### Nuprl Lemma : no_repeats-remove-first

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

Proof

Definitions occuring in Statement :  remove-first: `remove-first(P;L)` no_repeats: `no_repeats(T;l)` l_member: `(x ∈ l)` list: `T List` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  no_repeats: `no_repeats(T;l)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` prop: `ℙ` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_apply: `x[s]` gt: `i > j` subtype_rel: `A ⊆r B` l_all: `(∀x∈L.P[x])` less_than: `a < b`
Lemmas referenced :  select-remove-first lelt_wf length_wf remove-first_wf l_member_wf length-remove-first-le decidable__all_int_seg not_wf assert_wf select_wf list-subtype int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma int_seg_wf decidable__not decidable__assert equal_wf nat_wf less_than_wf no_repeats_wf bool_wf list_wf no_repeats_witness decidable__equal_int intformeq_wf int_formula_prop_eq_lemma le_wf decidable__or equal-wf-base int_subtype_base or_wf intformor_wf int_formula_prop_or_lemma length-remove-first itermSubtract_wf int_term_value_subtract_lemma
Rules used in proof :  sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin independent_functionElimination extract_by_obid isectElimination hypothesisEquality hypothesis setElimination rename dependent_set_memberEquality independent_pairFormation productElimination cumulativity because_Cache functionExtensionality applyEquality setEquality instantiate dependent_functionElimination natural_numberEquality addEquality sqequalRule lambdaEquality equalityTransitivity equalitySymmetry independent_isectElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality universeEquality applyLambdaEquality

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].
no\_repeats(T;remove-first(P;L))  supposing  no\_repeats(T;L)

Date html generated: 2017_04_17-AM-08_34_28
Last ObjectModification: 2017_02_27-PM-04_56_02

Theory : list_1

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