Nuprl Lemma : strong-continuity2-no-inner-squash-unique-bound
∀F:(ℕ ⟶ ℕ) ⟶ ℕ
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
∀f:ℕ ⟶ ℕ. ∃n:ℕ. (F f < n ∧ ((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ (m = n ∈ ℕ)))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
unit: Unit,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
true: True,
outl: outl(x)
Lemmas referenced :
strong-continuity2-no-inner-squash-bound,
implies-quotient-true,
exists_wf,
nat_wf,
int_seg_wf,
unit_wf2,
all_wf,
less_than_wf,
equal_wf,
subtype_rel_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
subtype_rel_union,
assert_wf,
isl_wf,
strong-continuity-test-bound_wf,
decidable__assert,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
lelt_wf,
unit_subtype_base,
int_subtype_base,
le_wf,
set_subtype_base,
union_subtype_base,
subtype_base_sq,
satisfiable-full-omega-tt,
subtype_rel_dep_function,
strong-continuity-test-bound-prop1,
and_wf,
btrue_wf,
bool_wf,
bool_subtype_base,
outl_wf,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
strong-continuity-test-bound-prop3,
not-isl-assert-isr,
strong-continuity-test-bound-prop4,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
functionEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
unionEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
unionElimination,
functionExtensionality,
inrEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
cumulativity,
instantiate,
computeAll,
hyp_replacement,
promote_hyp
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. (F f < n \mwedge{} ((M n f) = (inl (F f))) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} (m = n)))))
Date html generated:
2019_06_20-PM-02_53_52
Last ObjectModification:
2018_08_22-AM-00_06_08
Theory : continuity
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