Nuprl Lemma : not-rpositive
∀[x:ℝ]. rnonneg(-(x)) supposing ¬rpositive(x)
Proof
Definitions occuring in Statement :
rnonneg: rnonneg(x),
rpositive: rpositive(x),
rminus: -(x),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
real: ℝ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
rpositive: rpositive(x),
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
prop: ℙ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rnonneg2: rnonneg2(x),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
int_upper: {i...},
guard: {T},
so_apply: x[s],
rminus: -(x)
Lemmas referenced :
int_term_value_minus_lemma,
itermMinus_wf,
nat_plus_subtype_nat,
mul_preserves_le,
int_term_value_mul_lemma,
itermMultiply_wf,
int_upper_properties,
subtype_rel_sets,
less_than_transitivity1,
le_wf,
all_wf,
int_upper_wf,
mul_nat_plus,
rnonneg-iff,
rpositive_wf,
not_wf,
real_wf,
rminus_wf,
less_than'_wf,
nat_plus_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
less_than_wf,
decidable__lt,
nat_plus_properties,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
applyEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
independent_functionElimination,
dependent_set_memberFormation,
isectElimination,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
productElimination,
independent_pairEquality,
because_Cache,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
multiplyEquality,
setEquality
Latex:
\mforall{}[x:\mBbbR{}]. rnonneg(-(x)) supposing \mneg{}rpositive(x)
Date html generated:
2016_05_18-AM-07_13_02
Last ObjectModification:
2016_01_17-AM-01_52_39
Theory : reals
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