Nuprl Lemma : pseudo-positive-is-positive
∀x:ℝ. r0 < x supposing pseudo-positive(x)
Proof
Definitions occuring in Statement :
pseudo-positive: pseudo-positive(x),
rless: x < y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
true: True,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
pseudo-positive: pseudo-positive(x),
int-to-real: r(n),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
label: ...$L... t,
so_lambda: λ2x.t[x],
so_apply: x[s],
regular-int-seq: k-regular-seq(f),
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
rabs: |x|,
absval: |i|,
subtract: n - m,
bdd-diff: bdd-diff(f;g),
gt: i > j
Lemmas referenced :
rlessw_wf,
int-to-real_wf,
rless_wf,
pseudo-positive_wf,
real_wf,
weak-Markov-principle-alt,
false_wf,
le_wf,
nat_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
lt_int_wf,
rabs_wf,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel,
less_than_wf,
assert_of_lt_int,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
regularize-k-regular,
accelerate_wf,
regular-int-seq_wf,
nat_plus_wf,
equal-wf-base-T,
not_wf,
nat_plus_properties,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
subtype_rel_dep_function,
nat_plus_subtype_nat,
squash_wf,
true_wf,
regularize_wf,
iff_weakening_equal,
set_wf,
regularize-real,
subtype_rel_sets,
real-regular,
accelerate-req,
itermAdd_wf,
int_term_value_add_lemma,
rless_functionality,
req_weakening,
req-iff-bdd-diff,
bdd-diff_transitivity,
accelerate-bdd-diff,
eq_int_eq_false,
equal-wf-base,
int_subtype_base,
bfalse_wf,
sq_stable__regular-int-seq,
absval_wf,
subtract_wf,
multiply-is-int-iff,
int-triangle-inequality,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_functionality,
le_weakening,
absval_pos,
mul_bounds_1a,
mul_preserves_le,
absval-diff-symmetry,
mul-distributes,
minus-one-mul,
mul-commutes,
mul-associates,
zero-mul,
regularize-regular,
bdd-diff_wf,
all_wf,
add-mul-special,
zero-le-nat,
itermMinus_wf,
int_term_value_minus_lemma,
absval-minus,
sq_stable__less_than,
rabs-rless-iff,
assert_wf,
bnot_wf,
equal-wf-T-base,
not-equal-2,
intformor_wf,
int_formula_prop_or_lemma,
add-associates,
add-zero,
condition-implies-le,
minus-add,
minus-zero,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
absval_lbound,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
independent_pairFormation,
setElimination,
rename,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
imageElimination,
approximateComputation,
int_eqEquality,
functionEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
inlFormation,
inrFormation,
multiplyEquality,
universeEquality,
setEquality,
baseApply,
closedConclusion,
addEquality,
minusEquality,
applyLambdaEquality,
impliesFunctionality,
dependent_set_memberFormation
Latex:
\mforall{}x:\mBbbR{}. r0 < x supposing pseudo-positive(x)
Date html generated:
2017_10_03-AM-09_09_52
Last ObjectModification:
2017_09_20-PM-06_04_48
Theory : reals
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