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Property name | Formalization |
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FVECTOR_NEG | ∣- !fv. ~fv = −1 ** fv |
FVECTOR_ADD_MUL_LK | ∣- !fv1 fv2 k. k ** (fv1 + fv2) = k ** fv1 + k ** fv2 |
FVECTOR_ADD_RDISTRIB | ∣- !fv1 fv2 fv3. (fv1 + fv2) ** fv3 = (∖x. (fv1 ** fv3) x + (fv2 ** fv3) x) |
FVECTOR_ADD_ASSOC | ∣- !fv1 fv2 fv3. fv1 + fv2 + fv3 = fv1 + (fv2 + fv3) |
FVECTOR_SUB_LZERO | ∣- !fv. fvector_0 − fv = ~fv |
FVECTOR_DOT_FBASIS | ∣- !fv k x. k < dimindex (:'n) ==> (fv ** fvector_basis k = fv ' k) |
FVECTOR_DOT_FCP | ∣- ($FCP fv1 ** fv2 = (∖x. sum (0,dimindex (:'n)) (∖i. fv1 i x * fv2 ' i x))) ∧ (fv2 ** $FCP fv1 = (∖x. sum (0,dimindex (:'n)) (∖i. fv2 ' i x * fv1 i x))) |
FVECTOR_ADD_INDEX | ∣- !fv1 fv2 i. i < dimindex (:'n) ==> ((fv1 + fv2) ' i = (∖x. fv1 ' i x + fv2 ' i x)) |
FVECTOR_SUB_INDEX | ∣- !fv1 fv2 i. i < dimindex (:'n) ==> ((fv1 − fv2) ' i = (∖x. fv1 ' i x − fv2 ' i x)) |
FVECTOR_NEG_NEG | ∣- !fv. ~~fv = fv |
FVECTOR_ADD_MUL_LKX | ∣- !fv1 fv2 kx. kx ** (fv1 + fv2) = kx ** fv1 + kx ** fv2 |
FVECTOR_ADD_MUL_RKX | ∣- !fv1 fv2 kx. (fv1 + fv2) ** kx = fv1 ** kx + fv2 ** kx |
FVECTOR_MUL_LRADD | ∣- !fv k l. (k + l) ** fv = k ** fv + l ** fv |
FVECTOR_MUL_RRADD | ∣- !fv k l. fv ** (k + l) = fv ** k + fv ** l |
FVECTOR_MUL_LFADD | ∣- !fv f g. (∖x. f x + g x) ** fv = f ** fv + g ** fv |
FVECTOR_ADD_RDISTRIB | ∣- !fv1 fv2 fv3. (fv1 + fv2) ** fv3 = (∖x. (fv1 ** fv3) x + (fv2 ** fv3) x) |
FVECTOR_SUB_LDISTRIB | ∣- !fv1 fv2 fv3. fv1 ** (fv2 − fv3) = (∖x. (fv1 ** fv2) x − (fv1 ** fv3) x) |
FVECTOR_SUB_RDISTRIB | ∣- !fv1 fv2 fv3. (fv1 − fv2) ** fv3 = (∖x. (fv1 ** fv3) x − (fv2 ** fv3) x) |
FVECTOR_DOT_LMUL_K | ∣- !fv1 fv2 k. (∖x. k * (fv1 ** fv2) x) = (k ** fv1) ** fv2 |
FVECTOR_DOT_LMUL_KX | ∣- !fv1 fv2 k. (∖x. k x * (fv1 ** fv2) x) = (k ** fv1) ** fv2 |
FVECTOR_MUL_LK_ASSOC | ∣- !fv k l. k ** l ** fv = (k * l) ** fv |
FVECTOR_MUL_LKX_ASSOC | ∣- !fv f g. f ** g ** fv = (∖x. f x * g x) ** fv |
FVECTOR_DOT_COMM | ∣- !fv1 fv2. fv1 ** fv2 = fv2 ** fv1 |
FVECTOR_EQ | ∣- !fv1 fv2. (fv1 = fv2) <=> (fv1 − fv2 = fvector_0) |
FVECTOR_EQ2 | ∣- !fv1 fv2. (fv1 = fv2) <=> !i. i < dimindex (:'n) ==> (fv1 ' i = fv2 ' i) |
FVECTOR_ADD_LID | ∣- !fv. fvector_0 + fv = fv |
FVECTOR_ADD_RID | ∣- !fv. fv + fvector_0 = fv |
FVECTOR_ADD_NEG | ∣- !fv. fv + ~fv = fvector_0 |
FVECTOR_ADD_NEG2 | ∣- !fv1 fv2. fv1 + ~fv2 = fv1 − fv2 |
FVECTOR_SUB_ADD | ∣- !fv1 fv2. fv1 − fv2 + fv2 = fv1 |
FVECTOR_MUL_L1 | ∣- !fv. 1 ** fv = fv |
FVECTOR_LNEG_UNIQ | ∣- !fv1 fv2. (fv1 + fv2 = fvector_0) <=> (fv1 = ~fv2) |
FVECTOR_RNEG_UNIQ | ∣- !fv1 fv2. (fv1 + fv2 = fvector_0) <=> (fv2 = ~fv1) |
FVECTOR_MULK_COMM | ∣- !fv k. fv ** k = k ** fv |
FVECTOR_MULKX_COMM | ∣- !fv f. fv ** f = f ** fv |
FVECTOR_EXIST_NEG | ∣- !fv. ?fv'. fv + fv' = fvector_0 |
FVECTOR_FVECTOR_0_DOT | ∣- !fv. fvector_0 ** fv = (∖x. 0) |
COMPUTE_FVEC_MUL_MATRIX | ∣- !fv A x. compute_fvector fv x ** A = compute_fvector (fv ** A) x |
COMPUTE_VEC_MUL_FVEC | ∣- !fv v x. v ** compute_fvector fv x = (v ** fv) x |
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