Nuprl Lemma : expand-det-by-row
∀[n:ℕ]. ∀[i:ℕn]. ∀[r:CRng]. ∀[M:Matrix(n;n;r)].
(|M| = (Σ(r) 0 ≤ j < n. if isEven(i + j) then M[i,j] else -r M[i,j] fi * |matrix-minor(i;j;M)|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-minor: matrix-minor(i;j;m),
matrix-det: |M|,
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
isEven: isEven(n),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T,
rng_sum: rng_sum,
crng: CRng,
rng_times: *,
rng_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
true: True,
squash: ↓T,
prop: ℙ,
rng: Rng,
crng: CRng,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
infix_ap: x f y,
so_lambda: λ2x.t[x],
mx: matrix(M[x; y]),
matrix-transpose: M',
matrix-ap: M[i,j],
top: Top,
int_seg: {i..j-},
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
lelt: i ≤ j < k
Lemmas referenced :
nat_wf,
int_seg_wf,
crng_wf,
matrix_wf,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
matrix-transpose_wf,
expand-det-by-column,
iff_weakening_equal,
det-transpose,
matrix-det_wf,
rng_times_wf,
rng_wf,
rng_sum_wf,
rng_minus_wf,
matrix-ap_wf,
add-commutes,
isEven_wf,
bool_wf,
ifthenelse_wf,
matrix-minor_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
subtract_wf,
transpose-matrix-minor
Rules used in proof :
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
because_Cache,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
independent_functionElimination,
productElimination,
independent_isectElimination,
intEquality,
functionEquality,
addEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
dependent_functionElimination,
dependent_set_memberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}n]. \mforall{}[r:CRng]. \mforall{}[M:Matrix(n;n;r)].
(|M| = (\mSigma{}(r) 0 \mleq{} j < n. if isEven(i + j) then M[i,j] else -r M[i,j] fi * |matrix-minor(i;j;M)|))
Date html generated:
2018_05_21-PM-09_39_27
Last ObjectModification:
2018_01_02-PM-02_16_02
Theory : matrices
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