Nuprl Lemma : co-W-ext
∀[A:Type]. ∀[B:A ⟶ Type]. co-W(A;a.B[a]) ≡ a:A × (B[a] ⟶ co-W(A;a.B[a]))
Proof
Definitions occuring in Statement :
co-W: co-W(A;a.B[a]),
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
co-W: co-W(A;a.B[a]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
ext-eq: A ≡ B
Lemmas referenced :
corec-ext,
continuous-monotone-depproduct,
continuous-monotone-constant,
subtype_rel_dep_function,
subtype_rel_wf,
false_wf,
le_wf,
equal_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
hypothesisEquality,
functionEquality,
applyEquality,
functionExtensionality,
universeEquality,
independent_isectElimination,
hypothesis,
lambdaFormation,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isectEquality,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. co-W(A;a.B[a]) \mequiv{} a:A \mtimes{} (B[a] {}\mrightarrow{} co-W(A;a.B[a]))
Date html generated:
2018_05_21-PM-10_18_24
Last ObjectModification:
2017_07_26-PM-06_36_38
Theory : bar!induction
Home
Index