Nuprl Lemma : det-multiple-row-ops
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ∀[a:ℕn]. ∀[k:|r|].
(|matrix(if x=a then M[x,y] else (M[x,y] +r (k * M[a,y])))| = |M| ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-det: |M|,
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
int_eq: if a=b then c else d,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
nat: ℕ,
rng: Rng,
crng: CRng,
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
all: ∀x:A. B[x],
true: True,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
lelt: i ≤ j < k,
guard: {T},
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
squash: ↓T,
nequal: a ≠ b ∈ T ,
row-op: row-op(r;a;b;k;M)
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
int_seg_wf,
rng_car_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
matrix-ap_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_seg_properties,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
matrix_ap_mx_lemma,
rng_wf,
true_wf,
squash_wf,
matrix-det_wf,
decidable__lt,
decidable__equal_int,
rng_times_wf,
rng_plus_wf,
infix_ap_wf,
neg_assert_of_eq_int,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_of_eq_int,
eq_int_wf,
rng_sig_wf,
mx_wf,
lelt_wf,
det-row-op
Rules used in proof :
natural_numberEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
independent_pairFormation,
voidEquality,
voidElimination,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
intWeakElimination,
lambdaFormation,
baseClosed,
imageMemberEquality,
cumulativity,
instantiate,
promote_hyp,
productElimination,
equalityElimination,
functionExtensionality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
int_eqReduceFalseSq,
int_eqReduceTrueSq,
functionEquality,
dependent_set_memberEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. \mforall{}[a:\mBbbN{}n]. \mforall{}[k:|r|].
(|matrix(if x=a then M[x,y] else (M[x,y] +r (k * M[a,y])))| = |M|)
Date html generated:
2018_05_21-PM-09_37_11
Last ObjectModification:
2018_01_02-PM-03_59_43
Theory : matrices
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