Nuprl Lemma : weak-continuity-principle-nat-int
∀F:(ℕ ⟶ ℤ) ⟶ ℕ. ∀f:ℕ ⟶ ℤ. ∀G:n:ℕ ⟶ {g:ℕ ⟶ ℤ| f = g ∈ (ℕn ⟶ ℤ)} .  ∃n:ℕ. ((F f) = (F (G n)) ∈ ℕ)
Proof
Definitions occuring in Statement : 
int_seg: {i..j-}
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
natural_number: $n
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
nat: ℕ
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
uimplies: b supposing a
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
squash: ↓T
, 
exists: ∃x:A. B[x]
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
Lemmas referenced : 
weak-continuity-nat-int, 
nat_wf, 
equal_wf, 
int_seg_wf, 
subtype_rel_dep_function, 
int_seg_subtype_nat, 
false_wf, 
subtype_rel_self, 
squash-from-quotient, 
exists_wf, 
all_wf, 
set_wf, 
and_wf, 
mu_wf, 
eq_int_wf, 
assert_of_eq_int, 
nat_properties, 
decidable__equal_int, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformeq_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
assert_wf, 
mu-property, 
decidable__le, 
intformle_wf, 
itermConstant_wf, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
le_wf
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
hypothesis, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
functionEquality, 
setEquality, 
because_Cache, 
intEquality, 
isectElimination, 
natural_numberEquality, 
setElimination, 
rename, 
applyEquality, 
sqequalRule, 
lambdaEquality, 
independent_isectElimination, 
independent_pairFormation, 
functionExtensionality, 
independent_functionElimination, 
imageElimination, 
productElimination, 
dependent_pairFormation, 
imageMemberEquality, 
baseClosed, 
addLevel, 
levelHypothesis, 
equalitySymmetry, 
dependent_set_memberEquality, 
equalityTransitivity, 
applyLambdaEquality, 
unionElimination, 
approximateComputation, 
int_eqEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality
Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbZ{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}G:n:\mBbbN{}  {}\mrightarrow{}  \{g:\mBbbN{}  {}\mrightarrow{}  \mBbbZ{}|  f  =  g\}  .    \mexists{}n:\mBbbN{}.  ((F  f)  =  (F  (G  n)))
Date html generated:
2017_09_29-PM-06_06_03
Last ObjectModification:
2017_07_08-AM-11_36_19
Theory : continuity
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