Nuprl Lemma : afcs-contradicts-markov
(↓∃a:ℕ ⟶ ℕ. (is-absolutely-free{i:l}(a) ∧ init0(a) ∧ increasing-sequence(a)))
⇒ (¬(∀A:ℕ ⟶ ℙ. ((∀n:ℕ. ((A n) ∨ (¬(A n)))) ⇒ (¬¬(∃n:ℕ. (A n))) ⇒ (∃n:ℕ. (A n)))))
Proof
Definitions occuring in Statement :
is-absolutely-free: is-absolutely-free{i:l}(f),
init0: init0(a),
increasing-sequence: increasing-sequence(a),
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
implies: P ⇒ Q,
not: ¬A,
squash: ↓T,
false: False,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
istype-nat,
subtype_rel_self,
istype-void,
squash_wf,
nat_wf,
is-absolutely-free_wf,
init0_wf,
increasing-sequence_wf,
ge_wf,
nat_properties,
decidable__or,
le_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_wf,
istype-le,
not_wf,
decidable__not,
intformor_wf,
int_formula_prop_or_lemma,
Kripke2a,
Kripke2b
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
imageElimination,
cut,
hypothesis,
sqequalRule,
Error :functionIsType,
introduction,
extract_by_obid,
Error :universeIsType,
universeEquality,
because_Cache,
Error :unionIsType,
applyEquality,
hypothesisEquality,
thin,
instantiate,
isectElimination,
Error :productIsType,
productEquality,
cumulativity,
functionEquality,
dependent_functionElimination,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
independent_functionElimination,
productElimination,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation
Latex:
(\mdownarrow{}\mexists{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (is-absolutely-free\{i:l\}(a) \mwedge{} init0(a) \mwedge{} increasing-sequence(a)))
{}\mRightarrow{} (\mneg{}(\mforall{}A:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}n:\mBbbN{}. ((A n) \mvee{} (\mneg{}(A n)))) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. (A n))) {}\mRightarrow{} (\mexists{}n:\mBbbN{}. (A n)))))
Date html generated:
2019_06_20-PM-03_08_08
Last ObjectModification:
2019_01_17-PM-10_28_05
Theory : continuity
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