Nuprl Lemma : strong-continuity-test-prop2
∀[T:Type]
∀M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕ?). ∀n:ℕ. ∀f:ℕn ⟶ T. ∀b:ℕ?.
((↑isl(b))
⇒ (↑isr(strong-continuity-test(M;n;f;b)))
⇒ (∃m:ℕ. (m < n ∧ (↑isl(strong-continuity-test(M;m;f;M m f))))))
Proof
Definitions occuring in Statement :
strong-continuity-test: strong-continuity-test(M;n;f;b),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isr: isr(x),
isl: isl(x),
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
ge: i ≥ j ,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
isr: isr(x),
cand: A c∧ B
Lemmas referenced :
assert_wf,
isr_wf,
nat_wf,
unit_wf2,
strong-continuity-test_wf,
int_seg_wf,
isl_wf,
false_wf,
le_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
all_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
exists_wf,
less_than_wf,
subtype_rel_dep_function,
int_seg_subtype,
nat_properties,
set_wf,
primrec-wf2,
strong-continuity-test-unroll,
isr-not-isl,
subtype_rel_union,
top_wf,
eq_int_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
decidable__assert,
decidable__lt,
not-isl-assert-isr
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
unionEquality,
functionEquality,
universeEquality,
because_Cache,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
instantiate,
productElimination,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (\mBbbN{}?). \mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} T. \mforall{}b:\mBbbN{}?.
((\muparrow{}isl(b))
{}\mRightarrow{} (\muparrow{}isr(strong-continuity-test(M;n;f;b)))
{}\mRightarrow{} (\mexists{}m:\mBbbN{}. (m < n \mwedge{} (\muparrow{}isl(strong-continuity-test(M;m;f;M m f))))))
Date html generated:
2016_12_12-AM-09_22_17
Last ObjectModification:
2016_11_22-AM-11_48_24
Theory : continuity
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