Nuprl Lemma : strong-continuity-test-prop1
∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕ?)]. ∀[n:ℕ]. ∀[f:ℕn ⟶ T]. ∀[b:ℕ?].
((↑isl(strong-continuity-test(M;n;f;b)))
⇒ ((↑isl(b)) ∧ (∀i:ℕ. (i < n ⇒ (↑isr(M i f)))) ∧ (strong-continuity-test(M;n;f;b) = b ∈ (ℕ?))))
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],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
primrec: primrec(n;b;c),
strong-continuity-test: strong-continuity-test(M;n;f;b),
cand: A c∧ B,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
sq_type: SQType(T),
assert: ↑b,
isl: isl(x),
rev_uimplies: rev_uimplies(P;Q),
true: True,
label: ...$L... t,
squash: ↓T,
nequal: a ≠ b ∈ T ,
bnot: ¬bb,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-it: exposed-it
Lemmas referenced :
assert_witness,
less_than_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
false_wf,
int_seg_subtype,
subtype_rel_dep_function,
isr_wf,
int_seg_wf,
strong-continuity-test_wf,
unit_wf2,
nat_wf,
isl_wf,
assert_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le_wf,
less_than_irreflexivity,
less_than_transitivity1,
ge_wf,
int_term_value_constant_lemma,
itermConstant_wf,
full-omega-unsat,
strong-continuity-test-unroll,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_subtype_base,
equal-wf-base,
not_wf,
bnot_wf,
eq_int_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__lt,
decidable__equal_int,
true_wf,
squash_wf,
assert_functionality_wrt_uiff,
not-isl-assert-isr,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
equal_wf
Rules used in proof :
axiomEquality,
independent_functionElimination,
independent_pairEquality,
productElimination,
isect_memberFormation,
universeEquality,
functionEquality,
unionEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
lambdaEquality,
sqequalRule,
because_Cache,
rename,
setElimination,
natural_numberEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality,
intWeakElimination,
approximateComputation,
impliesFunctionality,
instantiate,
baseClosed,
equalitySymmetry,
equalityTransitivity,
imageMemberEquality,
imageElimination,
promote_hyp,
equalityElimination
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(strong-continuity-test(M;n;f;b)))
{}\mRightarrow{} ((\muparrow{}isl(b)) \mwedge{} (\mforall{}i:\mBbbN{}. (i < n {}\mRightarrow{} (\muparrow{}isr(M i f)))) \mwedge{} (strong-continuity-test(M;n;f;b) = b)))
Date html generated:
2018_05_21-PM-01_17_44
Last ObjectModification:
2018_05_18-PM-04_03_28
Theory : continuity
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