Nuprl Lemma : bag-member-implies-hd-append
∀[T:Type]. ∀[x:T]. ∀[b:bag(T)]. ↓∃c:bag(T). (b = ({x} + c) ∈ bag(T)) supposing x ↓∈ b
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-append: as + bs,
single-bag: {x},
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
bag-member: x ↓∈ bs,
and: P ∧ Q,
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
subtype_rel: A ⊆r B,
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
int_iseg: {i...j}
Lemmas referenced :
bag_to_squash_list,
bag-member_wf,
reject_wf,
list-subtype-bag,
list_ind_cons_lemma,
list_ind_nil_lemma,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
cons_wf,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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_wf,
equal_wf,
bag_wf,
bag-append_wf,
single-bag_wf,
squash_wf,
exists_wf,
permutation_inversion,
permutation-cons,
firstn_nth_tl_decomp,
lelt_wf,
length_wf,
firstn_wf,
nth_tl_wf,
append_wf,
permutation_weakening,
reject_eq_firstn_nth_tl,
length-append,
length_firstn,
length_of_cons_lemma,
intformless_wf,
int_formula_prop_less_lemma,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
rename,
dependent_pairFormation,
setElimination,
applyEquality,
independent_isectElimination,
lambdaEquality,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
universeEquality,
dependent_set_memberEquality,
addEquality,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. \mdownarrow{}\mexists{}c:bag(T). (b = (\{x\} + c)) supposing x \mdownarrow{}\mmember{} b
Date html generated:
2017_10_01-AM-08_55_47
Last ObjectModification:
2017_07_26-PM-04_37_53
Theory : bags
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