### Nuprl Lemma : remove-first-member-implies

`∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹. ∀x:T.  ((x ∈ remove-first(P;L)) `` (x ∈ L))`

Proof

Definitions occuring in Statement :  remove-first: `remove-first(P;L)` l_member: `(x ∈ l)` list: `T List` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `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]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` prop: `ℙ` so_apply: `x[s]` remove-first: `remove-first(P;L)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` iff: `P `⇐⇒` Q` and: `P ∧ Q` false: `False` list_ind: list_ind nil: `[]` it: `⋅` rev_implies: `P `` Q` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` subtype_rel: `A ⊆r B`
Lemmas referenced :  list_induction all_wf l_member_wf bool_wf remove-first_wf list_wf list_ind_nil_lemma nil_member nil_wf list_ind_cons_lemma cons_wf cons_member bool_cases subtype_base_sq bool_subtype_base eqtt_to_assert equal_wf eqff_to_assert assert_of_bnot subtype_rel_dep_function subtype_rel_sets subtype_rel_self set_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality functionEquality setEquality cumulativity hypothesis functionExtensionality applyEquality independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality productElimination rename because_Cache inlFormation dependent_set_memberEquality equalityTransitivity equalitySymmetry unionElimination instantiate independent_isectElimination inrFormation setElimination hyp_replacement Error :applyLambdaEquality,  universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.  \mforall{}x:T.    ((x  \mmember{}  remove-first(P;L))  {}\mRightarrow{}  (x  \mmember{}  L))

Date html generated: 2016_10_21-AM-10_27_23
Last ObjectModification: 2016_07_12-AM-05_39_43

Theory : list_1

Home Index