Nuprl Lemma : length-remove-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L1,L2:T List].
||remove-repeats(eq;L1)|| = ||remove-repeats(eq;L2)|| ∈ ℤ supposing ∀x:T. ((x ∈ L1) ⇐⇒ (x ∈ L2))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
l_member: (x ∈ l),
length: ||as||,
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
set-equal-no_repeats-length,
remove-repeats_wf,
remove-repeats_property,
all_wf,
iff_wf,
l_member_wf,
member-remove-repeats
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
productElimination,
sqequalRule,
lambdaEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
allFunctionality,
independent_pairFormation,
impliesFunctionality,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L1,L2:T List].
||remove-repeats(eq;L1)|| = ||remove-repeats(eq;L2)|| supposing \mforall{}x:T. ((x \mmember{} L1) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L2))
Date html generated:
2016_05_14-PM-03_26_38
Last ObjectModification:
2015_12_26-PM-06_23_09
Theory : decidable!equality
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