Nuprl Lemma : l_all2_cons
∀[T:Type]. ∀L:T List. ∀[P:T ⟶ T ⟶ ℙ]. ∀u:T. ((∀x<y∈[u / L].P[x;y]) ⇐⇒ (∀y∈L.P[u;y]) ∧ (∀x<y∈L.P[x;y]))
Proof
Definitions occuring in Statement :
l_all2: (∀x<y∈L.P[x; y]),
l_all: (∀x∈L.P[x]),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
l_all2: (∀x<y∈L.P[x; y]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
l_before_wf,
l_member_wf,
equal_wf,
all_wf,
or_wf,
cons_before,
l_all_iff,
cons_wf,
l_all_wf,
iff_wf,
list_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
inlFormation,
because_Cache,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
inrFormation,
productEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
productElimination,
comment,
addLevel,
impliesFunctionality,
allFunctionality,
setElimination,
rename,
setEquality,
allLevelFunctionality,
impliesLevelFunctionality,
andLevelFunctionality,
universeEquality,
unionElimination,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}u:T. ((\mforall{}x<y\mmember{}[u / L].P[x;y]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}y\mmember{}L.P[u;y]) \mwedge{} (\mforall{}x<y\mmember{}L.P[x;y]))
Date html generated:
2017_10_01-AM-08_34_37
Last ObjectModification:
2017_07_26-PM-04_25_30
Theory : list!
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