Nuprl Lemma : old-Kripke2a
(∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ∀f:ℕ ⟶ ℕ. ((P f) ⇒ ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g)))))
⇒ (∀a:{a:ℕ ⟶ ℕ| increasing-sequence(a)} . ∀m:ℕ. (¬¬(∃n:ℕ. ((a n) ≥ m ))))
Proof
Definitions occuring in Statement :
increasing-sequence: increasing-sequence(a),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
true: True,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
squash: ↓T
Lemmas referenced :
not_wf,
exists_wf,
nat_wf,
ge_wf,
set_wf,
increasing-sequence_wf,
all_wf,
quotient_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
equiv_rel_true,
not-quotient-true,
decidable__lt,
nat_properties,
decidable__le,
le_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
increasing-sequence-prop1,
int_seg_properties,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtract_wf,
add_nat_wf,
itermConstant_wf,
itermSubtract_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
decidable__equal_int,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
functionEquality,
instantiate,
cumulativity,
universeEquality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
productElimination,
unionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
addEquality,
applyLambdaEquality,
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{} (\mforall{}a:\{a:\mBbbN{} {}\mrightarrow{} \mBbbN{}| increasing-sequence(a)\} . \mforall{}m:\mBbbN{}. (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}. ((a n) \mgeq{} m ))))
Date html generated:
2017_09_29-PM-06_09_25
Last ObjectModification:
2017_04_22-PM-05_25_48
Theory : continuity
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