Nuprl Lemma : determinant_wf
∀[r:CRng]. ∀[n:ℕ]. (determinant(n;r) ∈ Matrix(n;n;r) ⟶ |r|)
Proof
Definitions occuring in Statement :
determinant: determinant(n;r),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
assert: ↑b,
bnot: ¬bb,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
bfalse: ff,
sq_type: SQType(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
squash: ↓T,
or: P ∨ Q,
decidable: Dec(P),
rng: Rng,
crng: CRng,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
determinant: determinant(n;r),
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
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,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
nat_wf,
int_seg_wf,
matrix-minor_wf,
rng_minus_wf,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
decidable__lt,
subtract-add-cancel,
lelt_wf,
false_wf,
matrix-ap_wf,
eqtt_to_assert,
bool_wf,
isEven_wf,
rng_times_wf,
rng_car_wf,
infix_ap_wf,
rng_sum_wf,
iff_weakening_equal,
int_subtype_base,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
eq_int_eq_false,
equal_wf,
bool_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
matrix_wf,
rng_one_wf,
primrec-unroll,
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
Rules used in proof :
functionExtensionality,
promote_hyp,
dependent_set_memberEquality,
equalityElimination,
addEquality,
productElimination,
universeEquality,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
instantiate,
because_Cache,
unionElimination,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. (determinant(n;r) \mmember{} Matrix(n;n;r) {}\mrightarrow{} |r|)
Date html generated:
2018_05_21-PM-09_37_46
Last ObjectModification:
2017_12_12-AM-10_23_10
Theory : matrices
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