | ∣- !fm1 fm2. ~fm1 ** fm2 = fm1 ** ~fm2 |
| FMATRIX_NEG_PROD | ∣- !fm1 fm2. ~fm1 ** fm2 = ~(fm1 ** fm2) |
| FMATRIX_MUL_LMUL_KX | ∣- !fm1 fm2 kx. kx ** fm1 ** fm2 = (kx ** fm1) ** fm2 |
| FMATRIX_MUL_LK_ASSOC | ∣- !fm k l. k ** l ** fm = (k * l) ** fm |
| FMATRIX_MUL_LKX_ASSOC | ∣- !fm f g. f ** g ** fm = (∖x. f x * g x) ** fm |
| FMATRIX_ADD_LID | ∣- !fm. fmatrix_0 + fm = fm |
| FMATRIX_ADD_RID | ∣- !fm. fm + fmatrix_0 = fm |
| FMATRIX_ADD_NEG | ∣- !fm. fm + ~fm = fmatrix_0 |
| FMATRIX_ADD_NEG2 | ∣- !fm1 fm2. fm1 + ~fm2 = fm1 − fm2 |
| FMATRIX_SUB_ADD | ∣- !fm1 fm2. fm1 − fm2 + fm2 = fm1 |
| FMATRIX_SUB_LZERO | ∣- !fm. fmatrix_0 − fm = ~fm |
| FMATRIX_MUL_L1 | ∣- !fm. 1 ** fm = fm |
| FMATRIX_MULK_COMM | ∣- !fm k. fm ** k = k ** fm |
| FMATRIX_MULKX_COMM | ∣- !fm kx. fm ** kx = kx ** fm |
| FVECTOR_PROD_FMATRIX | ∣- !fm fv. fv ** fm = transp_fmatrix fm ** fv |
| FMATRIX_FVECTOR_0_PROD | ∣- !fm. fvector_0 ** fm = fvector_0 |
| FMATRIX_ROW_PROD | ∣- !fm. transp_fmatrix fm ** fm =
FCP i j. fun_column fm i ** fun_column fm j |
| TRANSP_FMATRIX_COLUMN | ∣- !fm i. i < dimindex (:'m) ==>
(fun_column (transp_fmatrix fm) i = fun_row fm i) |
| TRANSP_FMATRIX_FVECTOR_PROD | ∣- !fm fv. fm ** fv = fv ** transp_fmatrix fm |
| TRANSP_FMATRIX_PROD | ∣- !fm. transp_fmatrix (transp_fmatrix fm ** fm) =
transp_fmatrix fm ** fm |
| TRANSP_FMATRIX_ROW | ∣- !fm i. i < dimindex (:'n) ==>
(fun_row (transp_fmatrix fm) i = fun_column fm i) |
|
