### Nuprl Lemma : no_repeats_cons

`∀[T:Type]. ∀[l:T List]. ∀[x:T].  uiff(no_repeats(T;[x / l]);no_repeats(T;l) ∧ (¬(x ∈ l)))`

Proof

Definitions occuring in Statement :  no_repeats: `no_repeats(T;l)` l_member: `(x ∈ l)` cons: `[a / b]` list: `T List` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` not: `¬A` and: `P ∧ Q` universe: `Type`
Definitions unfolded in proof :  no_repeats: `no_repeats(T;l)` all: `∀x:A. B[x]` member: `t ∈ T` top: `Top` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` nat: `ℕ` sq_stable: `SqStable(P)` squash: `↓T` prop: `ℙ` true: `True` less_than': `less_than'(a;b)` le: `A ≤ B` subtype_rel: `A ⊆r B` subtract: `n - m` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)` guard: `{T}` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` select: `L[n]` cons: `[a / b]` sq_type: `SQType(T)` label: `...\$L... t`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin Error :isect_memberEquality_alt,  voidElimination hypothesis Error :isect_memberFormation_alt,  independent_pairFormation Error :lambdaFormation_alt,  independent_functionElimination Error :equalityIsType1,  Error :inhabitedIsType,  hypothesisEquality isectElimination cumulativity setElimination rename because_Cache independent_isectElimination natural_numberEquality imageMemberEquality baseClosed imageElimination Error :lambdaEquality_alt,  Error :universeIsType,  equalityTransitivity equalitySymmetry productElimination independent_pairEquality Error :isectIsType,  addEquality Error :productIsType,  universeEquality applyLambdaEquality minusEquality intEquality voidEquality isect_memberEquality lambdaEquality applyEquality lambdaFormation unionElimination dependent_set_memberEquality Error :dependent_set_memberEquality_alt,  Error :equalityIsType4,  instantiate Error :dependent_pairFormation_alt,  promote_hyp

Latex:
\mforall{}[T:Type].  \mforall{}[l:T  List].  \mforall{}[x:T].    uiff(no\_repeats(T;[x  /  l]);no\_repeats(T;l)  \mwedge{}  (\mneg{}(x  \mmember{}  l)))

Date html generated: 2019_06_20-PM-00_42_15
Last ObjectModification: 2018_10_01-PM-10_15_57

Theory : list_0

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