Nuprl Lemma : basic-implies-strong-continuity2
∀[T:Type]. ∀[F:(ℕ ⟶ T) ⟶ ℕ]. (basic-strong-continuity(T;F) ⇒ strong-continuity2(T;F))
Proof
Definitions occuring in Statement :
strong-continuity2: strong-continuity2(T;F),
basic-strong-continuity: basic-strong-continuity(T;F),
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
basic-strong-continuity: basic-strong-continuity(T;F),
member: t ∈ T,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
nat: ℕ,
bool: 𝔹,
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
b-union: A ⋃ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
false: False,
not: ¬A,
isl: isl(x),
int?: int?(x),
subtype_rel: A ⊆r B,
pi2: snd(t),
tunion: ⋃x:A.B[x],
unit: Unit,
sq_stable: SqStable(P),
squash: ↓T,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
sq_type: SQType(T),
has-value: (a)↓,
less_than': less_than'(a;b),
le: A ≤ B,
true: True,
assert: ↑b,
btrue: tt,
bfalse: ff,
strong-continuity2: strong-continuity2(T;F),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
basic-strong-continuity_wf,
istype-nat,
istype-universe,
int_seg_wf,
it_wf,
unit_wf2,
product_subtype_base,
int_subtype_base,
set_subtype_base,
ifthenelse_wf,
bool_wf,
tunion_subtype_base,
product-value-type,
int-value-type,
istype-int,
le_wf,
set-value-type,
bunion-value-type,
nat_wf,
b-union_wf,
int?_wf,
btrue_neq_bfalse,
bfalse_wf,
btrue_wf,
istype-sqequal,
equal-wf-base,
sq_stable__all,
sq_stable__equal,
istype-le,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
subtype_base_sq,
value-type-has-value,
subtype_rel_self,
istype-false,
int_seg_subtype_nat,
subtype_rel_function,
istype-true,
istype-assert,
unit_subtype_base,
union_subtype_base,
equal-wf-base-T,
squash_wf,
iff_weakening_equal,
true_wf,
equal_wf,
mu_wf,
bool_subtype_base,
mu-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionIsType,
because_Cache,
instantiate,
universeEquality,
setElimination,
rename,
dependent_pairFormation_alt,
natural_numberEquality,
lambdaEquality_alt,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
inrEquality_alt,
inlEquality_alt,
applyEquality,
unionElimination,
independent_pairFormation,
inhabitedIsType,
intEquality,
sqequalRule,
independent_isectElimination,
productEquality,
closedConclusion,
voidElimination,
applyLambdaEquality,
productIsType,
dependent_set_memberEquality_alt,
productElimination,
sqequalBase,
setIsType,
unionIsType,
baseClosed,
baseApply,
sqequalIntensionalEquality,
setEquality,
unionEquality,
imageElimination,
equalityElimination,
axiomEquality,
functionIsTypeImplies,
imageMemberEquality,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
cumulativity,
isintReduceTrue,
callbyvalueReduce,
independent_pairEquality,
isectIsType,
dependent_pairEquality_alt
Latex:
\mforall{}[T:Type]. \mforall{}[F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} \mBbbN{}]. (basic-strong-continuity(T;F) {}\mRightarrow{} strong-continuity2(T;F))
Date html generated:
2020_05_19-PM-10_04_31
Last ObjectModification:
2020_01_04-PM-08_04_32
Theory : continuity
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