Nuprl Lemma : l_all-remove-repeats
∀[T:Type]. ∀eq:EqDecider(T). ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ ℙ. ((∀x∈remove-repeats(eq;L).P[x]) ⇐⇒ (∀x∈L.P[x]))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
member-remove-repeats,
l_member_wf,
remove-repeats_wf,
subtype_rel_self,
l_all_iff,
l_all_wf,
list_wf,
deq_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
lambdaEquality_alt,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
universeIsType,
setIsType,
because_Cache,
independent_pairFormation,
sqequalRule,
functionIsType,
applyEquality,
instantiate,
promote_hyp,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}.
((\mforall{}x\mmember{}remove-repeats(eq;L).P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L.P[x]))
Date html generated:
2020_05_19-PM-09_52_30
Last ObjectModification:
2019_10_18-PM-00_14_07
Theory : decidable!equality
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