Nuprl Lemma : KleeneSearch_wf
∀[T:{T:Type| (T ⊆r ℕ) ∧ (↓T)} ]. ∀[F:(ℕ ⟶ T) ⟶ ℕ]. ∀[M:⇃(basic-strong-continuity(T;F))]. ∀[f:ℕ ⟶ T]. ∀[start:ℕ].
(KleeneSearch(M;f;start) ∈ ⇃({m:ℕ| (start ≤ m) ∧ (∀g:ℕ ⟶ T. ((g = f ∈ (ℕm ⟶ T)) ⇒ ((F g) = (F f) ∈ ℤ)))} ))
Proof
Definitions occuring in Statement :
KleeneSearch: KleeneSearch(M;f;n),
basic-strong-continuity: basic-strong-continuity(T;F),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
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: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
basic-strong-continuity: basic-strong-continuity(T;F),
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
exists: ∃x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
KleeneSearch: KleeneSearch(M;f;n),
less_than': less_than'(a;b),
has-value: (a)↓,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
cand: A c∧ B,
label: ...$L... t,
pi1: fst(t),
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
value-type-has-value,
b-union_wf,
bunion-value-type,
set-value-type,
le_wf,
int-value-type,
product-value-type,
itermAdd_wf,
int_term_value_add_lemma,
basic-strong-continuity_wf,
quotient_wf,
all_wf,
equal_wf,
equal-wf-base,
true_wf,
equiv_rel_true,
quotient-member-eq,
istype-true,
istype-nat,
istype-universe,
subtype_rel_wf,
squash_wf,
false_wf,
istype-top,
top_wf,
pi2_wf,
isint-int,
iff_weakening_equal,
subtype_rel-equal,
trivial-equal,
member_wf,
ext-eq_weakening,
subtype_rel_weakening,
subtype_rel_b-union-left,
product_subtype_base,
ifthenelse_wf,
bool_wf,
tunion_subtype_base,
subtype_rel_b-union-right,
imax_wf,
imax_nat,
add_nat_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
add_functionality_wrt_eq,
imax_unfold,
imax_ub,
le_functionality,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
productElimination,
extract_by_obid,
isectElimination,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
applyLambdaEquality,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
Error :dependent_set_memberEquality_alt,
because_Cache,
Error :productIsType,
hypothesis_subsumption,
callbyvalueReduce,
productEquality,
imageElimination,
equalityElimination,
isintReduceTrue,
Error :equalityIstype,
addEquality,
pointwiseFunctionalityForEquality,
setEquality,
functionEquality,
closedConclusion,
Error :setIsType,
Error :functionIsType,
sqequalBase,
pertypeElimination,
promote_hyp,
Error :isectIsTypeImplies,
universeEquality,
independent_pairEquality,
baseClosed,
imageMemberEquality,
Error :inrFormation_alt
Latex:
\mforall{}[T:\{T:Type| (T \msubseteq{}r \mBbbN{}) \mwedge{} (\mdownarrow{}T)\} ]. \mforall{}[F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}]. \mforall{}[M:\00D9(basic-strong-continuity(T;F))].
\mforall{}[f:\mBbbN{} {}\mrightarrow{} T]. \mforall{}[start:\mBbbN{}].
(KleeneSearch(M;f;start) \mmember{} \00D9(\{m:\mBbbN{}| (start \mleq{} m) \mwedge{} (\mforall{}g:\mBbbN{} {}\mrightarrow{} T. ((g = f) {}\mRightarrow{} ((F g) = (F f))))\} ))
Date html generated:
2019_06_20-PM-02_51_01
Last ObjectModification:
2019_03_06-AM-10_52_09
Theory : continuity
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