Nuprl Lemma : function-valueall-type
∀[A:Type]. ∀[B:A ⟶ Type]. valueall-type(a:A ⟶ B[a]) supposing ↓∃a:A. value-type(B[a])
Proof
Definitions occuring in Statement :
valueall-type: valueall-type(T),
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
so_apply: x[s],
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
valueall-type: valueall-type(T),
has-value: (a)↓,
evalall: evalall(t),
all: ∀x:A. B[x],
or: P ∨ Q,
top: Top,
not: ¬A,
false: False,
outl: outl(x),
outr: outr(x),
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
sq_stable__valueall-type,
value-type-has-value,
bottom_diverge,
has-value-implies-dec-ispair,
bottom-sqle,
has-value-implies-dec-isinl,
has-value-implies-dec-isinr,
equal-wf-base,
base_wf,
squash_wf,
exists_wf,
value-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
thin,
functionEquality,
hypothesisEquality,
applyEquality,
hypothesis,
independent_functionElimination,
productElimination,
rename,
independent_isectElimination,
callbyvalueApply,
sqequalRule,
callbyvalueReduce,
dependent_functionElimination,
baseApply,
closedConclusion,
baseClosed,
unionElimination,
sqequalSqle,
applyPair,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
applyInl,
applyInr,
axiomSqleEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
lambdaEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. valueall-type(a:A {}\mrightarrow{} B[a]) supposing \mdownarrow{}\mexists{}a:A. value-type(B[a])
Date html generated:
2019_06_20-AM-11_21_28
Last ObjectModification:
2018_08_16-AM-11_09_04
Theory : call!by!value_1
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