Nuprl Lemma : strong-continuity-test-bound-prop3
∀[M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)]. ∀[n,m:ℕ]. ∀[f:ℕ ⟶ ℕ]. ∀[b:ℕn].
(b < m
⇒ (↑isl(M n f))
⇒ (↑isl(M m f))
⇒ (↑isl(strong-continuity-test-bound(M;n;f;b)))
⇒ (↑isl(strong-continuity-test-bound(M;m;f;b)))
⇒ (m = n ∈ ℕ))
Proof
Definitions occuring in Statement :
strong-continuity-test-bound: strong-continuity-test-bound(M;n;f;b),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
less_than: a < b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
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,
prop: ℙ,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
le_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
isr-not-isl,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
int_seg_properties,
strong-continuity-test-bound-prop2,
decidable__lt,
less_than_wf,
lelt_wf,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
nat_wf,
strong-continuity-test-bound_wf,
unit_wf2,
int_seg_wf,
isl_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
functionExtensionality,
applyEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
productElimination,
functionEquality,
unionEquality,
isect_memberFormation,
introduction,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
unionElimination,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[b:\mBbbN{}n].
(b < m
{}\mRightarrow{} (\muparrow{}isl(M n f))
{}\mRightarrow{} (\muparrow{}isl(M m f))
{}\mRightarrow{} (\muparrow{}isl(strong-continuity-test-bound(M;n;f;b)))
{}\mRightarrow{} (\muparrow{}isl(strong-continuity-test-bound(M;m;f;b)))
{}\mRightarrow{} (m = n))
Date html generated:
2016_05_19-AM-11_59_55
Last ObjectModification:
2016_05_17-PM-05_02_57
Theory : continuity
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