Nuprl Lemma : strong-continuity-test-prop3
∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕ?)]. ∀[n,m:ℕ]. ∀[f:ℕ ⟶ T].
((↑isl(strong-continuity-test(M;n;f;M n f))) ⇒ (↑isl(strong-continuity-test(M;m;f;M m f))) ⇒ (n = m ∈ ℤ))
Proof
Definitions occuring in Statement :
strong-continuity-test: strong-continuity-test(M;n;f;b),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
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],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
top_wf,
subtype_rel_union,
isr-not-isl,
decidable__lt,
decidable__equal_int,
strong-continuity-test-prop1,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_dep_function,
strong-continuity-test_wf,
unit_wf2,
nat_wf,
isl_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
functionEquality,
unionEquality,
universeEquality,
isect_memberFormation,
introduction,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (\mBbbN{}?)]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} T].
((\muparrow{}isl(strong-continuity-test(M;n;f;M n f)))
{}\mRightarrow{} (\muparrow{}isl(strong-continuity-test(M;m;f;M m f)))
{}\mRightarrow{} (n = m))
Date html generated:
2016_05_19-AM-11_59_28
Last ObjectModification:
2016_05_16-PM-05_42_13
Theory : continuity
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