Nuprl Lemma : rpositive-int
∀[n:ℤ]. uiff(rpositive(r(n));0 < n)
Proof
Definitions occuring in Statement :
rpositive: rpositive(x),
int-to-real: r(n),
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
int-to-real: r(n),
rpositive: rpositive(x),
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
nat_plus_subtype_nat,
mul_preserves_le,
decidable__lt,
member-less_than,
less_than_wf,
nat_plus_wf,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
lemma_by_obid,
isectElimination,
lambdaEquality,
natural_numberEquality,
multiplyEquality,
hypothesisEquality,
introduction,
independent_isectElimination,
intEquality,
dependent_functionElimination,
unionElimination,
because_Cache,
applyEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberFormation,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n:\mBbbZ{}]. uiff(rpositive(r(n));0 < n)
Date html generated:
2016_05_18-AM-07_02_04
Last ObjectModification:
2016_01_17-AM-01_49_02
Theory : reals
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