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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