Nuprl Lemma : strong-continuity3-half-squash-equipollent
∀[T:Type]. ∀[B:Type]. (T ~ B ⇒ (∀F:(ℕ ⟶ B) ⟶ ℕ. ⇃(strong-continuity3(B;F)))) supposing (T ⊆r ℕ) ∧ (↓T)
Proof
Definitions occuring in Statement :
strong-continuity3: strong-continuity3(T;F),
equipollent: A ~ B,
quotient: x,y:A//B[x; y],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
prop: ℙ,
pi1: fst(t),
exists: ∃x:A. B[x],
equipollent: A ~ B,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
strong-continuity3_functionality,
implies-quotient-true2,
trivial-quotient-true,
strong-continuity3_wf,
squash_wf,
subtype_rel_wf,
equipollent_wf,
compose_wf,
nat_wf,
strong-continuity3-half-squash
Rules used in proof :
independent_functionElimination,
productEquality,
universeEquality,
because_Cache,
cumulativity,
functionEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_functionElimination,
independent_pairFormation,
independent_isectElimination,
isectElimination,
extract_by_obid,
lambdaFormation,
rename,
baseClosed,
hypothesisEquality,
imageMemberEquality,
imageElimination,
hypothesis,
axiomEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]
\mforall{}[B:Type]. (T \msim{} B {}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} B) {}\mrightarrow{} \mBbbN{}. \00D9(strong-continuity3(B;F)))) supposing (T \msubseteq{}r \mBbbN{}) \mwedge{} (\mdownarrow{}T)
Date html generated:
2017_09_29-PM-06_05_35
Last ObjectModification:
2017_09_03-PM-09_41_34
Theory : continuity
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