Nuprl Lemma : canonicalizable-product
∀[T:Type]. ∀[B:T ⟶ Type]. (canonicalizable(T) ⇒ (∀x:T. canonicalizable(B[x])) ⇒ canonicalizable(x:T × B[x]))
Proof
Definitions occuring in Statement :
canonicalizable: canonicalizable(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
istype-base,
canonicalizable-iff,
canonicalizable_wf,
istype-universe,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
productElimination,
thin,
productIsType,
because_Cache,
universeIsType,
applyEquality,
hypothesisEquality,
sqequalRule,
functionIsType,
introduction,
extract_by_obid,
hypothesis,
equalityIstype,
sqequalBase,
equalitySymmetry,
sqequalHypSubstitution,
independent_functionElimination,
isectElimination,
dependent_functionElimination,
productEquality,
inhabitedIsType,
instantiate,
universeEquality,
rename,
dependent_pairFormation_alt,
baseApply,
closedConclusion,
baseClosed,
dependent_pairEquality_alt,
lambdaEquality_alt,
equalityTransitivity,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[B:T {}\mrightarrow{} Type].
(canonicalizable(T) {}\mRightarrow{} (\mforall{}x:T. canonicalizable(B[x])) {}\mRightarrow{} canonicalizable(x:T \mtimes{} B[x]))
Date html generated:
2019_10_15-AM-10_20_03
Last ObjectModification:
2019_08_29-AM-11_01_39
Theory : call!by!value_2
Home
Index