Nuprl Lemma : single-valued-bag-hd

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


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: 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: supposing a bag: bag(T) all: x:A. B[x] prop: quotient: x,y:A//B[x; y] and: P ∧ Q bag-size: #(bs) listp: List+ squash: T true: True subtype_rel: A ⊆B nat: 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:  Q int_seg: {i..j-} lelt: i ≤ j < k guard: {T} iff: ⇐⇒ Q rev_implies:  Q single-valued-bag: single-valued-bag(b;T) ge: i ≥  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

\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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