Nuprl Lemma : smallbag-subtype-list
∀T:Type. ({b:bag(T)| #(b) < 2} ⊆r {b:T List| ||b|| < 2} )
Proof
Definitions occuring in Statement :
bag-size: #(bs),
bag: bag(T),
length: ||as||,
list: T List,
less_than: a < b,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
bag-size: #(bs),
uall: ∀[x:A]. B[x],
prop: ℙ,
bag: bag(T),
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
true: True,
implies: P ⇒ Q,
or: P ∨ Q,
cons: [a / b],
top: Top,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
permutation: permutation(T;L1;L2),
permute_list: (L o f),
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
lelt: i ≤ j < k,
decidable: Dec(P),
guard: {T},
less_than: a < b,
sq_type: SQType(T),
select: L[n]
Lemmas referenced :
less_than_wf,
length_wf,
bag_wf,
bag-size_wf,
list_wf,
permutation_wf,
permutation_weakening,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
equal_wf,
permutation-length,
list-cases,
nil_wf,
product_subtype_list,
length_of_nil_lemma,
length_of_cons_lemma,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
intformless_wf,
int_formula_prop_less_lemma,
length_wf_nat,
nat_wf,
cons_wf,
mklist-single,
subtype_base_sq,
int_seg_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
decidable__le,
int_seg_properties,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
lambdaEquality,
setElimination,
thin,
rename,
cut,
sqequalHypSubstitution,
dependent_set_memberEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
natural_numberEquality,
setEquality,
applyEquality,
because_Cache,
sqequalRule,
universeEquality,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_isectElimination,
pointwiseFunctionality,
pertypeElimination,
productElimination,
productEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination,
hypothesis_subsumption,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
hyp_replacement,
applyLambdaEquality,
instantiate
Latex:
\mforall{}T:Type. (\{b:bag(T)| \#(b) < 2\} \msubseteq{}r \{b:T List| ||b|| < 2\} )
Date html generated:
2017_10_01-AM-08_45_52
Last ObjectModification:
2017_07_26-PM-04_30_57
Theory : bags
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