### Nuprl Lemma : bag-count-drop-trivial

`∀[T:Type]. ∀eq:EqDecider(T). ∀[x,y:T]. ∀[bs:bag(T)].  (#y in bag-drop(eq;bs;x)) = (#y in bs) ∈ ℕ supposing ¬(x = y ∈ T)`

Proof

Definitions occuring in Statement :  bag-drop: `bag-drop(eq;bs;a)` bag-count: `(#x in bs)` bag: `bag(T)` deq: `EqDecider(T)` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` uimplies: `b supposing a` bag-drop: `bag-drop(eq;bs;a)` or: `P ∨ Q` exists: `∃x:A. B[x]` and: `P ∧ Q` squash: `↓T` prop: `ℙ` label: `...\$L... t` decidable: `Dec(P)` false: `False` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` implies: `P `` Q` not: `¬A` top: `Top` nat: `ℕ` guard: `{T}` ge: `i ≥ j ` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` bag-count: `(#x in bs)` count: `count(P;L)` reduce: `reduce(f;k;as)` list_ind: list_ind single-bag: `{x}` cons: `[a / b]` ifthenelse: `if b then t else f fi ` nil: `[]` it: `⋅` deq: `EqDecider(T)` bool: `𝔹` unit: `Unit` btrue: `tt` eqof: `eqof(d)` le: `A ≤ B` less_than': `less_than'(a;b)` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  bag-remove1-property equal_wf squash_wf true_wf nat_wf bag-count_wf bag-count-append single-bag_wf decidable__equal_int add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_wf false_wf decidable__le nat_properties intformle_wf itermConstant_wf int_formula_prop_le_lemma int_term_value_constant_lemma le_wf iff_weakening_equal subtype_base_sq set_subtype_base int_subtype_base not_wf bool_wf eqtt_to_assert safe-assert-deq eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot decide_bfalse_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache dependent_functionElimination hypothesisEquality unionElimination productElimination hypothesis sqequalRule applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry cumulativity pointwiseFunctionality rename promote_hyp baseApply closedConclusion baseClosed independent_isectElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll dependent_set_memberEquality applyLambdaEquality setElimination imageMemberEquality universeEquality independent_functionElimination instantiate addEquality hyp_replacement axiomEquality equalityElimination

Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T)
\mforall{}[x,y:T].  \mforall{}[bs:bag(T)].    (\#y  in  bag-drop(eq;bs;x))  =  (\#y  in  bs)  supposing  \mneg{}(x  =  y)

Date html generated: 2018_05_21-PM-09_48_38
Last ObjectModification: 2017_07_26-PM-06_30_43

Theory : bags_2

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