Nuprl Lemma : KleeneM_wf
∀[T:{T:Type| (T ⊆r ℕ) ∧ (↓T)} ]. ∀[F:(ℕ ⟶ T) ⟶ ℕ]. ∀[f:ℕ ⟶ T].
(KleeneM(F;f) ∈ ⇃({m:ℕ+| ∀g:ℕ ⟶ T. ((g = f ∈ (ℕm ⟶ T)) ⇒ ((F g) = (F f) ∈ ℤ))} ))
Proof
Definitions occuring in Statement :
KleeneM: KleeneM(F;f),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
KleeneM: KleeneM(F;f),
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
cand: A c∧ B,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True
Lemmas referenced :
KleeneSearch_wf,
subtype_rel_wf,
nat_wf,
squash_wf,
Kleene-M_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
quotient_subtype_quotient,
subtype_rel_sets,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
subtype_rel_self,
set_subtype_base,
le_wf,
int_subtype_base,
istype-nat,
equiv_rel_true,
true_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
Error :dependent_set_memberEquality_alt,
hypothesisEquality,
independent_pairFormation,
hypothesis,
Error :productIsType,
Error :universeIsType,
because_Cache,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
voidElimination,
applyEquality,
Error :lambdaFormation_alt,
Error :equalityIstype,
Error :functionIsType,
Error :inhabitedIsType,
intEquality,
sqequalBase,
equalitySymmetry,
Error :setIsType,
axiomEquality,
equalityTransitivity,
Error :isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[T:\{T:Type| (T \msubseteq{}r \mBbbN{}) \mwedge{} (\mdownarrow{}T)\} ]. \mforall{}[F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} T].
(KleeneM(F;f) \mmember{} \00D9(\{m:\mBbbN{}\msupplus{}| \mforall{}g:\mBbbN{} {}\mrightarrow{} T. ((g = f) {}\mRightarrow{} ((F g) = (F f)))\} ))
Date html generated:
2019_06_20-PM-02_51_12
Last ObjectModification:
2019_02_11-AM-11_26_26
Theory : continuity
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