Nuprl Lemma : dot-product-nonneg
∀[n:ℕ]. ∀[x:ℝ^n]. (r0 ≤ x ⋅ x)
Proof
Definitions occuring in Statement :
dot-product: x ⋅ y,
real-vec: ℝ^n,
rleq: x ≤ y,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
dot-product: x ⋅ y,
real-vec: ℝ^n,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
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],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
so_apply: x[s],
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rleq: x ≤ y,
rnonneg: rnonneg(x),
nat_plus: ℕ+,
subtype_rel: A ⊆r B
Lemmas referenced :
nat_wf,
real_wf,
nat_plus_wf,
int-to-real_wf,
nat_plus_properties,
rsum_wf,
rsub_wf,
less_than'_wf,
le_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
itermSubtract_wf,
intformle_wf,
square-nonneg,
int_seg_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
subtract-add-cancel,
rmul_wf,
subtract_wf,
rsum_nonneg
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
addEquality,
lambdaFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. (r0 \mleq{} x \mcdot{} x)
Date html generated:
2016_05_18-AM-09_47_56
Last ObjectModification:
2016_01_17-AM-02_52_15
Theory : reals
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