### Nuprl Lemma : bag-count-sqequal

`∀[T:Type]. ∀[bs:bag(T)]. ∀[eq:EqDecider(T)]. ∀[x:T].  ((#x in bs) ~ #([y∈bs|eq x y]))`

Proof

Definitions occuring in Statement :  bag-count: `(#x in bs)` bag-size: `#(bs)` bag-filter: `[x∈b|p[x]]` bag: `bag(T)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` apply: `f a` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` exists: `∃x:A. B[x]` bag-filter: `[x∈b|p[x]]` bag-size: `#(bs)` bag-count: `(#x in bs)` deq: `EqDecider(T)` subtype_rel: `A ⊆r B` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` sq_type: `SQType(T)` guard: `{T}`
Lemmas referenced :  subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base bag_to_squash_list count-length-filter non_neg_length filter_wf5 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 filter_functionality eta_conv bool_wf equal_wf bag-count_wf bag-size_wf assert_wf bag-filter_wf deq_wf bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesis independent_isectElimination sqequalRule intEquality lambdaEquality natural_numberEquality hypothesisEquality because_Cache imageElimination productElimination promote_hyp rename applyEquality setElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll dependent_set_memberEquality hyp_replacement equalitySymmetry Error :applyLambdaEquality,  setEquality equalityTransitivity independent_functionElimination sqequalAxiom universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].    ((\#x  in  bs)  \msim{}  \#([y\mmember{}bs|eq  x  y]))

Date html generated: 2016_10_25-AM-11_25_17
Last ObjectModification: 2016_07_12-AM-07_29_33

Theory : bags_2

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