Nuprl Lemma : bag-append-is-single
∀[T:Type]. ∀[x:T].
∀as,bs:bag(T).
↓((as = {x} ∈ bag(T)) ∧ (bs = {} ∈ bag(T))) ∨ ((bs = {x} ∈ bag(T)) ∧ (as = {} ∈ bag(T)))
supposing (as + bs) = {x} ∈ bag(T)
Proof
Definitions occuring in Statement :
bag-append: as + bs,
single-bag: {x},
empty-bag: {},
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
nil: [],
it: ⋅,
empty-bag: {},
subtype_rel: A ⊆r B,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
respects-equality: respects-equality(S;T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
single-bag: {x},
bag-size: #(bs),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
bag_to_squash_list,
equal_wf,
bag_wf,
bag-append_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
empty-bag_wf,
nil_wf,
list-subtype-bag,
single-bag_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
cons_wf,
squash_wf,
equal-wf-T-base,
equal-wf-base,
equal-wf-base-T,
istype-nat,
istype-universe,
subtype-respects-equality,
bag-append-ident,
true_wf,
bag-size_wf,
subtype_rel_self,
iff_weakening_equal,
list_ind_cons_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
length-append,
non_neg_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity,
rename,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
inrFormation_alt,
productIsType,
equalityIstype,
sqequalBase,
because_Cache,
closedConclusion,
voidEquality,
applyEquality,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
instantiate,
baseApply,
intEquality,
functionEquality,
unionEquality,
productEquality,
functionIsType,
isectIsTypeImplies,
universeEquality,
inlFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[x:T].
\mforall{}as,bs:bag(T). \mdownarrow{}((as = \{x\}) \mwedge{} (bs = \{\})) \mvee{} ((bs = \{x\}) \mwedge{} (as = \{\})) supposing (as + bs) = \{x\}
Date html generated:
2019_10_15-AM-11_00_16
Last ObjectModification:
2018_11_30-AM-09_54_48
Theory : bags
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