Nuprl Lemma : l_all_implies_b_all
∀[A:Type]. ∀as:A List. ∀P:A ⟶ ℙ. ((∀x,y:A. Dec(x = y ∈ A)) ⇒ (∀x∈as.P[x]) ⇒ b_all(A;as;x.P[x]))
Proof
Definitions occuring in Statement :
b_all: b_all(T;b;x.P[x]),
l_all: (∀x∈L.P[x]),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
b_all: b_all(T;b;x.P[x]),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
l_all_iff,
l_member_wf,
bag-member-list,
bag-member_wf,
list-subtype-bag,
l_all_wf,
all_wf,
decidable_wf,
equal_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
hypothesis,
productElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}as:A List. \mforall{}P:A {}\mrightarrow{} \mBbbP{}. ((\mforall{}x,y:A. Dec(x = y)) {}\mRightarrow{} (\mforall{}x\mmember{}as.P[x]) {}\mRightarrow{} b\_all(A;as;x.P[x]))
Date html generated:
2016_05_15-PM-02_41_55
Last ObjectModification:
2015_12_27-AM-09_40_06
Theory : bags
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