### Nuprl Lemma : list-diff-cons-single

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as:T List]. ∀[b,x:T].  [x / as]-[b] = [x / as-[b]] ∈ (T List) supposing ¬(x = b ∈ T)`

Proof

Definitions occuring in Statement :  list-diff: `as-bs` cons: `[a / b]` nil: `[]` list: `T List` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` not: `¬A` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` all: `∀x:A. B[x]` top: `Top` deq: `EqDecider(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A`
Lemmas referenced :  equal_wf squash_wf true_wf list-diff-cons cons_wf nil_wf list-diff_wf iff_weakening_equal deq_member_cons_lemma deq_member_nil_lemma bor_wf bfalse_wf bool_wf eqtt_to_assert assert-deq-member eqff_to_assert deq-member_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot l_member_wf not_wf list_wf deq_wf member_singleton
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut applyEquality thin lambdaEquality sqequalHypSubstitution imageElimination extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry because_Cache cumulativity natural_numberEquality sqequalRule imageMemberEquality baseClosed universeEquality independent_isectElimination productElimination independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality setElimination rename lambdaFormation unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate axiomEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[as:T  List].  \mforall{}[b,x:T].
[x  /  as]-[b]  =  [x  /  as-[b]]  supposing  \mneg{}(x  =  b)

Date html generated: 2017_04_17-AM-09_12_58
Last ObjectModification: 2017_02_27-PM-05_19_52

Theory : decidable!equality

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