Nuprl Lemma : det-kI+J
∀[r:CRng]. ∀[n:ℕ]. ∀[a:|r|]. (|a*I + J| = if (n =z 0) then 1 else (a +r int-to-ring(r;n)) * (a ↑r (n - 1)) fi ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-scalar-mul: k*M,
matrix-det: |M|,
J-matrix: J,
identity-matrix: I,
matrix-plus: M + N,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T,
int-to-ring: int-to-ring(r;n),
rng_nexp: e ↑r n,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
crng: CRng,
rng: Rng,
squash: ↓T,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
infix_ap: x f y,
lt_int: i <z j,
int-to-ring: int-to-ring(r;n),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
matrix-plus: M + N,
matrix-scalar-mul: k*M,
identity-matrix: I,
J-matrix: J,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
assert: ↑b,
bnot: ¬bb,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
ringeq_int_terms: t1 ≡ t2,
less_than: a < b,
matrix-minor: matrix-minor(i;j;m),
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
diagonal-matrix: diagonal-matrix(r;x.F[x]),
nat_plus: ℕ+,
nat_op: n x(op;id) e,
mon_nat_op: n ⋅ e,
rng_nexp: e ↑r n,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
mon_itop: Π lb ≤ i < ub. E[i],
rng_prod: rng_prod
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
matrix-det-dim0,
rng_car_wf,
subtract-1-ge-0,
subtype_base_sq,
bool_wf,
bool_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
eq_int_eq_false,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
bfalse_wf,
subtype_rel_self,
iff_weakening_equal,
istype-nat,
crng_wf,
decidable__equal_int,
rng_nat_op_one,
rng_nexp_zero,
rng_nat_op_wf,
rng_one_wf,
matrix_ap_mx_lemma,
subtract_wf,
rng_nexp_wf,
int-to-ring_wf,
rng_plus_wf,
rng_times_wf,
infix_ap_wf,
J-matrix_wf,
identity-matrix_wf,
le_wf,
false_wf,
matrix-scalar-mul_wf,
matrix-plus_wf,
matrix-det-dim1,
rng_times_over_plus,
rng_times_one,
rng_plus_comm,
lelt_wf,
decidable__lt,
int_term_value_subtract_lemma,
itermSubtract_wf,
int_seg_wf,
matrix-ap_wf,
rng_zero_wf,
mx_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
det-add-row,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
rng_sig_wf,
rng_wf,
nat_wf,
matrix_wf,
matrix-det_wf,
equal-wf-base,
int_seg_properties,
rng_times_zero,
rng_plus_zero,
expand-det-by-row,
matrix-minor_wf,
rng_minus_wf,
isEven_wf,
rng_sum_unroll_hi,
ringeq-iff-rsub-is-0,
itermMinus_wf,
itermMultiply_wf,
itermAdd_wf,
rng_sum_is_0,
ring_polynomial_null,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
ring_term_value_var_lemma,
ring_term_value_minus_lemma,
assert-isEven,
btrue_wf,
two-mul,
less_than_wf,
top_wf,
assert_of_lt_int,
lt_int_wf,
det-multiple-row-ops,
rng_times_over_minus,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_inv_assoc,
det-multiple-col-ops,
rng_plus_inv,
rng_minus_zero,
det-diagonal,
rng_prod_unroll_hi,
rng_prod_wf,
istype-le,
subtract-add-cancel,
int-to-ring-add,
int-to-ring-one,
rng_nexp_unroll,
istype-false,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
le-add-cancel2,
condition-implies-le,
minus-add,
add-swap,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
le-add-cancel,
int_term_value_add_lemma,
crng_times_comm,
crng_times_ac_1,
rng_times_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
instantiate,
cumulativity,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality,
because_Cache,
productElimination,
unionElimination,
intEquality,
voidEquality,
isect_memberEquality,
lambdaFormation,
dependent_set_memberEquality,
lambdaEquality,
hyp_replacement,
dependent_pairFormation,
int_eqReduceFalseSq,
promote_hyp,
int_eqReduceTrueSq,
equalityElimination,
functionEquality,
addEquality,
multiplyEquality,
axiomSqEquality,
isect_memberFormation,
lessCases,
functionExtensionality,
dependent_set_memberEquality_alt,
applyLambdaEquality,
minusEquality,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[a:|r|].
(|a*I + J| = if (n =\msubz{} 0) then 1 else (a +r int-to-ring(r;n)) * (a \muparrow{}r (n - 1)) fi )
Date html generated:
2019_10_16-AM-11_28_33
Last ObjectModification:
2018_10_18-PM-11_52_47
Theory : matrices
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