Nuprl Lemma : pairwise-map1
∀[T,T':Type]. ∀f:T ⟶ T'. ∀L:T List. ∀[P:T' ⟶ T' ⟶ ℙ']. ((∀x,y∈map(f;L). P[x;y]) ⇐⇒ (∀x,y∈L. P[f x;f y]))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
pairwise-map,
subtype_rel_dep_function,
l_member_wf,
set_wf,
list_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T,T':Type].
\mforall{}f:T {}\mrightarrow{} T'. \mforall{}L:T List. \mforall{}[P:T' {}\mrightarrow{} T' {}\mrightarrow{} \mBbbP{}']. ((\mforall{}x,y\mmember{}map(f;L). P[x;y]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y\mmember{}L. P[f x;f y]))
Date html generated:
2016_05_15-PM-03_59_22
Last ObjectModification:
2015_12_27-PM-03_05_58
Theory : general
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