Nuprl Lemma : gen-continuity-contradicts-markov
(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ. ((P f) ⇒ ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g)))))
⇒ (¬(∀A:ℕ ⟶ ℙ. ((∀n:ℕ. ((A n) ∨ (¬(A n)))) ⇒ (¬¬(∃n:ℕ. (A n))) ⇒ (∃n:ℕ. (A n)))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
not: ¬A,
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],
nat: ℕ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
is-absolutely-free: is-absolutely-free{i:l}(f),
increasing-sequence: increasing-sequence(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
istype-nat,
subtype_rel_self,
istype-void,
quotient_wf,
nat_wf,
equal_wf,
int_seg_wf,
true_wf,
equiv_rel_true,
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,
sq_stable_from_decidable,
equal-wf-base,
itermAdd_wf,
int_term_value_add_lemma,
set_subtype_base,
int_subtype_base,
decidable__equal_nat,
add_nat_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
Kripke2b,
init0-zero-seq,
increasing-zero-seq,
zero-seq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
sqequalRule,
functionIsType,
introduction,
extract_by_obid,
universeIsType,
universeEquality,
unionIsType,
applyEquality,
hypothesisEquality,
instantiate,
isectElimination,
productIsType,
inhabitedIsType,
productEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
lambdaEquality_alt,
equalityIstype,
independent_isectElimination,
setIsType,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
unionEquality,
productElimination,
addEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
(\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((P f) {}\mRightarrow{} \00D9(\mexists{}k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (P g)))))
{}\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:
2020_05_19-PM-10_06_04
Last ObjectModification:
2020_01_04-PM-08_04_11
Theory : continuity
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