Nuprl Lemma : weak-continuity-skolem_functionality
∀[T,S:Type].
∀e:T ~ S. ∀F:(ℕ ⟶ S) ⟶ ℕ. (weak-continuity-skolem(T;λf.(F ((fst(e)) o f))) ⇒ weak-continuity-skolem(S;F))
Proof
Definitions occuring in Statement :
weak-continuity-skolem: weak-continuity-skolem(T;F),
equipollent: A ~ B,
compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
pi1: fst(t),
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
weak-continuity-skolem: weak-continuity-skolem(T;F),
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
compose: f o g,
nat: ℕ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject-inverse,
weak-continuity-skolem_wf,
nat_wf,
compose_wf,
equipollent_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
all_wf,
and_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
cumulativity,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
dependent_pairFormation,
natural_numberEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
setElimination,
applyLambdaEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T,S:Type].
\mforall{}e:T \msim{} S. \mforall{}F:(\mBbbN{} {}\mrightarrow{} S) {}\mrightarrow{} \mBbbN{}.
(weak-continuity-skolem(T;\mlambda{}f.(F ((fst(e)) o f))) {}\mRightarrow{} weak-continuity-skolem(S;F))
Date html generated:
2017_04_17-AM-09_54_00
Last ObjectModification:
2017_02_27-PM-05_49_00
Theory : continuity
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