Nuprl Lemma : square-nonneg
∀[x:ℝ]. (r0 ≤ (x * x))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uiff: uiff(P;Q),
top: Top,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
uimplies: b supposing a,
rnonneg2: rnonneg2(x),
reg-seq-mul: reg-seq-mul(x;y),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
member: t ∈ T,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
guard: {T},
int_upper: {i...},
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
exists: ∃x:A. B[x]
Lemmas referenced :
rmul-bdd-diff-reg-seq-mul,
rnonneg2_functionality,
req_weakening,
rmul-identity1,
rmul_functionality,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
real_term_polynomial,
req_transitivity,
rnonneg_functionality,
reg-seq-mul_wf,
rnonneg-iff,
nat_plus_wf,
real_wf,
int-to-real_wf,
rmul_wf,
rsub_wf,
less_than'_wf,
less_than_wf,
int_upper_wf,
all_wf,
le_wf,
less_than_transitivity1,
int_upper_properties,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal_wf,
nat_plus_subtype_nat,
false_wf,
square_non_neg,
mul_nat_plus,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_functionality,
le_weakening,
multiply_functionality_wrt_le,
div_bounds_1
Rules used in proof :
voidEquality,
isect_memberEquality,
intEquality,
int_eqEquality,
computeAll,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
because_Cache,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
minusEquality,
rename,
setElimination,
natural_numberEquality,
hypothesis,
applyEquality,
isectElimination,
extract_by_obid,
voidElimination,
independent_pairEquality,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
divideEquality,
multiplyEquality,
lemma_by_obid,
baseClosed,
imageMemberEquality,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairFormation
Latex:
\mforall{}[x:\mBbbR{}]. (r0 \mleq{} (x * x))
Date html generated:
2017_10_03-AM-08_29_24
Last ObjectModification:
2017_08_02-PM-00_31_37
Theory : reals
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