### Nuprl Lemma : bag-append-is-single

`∀[T:Type]. ∀[x:T].`
`  ∀as,bs:bag(T).`
`    ↓((as = {x} ∈ bag(T)) ∧ (bs = {} ∈ bag(T))) ∨ ((bs = {x} ∈ bag(T)) ∧ (as = {} ∈ bag(T))) `
`    supposing (as + bs) = {x} ∈ bag(T)`

Proof

Definitions occuring in Statement :  bag-append: `as + bs` single-bag: `{x}` empty-bag: `{}` bag: `bag(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` squash: `↓T` or: `P ∨ Q` and: `P ∧ Q` 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` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` and: `P ∧ Q` or: `P ∨ Q` guard: `{T}` cand: `A c∧ B` bag-append: `as + bs` append: `as @ bs` list_ind: list_ind nil: `[]` it: `⋅` empty-bag: `{}` subtype_rel: `A ⊆r B` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` respects-equality: `respects-equality(S;T)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` single-bag: `{x}` bag-size: `#(bs)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]`
Lemmas referenced :  bag_to_squash_list equal_wf bag_wf bag-append_wf 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 istype-less_than list-cases empty-bag_wf nil_wf list-subtype-bag single-bag_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-false istype-le list_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 le_wf cons_wf squash_wf equal-wf-T-base equal-wf-base equal-wf-base-T istype-nat istype-universe subtype-respects-equality bag-append-ident true_wf bag-size_wf subtype_rel_self iff_weakening_equal list_ind_cons_lemma length_of_cons_lemma length_of_nil_lemma length-append non_neg_length
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality equalityTransitivity rename setElimination 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 imageMemberEquality baseClosed functionIsTypeImplies inhabitedIsType unionElimination inrFormation_alt productIsType equalityIstype sqequalBase because_Cache closedConclusion voidEquality applyEquality hypothesis_subsumption dependent_set_memberEquality_alt instantiate baseApply intEquality functionEquality unionEquality productEquality functionIsType isectIsTypeImplies universeEquality inlFormation_alt

Latex:
\mforall{}[T:Type].  \mforall{}[x:T].
\mforall{}as,bs:bag(T).    \mdownarrow{}((as  =  \{x\})  \mwedge{}  (bs  =  \{\}))  \mvee{}  ((bs  =  \{x\})  \mwedge{}  (as  =  \{\}))  supposing  (as  +  bs)  =  \{x\}

Date html generated: 2019_10_15-AM-11_00_16
Last ObjectModification: 2018_11_30-AM-09_54_48

Theory : bags

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