Nuprl Lemma : weak-Markov-principle2
∀a:ℕ*. ((∀c:ℕ*. ((¬¬(∃n:ℕ. (¬((a n) = (c n) ∈ ℤ)))) ∨ (¬¬(∃n:ℕ. (¬(0 = (c n) ∈ ℤ)))))) ⇒ (∃n:ℕ. 0 < a n))
Proof
Definitions occuring in Statement :
nat-star: ℕ*,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat-star: ℕ*,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
or: P ∨ Q,
cand: A c∧ B,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
exists: ∃x:A. B[x],
true: True,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
pi1: fst(t),
decidable: Dec(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat-star-0: 0,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
all_wf,
nat-star_wf,
or_wf,
not_wf,
exists_wf,
nat_wf,
equal_wf,
equal-wf-base-T,
nat-star-retract_wf,
equal-wf-T-base,
equal-wf-base,
le_wf,
false_wf,
int_subtype_base,
subtype_base_sq,
strong-continuity2-implies-weak,
decidable__equal_int,
nat-star-0_wf,
squash_wf,
true_wf,
nat-star-retract-id,
subtype_rel_self,
iff_weakening_equal,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
quotient-implies-squash,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
decidable__not,
decidable__exists_int_seg,
decidable__equal_nat,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_seg_properties,
unit_wf2,
mu-dec-property,
it_wf,
less_than_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
mu-dec_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
intEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
baseClosed,
functionEquality,
unionElimination,
functionExtensionality,
dependent_functionElimination,
productEquality,
voidElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
cumulativity,
instantiate,
addLevel,
productElimination,
imageElimination,
universeEquality,
imageMemberEquality,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
applyLambdaEquality
Latex:
\mforall{}a:\mBbbN{}*. ((\mforall{}c:\mBbbN{}*. ((\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. (\mneg{}((a n) = (c n))))) \mvee{} (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. (\mneg{}(0 = (c n))))))) {}\mRightarrow{} (\mexists{}n:\mBbbN{}. 0 < a n))
Date html generated:
2018_05_21-PM-01_19_02
Last ObjectModification:
2018_05_15-PM-04_32_46
Theory : continuity
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