Nuprl Lemma : rpositive-rless
∀[x:ℝ]. (r0 < x ⇐⇒ rpositive(x))
Proof
Definitions occuring in Statement :
rless: x < y,
rpositive: rpositive(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
rpositive: rpositive(x),
int-to-real: r(n),
rless: x < y,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
real: ℝ,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
real_wf,
nat_plus_wf,
sq_exists_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_add_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,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
less_than_wf,
decidable__lt,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
introduction,
dependent_set_memberEquality,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
applyEquality,
unionElimination,
imageElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
multiplyEquality
Latex:
\mforall{}[x:\mBbbR{}]. (r0 < x \mLeftarrow{}{}\mRightarrow{} rpositive(x))
Date html generated:
2016_05_18-AM-07_03_23
Last ObjectModification:
2016_01_17-AM-01_49_33
Theory : reals
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