Nuprl Lemma : bag-no-repeats-count

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)].  uiff(bag-no-repeats(T;bs);∀[x:T]. uiff(1 ≤ (#x in bs);(#x in bs) = 1 ∈ ℤ))`

Proof

Definitions occuring in Statement :  bag-count: `(#x in bs)` bag-no-repeats: `bag-no-repeats(T;bs)` bag: `bag(T)` deq: `EqDecider(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` subtype_rel: `A ⊆r B` nat: `ℕ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag-no-repeats: `bag-no-repeats(T;bs)` squash: `↓T` deq: `EqDecider(T)` bag-filter: `[x∈b|p[x]]` bag-size: `#(bs)` guard: `{T}` istype: `istype(T)` cand: `A c∧ B` bag-count: `(#x in bs)` top: `Top` true: `True`
Lemmas referenced :  istype-le bag-count_wf decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf istype-int 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_eq_lemma int_formula_prop_wf le_witness_for_triv set_subtype_base le_wf int_subtype_base bag-no-repeats_wf bag_wf deq_wf istype-universe bag-count-sqequal bag-size_wf assert_wf bag-filter_wf equal-wf-base no-repeats-iff-count filter_functionality eta_conv bool_wf filter_wf5 subtype_rel_dep_function l_member_wf length_wf length_wf_nat list_wf exists_wf squash_wf no_repeats_wf equal_wf list-subtype-bag equal-wf-T-base nat_wf uiff_wf uall_wf bag_to_squash_list less_than'_wf satisfiable-full-omega-tt set_wf subtype_rel_self true_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut independent_pairFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality hypothesisEquality applyEquality lambdaEquality_alt setElimination rename inhabitedIsType equalityTransitivity equalitySymmetry sqequalRule dependent_functionElimination because_Cache unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality Error :memTop,  universeIsType voidElimination productElimination equalityIstype intEquality baseClosed sqequalBase independent_pairEquality isect_memberEquality_alt axiomEquality isectIsTypeImplies imageElimination imageMemberEquality isectIsType productIsType instantiate universeEquality promote_hyp hyp_replacement applyLambdaEquality setEquality setIsType lambdaFormation_alt isect_memberFormation productEquality dependent_pairFormation lambdaEquality cumulativity computeAll voidEquality isect_memberEquality lambdaFormation

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[bs:bag(T)].
uiff(bag-no-repeats(T;bs);\mforall{}[x:T].  uiff(1  \mleq{}  (\#x  in  bs);(\#x  in  bs)  =  1))

Date html generated: 2020_05_20-AM-09_04_13
Last ObjectModification: 2020_01_04-PM-10_27_42

Theory : bags_2

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