Nuprl Lemma : pair-list-set-type
∀[T:Type]. ∀[B:T ⟶ Type]. ∀[L:(t:T × B[t]) List]. (L ∈ (t:{t:T| (t ∈ map(λp.(fst(p));L))} × B[t]) List)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
member: t ∈ T,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
pi1: fst(t),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x]
Lemmas referenced :
list-set-type,
subtype_rel_list,
l_member_wf,
map_wf,
list-subtype,
list_wf,
member_map,
l_member-settype,
and_wf,
equal_wf,
pi1_wf_top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
applyEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
setEquality,
sqequalRule,
cumulativity,
because_Cache,
productElimination,
dependent_pairEquality,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
independent_isectElimination,
axiomEquality,
isect_memberEquality,
functionEquality,
universeEquality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[B:T {}\mrightarrow{} Type]. \mforall{}[L:(t:T \mtimes{} B[t]) List]. (L \mmember{} (t:\{t:T| (t \mmember{} map(\mlambda{}p.(fst(p));L))\} \mtimes{} B[t])\000C List)
Date html generated:
2016_05_15-PM-03_55_34
Last ObjectModification:
2015_12_27-PM-01_25_34
Theory : general
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