Nuprl Lemma : matrix-minor_wf
∀[n,m:ℤ]. ∀[r:RngSig]. ∀[i:ℕn]. ∀[j:ℕm]. ∀[M:Matrix(n;m;r)]. (matrix-minor(i;j;M) ∈ Matrix(n - 1;m - 1;r))
Proof
Definitions occuring in Statement :
matrix-minor: matrix-minor(i;j;m),
matrix: Matrix(n;m;r),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtract: n - m,
natural_number: $n,
int: ℤ,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
matrix: Matrix(n;m;r),
matrix-minor: matrix-minor(i;j;m),
int_seg: {i..j-},
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
lelt: i ≤ j < k,
guard: {T},
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
matrix-ap_wf,
less_than_wf,
int_seg_properties,
subtract_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
top_wf,
int_seg_wf,
matrix_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
lessCases,
independent_pairFormation,
baseClosed,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
imageElimination,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
addEquality,
axiomEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[r:RngSig]. \mforall{}[i:\mBbbN{}n]. \mforall{}[j:\mBbbN{}m]. \mforall{}[M:Matrix(n;m;r)].
(matrix-minor(i;j;M) \mmember{} Matrix(n - 1;m - 1;r))
Date html generated:
2019_10_16-AM-11_27_25
Last ObjectModification:
2018_08_20-PM-09_41_58
Theory : matrices
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