### Nuprl Lemma : count_remove1

`∀s:DSet. ∀as:|s| List. ∀b,c:|s|.  ((c #∈ (as \ b)) = ((c #∈ as) -- b2i(b (=b) c)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  count: `a #∈ as` remove1: `as \ a` ndiff: `a -- b` list: `T List` b2i: `b2i(b)` infix_ap: `x f y` all: `∀x:A. B[x]` int: `ℤ` equal: `s = t ∈ T` dset: `DSet` set_eq: `=b` set_car: `|p|`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` dset: `DSet` or: `P ∨ Q` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` infix_ap: `x f y` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True` b2i: `b2i(b)`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf set_car_wf list-cases remove1_nil_lemma count_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le remove1_cons_lemma count_cons_lemma nat_wf list_wf dset_wf ndiff_ann_l b2i_wf set_eq_wf b2i_bounds equal-wf-T-base bool_wf assert_wf equal_wf bnot_wf not_wf uiff_transitivity eqtt_to_assert assert_of_dset_eq iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot count_wf count_bounds istype-universe ndiff_inv iff_weakening_equal ndiff_wf add_com squash_wf true_wf add_functionality_wrt_eq subtype_rel_self equal-wf-base infix_ap_wf le_weakening2 non_neg_length remove1_wf decidable__lt ndiff_id_r zero-add
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality functionIsTypeImplies inhabitedIsType because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityIsType1 dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination equalityIsType4 baseApply closedConclusion baseClosed applyEquality intEquality equalityElimination universeEquality imageMemberEquality hyp_replacement addEquality

Latex:
\mforall{}s:DSet.  \mforall{}as:|s|  List.  \mforall{}b,c:|s|.    ((c  \#\mmember{}  (as  \mbackslash{}  b))  =  ((c  \#\mmember{}  as)  --  b2i(b  (=\msubb{})  c)))

Date html generated: 2019_10_16-PM-01_04_16
Last ObjectModification: 2018_10_08-AM-11_19_46

Theory : list_2

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