Nuprl Lemma : strong-continuity-test-bound-prop1
∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕn?)]. ∀[n:ℕ]. ∀[f:ℕn ⟶ T]. ∀[b:ℕn].
((↑isl(strong-continuity-test-bound(M;n;f;b))) ⇒ (strong-continuity-test-bound(M;n;f;b) = (inl b) ∈ (ℕn?)))
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),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
unit: Unit,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
isl: isl(x),
assert: ↑b,
bfalse: ff,
sq_type: SQType(T),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
assert_wf,
isl_wf,
int_seg_wf,
unit_wf2,
strong-continuity-test-bound_wf,
nat_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
strong-continuity-test-bound-unroll,
subtype_rel_dep_function,
subtype_rel_union,
int_seg_subtype_nat,
false_wf,
eq_int_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
lt_int_wf,
equal-wf-base-T,
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,
assert_of_lt_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
int_seg_subtype,
decidable__lt,
lelt_wf,
iff_weakening_equal
Rules used in proof :
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
productElimination,
cumulativity,
functionExtensionality,
applyEquality,
because_Cache,
functionEquality,
unionElimination,
dependent_set_memberEquality,
unionEquality,
universeEquality,
isect_memberFormation,
equalityTransitivity,
equalitySymmetry,
baseClosed,
baseApply,
closedConclusion,
instantiate,
impliesFunctionality,
equalityElimination,
promote_hyp,
inlEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (\mBbbN{}n?)]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} T]. \mforall{}[b:\mBbbN{}n].
((\muparrow{}isl(strong-continuity-test-bound(M;n;f;b)))
{}\mRightarrow{} (strong-continuity-test-bound(M;n;f;b) = (inl b)))
Date html generated:
2017_04_17-AM-10_00_44
Last ObjectModification:
2017_02_27-PM-05_53_24
Theory : continuity
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