### Nuprl Lemma : single-valued-bag-hd

`∀[T:Type]. ∀[b:bag(T)].  (hd(b) ∈ T) supposing (0 < #(b) and single-valued-bag(b;T))`

Proof

Definitions occuring in Statement :  single-valued-bag: `single-valued-bag(b;T)` bag-size: `#(bs)` bag: `bag(T)` hd: `hd(l)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` bag: `bag(T)` all: `∀x:A. B[x]` prop: `ℙ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` bag-size: `#(bs)` listp: `A List+` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` nat: `ℕ` permutation: `permutation(T;L1;L2)` exists: `∃x:A. B[x]` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` single-valued-bag: `single-valued-bag(b;T)` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  list_wf permutation_wf permutation_weakening hd_wf listp_properties less_than_wf length_wf equal-wf-base member_wf squash_wf true_wf bag-size_wf nat_wf single-valued-bag_wf bag_wf permutation-length length_wf_nat equal_wf select0 select_wf false_wf permute_list_select lelt_wf subtype_rel_self iff_weakening_equal int_seg_wf non_neg_length decidable__le nat_properties int_seg_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma bag-member-select subtype_rel_sets less_than_transitivity1 le_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution extract_by_obid isectElimination thin hypothesisEquality hypothesis promote_hyp lambdaFormation equalityTransitivity equalitySymmetry because_Cache dependent_functionElimination independent_isectElimination pointwiseFunctionality sqequalRule pertypeElimination productElimination cumulativity dependent_set_memberEquality natural_numberEquality productEquality applyEquality lambdaEquality imageElimination imageMemberEquality baseClosed axiomEquality setElimination rename isect_memberEquality universeEquality voidElimination voidEquality independent_pairFormation instantiate independent_functionElimination functionExtensionality unionElimination applyLambdaEquality approximateComputation dependent_pairFormation int_eqEquality intEquality hyp_replacement setEquality

Latex:
\mforall{}[T:Type].  \mforall{}[b:bag(T)].    (hd(b)  \mmember{}  T)  supposing  (0  <  \#(b)  and  single-valued-bag(b;T))

Date html generated: 2018_05_21-PM-06_24_57
Last ObjectModification: 2018_05_19-PM-05_15_26

Theory : bags

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