Nuprl Lemma : kripke2b-seq_wf
∀[a:ℕ ⟶ ℕ]. ∀[x:ℕ]. ∀[F:∀b:{b:ℕ ⟶ ℕ| a = b ∈ (ℕx ⟶ ℕ)} . ∃n:ℕ. ((b n) ≥ ((a x) + 1) )].
(kripke2b-seq(a;x;F) ∈ (ℕ ⟶ ℕ) ⟶ ℕ)
Proof
Definitions occuring in Statement :
kripke2b-seq: kripke2b-seq(a;x;F),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
exists: ∃x:A. B[x],
so_apply: x[s],
prop: ℙ,
ge: i ≥ j ,
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
kripke2b-seq: kripke2b-seq(a;x;F),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
eq-finite-seqs-implies-eq-upto,
subtype_rel_self,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
all_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
equal_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
decidable__le,
nat_properties,
le_wf,
false_wf,
add_nat_wf,
nat_wf,
exists_wf,
ge_wf,
pi1_wf,
min-inc-seq_wf,
eqtt_to_assert,
bool_wf,
eq-finite-seqs_wf
Rules used in proof :
functionEquality,
setEquality,
axiomEquality,
cumulativity,
instantiate,
independent_functionElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
baseClosed,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
dependent_functionElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
addEquality,
rename,
setElimination,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
hypothesis,
because_Cache,
hypothesisEquality,
applyEquality,
functionExtensionality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[x:\mBbbN{}]. \mforall{}[F:\mforall{}b:\{b:\mBbbN{} {}\mrightarrow{} \mBbbN{}| a = b\} . \mexists{}n:\mBbbN{}. ((b n) \mgeq{} ((a x) + 1) )].
(kripke2b-seq(a;x;F) \mmember{} (\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{})
Date html generated:
2017_04_20-AM-07_37_24
Last ObjectModification:
2017_04_19-PM-03_34_36
Theory : continuity
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