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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