Nuprl Lemma : strong-continuity-test-bound-prop4
∀M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?). ∀n:ℕ. ∀f:ℕn ⟶ ℕ. ∀b:ℕn.
((↑isr(strong-continuity-test-bound(M;n;f;b)))
⇒ (∃m:ℕ. (b < m ∧ m < n ∧ (↑isl(M m f)) ∧ (↑isl(strong-continuity-test-bound(M;m;f;b))))))
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,
isr: isr(x),
isl: isl(x),
less_than: a < b,
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
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
ifthenelse: if b then t else f fi ,
btrue: tt,
isr: isr(x),
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
cand: A c∧ B
Lemmas referenced :
not-isl-assert-isr,
decidable__lt,
decidable__assert,
assert_of_lt_int,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
assert_of_bnot,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
equal-wf-base-T,
lt_int_wf,
int_subtype_base,
equal-wf-base,
not_wf,
bnot_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
eq_int_wf,
int_seg_subtype_nat,
subtype_rel_union,
strong-continuity-test-bound-unroll,
primrec-wf2,
set_wf,
lelt_wf,
subtype_rel_self,
nat_properties,
false_wf,
int_seg_subtype,
subtype_rel_dep_function,
isl_wf,
less_than_wf,
exists_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
all_wf,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
nat_wf,
strong-continuity-test-bound_wf,
unit_wf2,
int_seg_wf,
isr_wf,
assert_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
functionEquality,
dependent_set_memberEquality,
unionElimination,
productEquality,
introduction,
unionEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
baseApply,
closedConclusion,
instantiate,
cumulativity,
independent_functionElimination,
impliesFunctionality
Latex:
\mforall{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?). \mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. \mforall{}b:\mBbbN{}n.
((\muparrow{}isr(strong-continuity-test-bound(M;n;f;b)))
{}\mRightarrow{} (\mexists{}m:\mBbbN{}. (b < m \mwedge{} m < n \mwedge{} (\muparrow{}isl(M m f)) \mwedge{} (\muparrow{}isl(strong-continuity-test-bound(M;m;f;b))))))
Date html generated:
2016_05_19-PM-00_00_03
Last ObjectModification:
2016_05_17-PM-05_52_21
Theory : continuity
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