Nuprl Lemma : strong-continuity3-implies-4
∀[T:Type]. ∀F:(ℕ ⟶ T) ⟶ ℕ. (strong-continuity3(T;F) ⇒ strong-continuity4(T;F))
Proof
Definitions occuring in Statement :
strong-continuity4: strong-continuity4(T;F),
strong-continuity3: strong-continuity3(T;F),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
strong-continuity3: strong-continuity3(T;F),
exists: ∃x:A. B[x],
strong-continuity4: strong-continuity4(T;F),
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
guard: {T},
isl: isl(x),
outl: outl(x),
so_apply: x[s],
pi1: fst(t),
bfalse: ff,
top: Top,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
sq_type: SQType(T),
cand: A c∧ B
Lemmas referenced :
strong-continuity3_wf,
istype-nat,
istype-universe,
decidable__exists_int_seg,
assert_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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_wf,
istype-le,
subtype_rel_function,
int_seg_wf,
int_seg_subtype,
istype-false,
sq_stable__le,
le_weakening2,
subtype_rel_self,
btrue_wf,
bfalse_wf,
less_than_wf,
assert_elim,
btrue_neq_bfalse,
decidable__and2,
istype-assert,
decidable__assert,
decidable__lt,
unit_wf2,
union_subtype_base,
set_subtype_base,
lelt_wf,
int_subtype_base,
unit_subtype_base,
nat_wf,
int_seg_subtype_nat,
subtype_rel_union,
it_wf,
le_wf,
true_wf,
istype-void,
imax_wf,
add_nat_wf,
imax_nat,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
pi1_wf,
decidable_wf,
iff_weakening_equal,
equal_wf,
subtype_base_sq,
squash_wf,
bool_subtype_base,
bool_wf,
isl_wf,
int_formula_prop_less_lemma,
intformless_wf,
imax_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
functionIsType,
because_Cache,
instantiate,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
productEquality,
applyEquality,
dependent_set_memberEquality_alt,
imageElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
productIsType,
applyLambdaEquality,
isect_memberEquality_alt,
unionIsType,
intEquality,
inlEquality_alt,
sqequalBase,
equalityIsType1,
inrEquality_alt,
functionExtensionality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
dependent_pairEquality_alt,
functionEquality,
unionEquality,
cumulativity,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}. (strong-continuity3(T;F) {}\mRightarrow{} strong-continuity4(T;F))
Date html generated:
2020_05_19-PM-10_04_36
Last ObjectModification:
2020_01_04-PM-08_04_04
Theory : continuity
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