Nuprl Lemma : weak-Markov-principle-alt
∀a,b:ℕ ⟶ ℕ. ((∀c:ℕ ⟶ ℕ. ((¬(a = c ∈ (ℕ ⟶ ℕ))) ∨ (¬(b = c ∈ (ℕ ⟶ ℕ))))) ⇒ (∃n:ℕ. (¬((a n) = (b n) ∈ ℤ))))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
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],
so_apply: x[s],
or: P ∨ Q,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
cand: A c∧ B,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
subtype_rel: A ⊆r B,
pi1: fst(t),
decidable: Dec(P),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
all_wf,
nat_wf,
or_wf,
not_wf,
equal_wf,
false_wf,
le_wf,
equal-wf-base,
equal-wf-T-base,
subtype_base_sq,
int_subtype_base,
strong-continuity2-implies-weak,
exists_wf,
decidable__equal_int,
quotient-implies-squash,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
subtype_rel_self,
equal-wf-base-T,
member_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__exists_int_seg,
decidable__not,
decidable__equal_nat,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mu-dec-property,
unit_wf2,
it_wf,
mu-dec_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
intEquality,
baseClosed,
because_Cache,
independent_functionElimination,
voidElimination,
productEquality,
setElimination,
rename,
addLevel,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
levelHypothesis,
promote_hyp,
productElimination,
applyEquality,
functionExtensionality,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}a,b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}c:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mneg{}(a = c)) \mvee{} (\mneg{}(b = c)))) {}\mRightarrow{} (\mexists{}n:\mBbbN{}. (\mneg{}((a n) = (b n)))))
Date html generated:
2018_05_21-PM-01_18_17
Last ObjectModification:
2018_05_19-AM-06_37_15
Theory : continuity
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