Nuprl Lemma : equal-bag-size-le1
∀[T:Type]. ∀[as,bs:bag(T)].
uiff(as = bs ∈ bag(T);(as = {} ∈ bag(T) ⇐⇒ bs = {} ∈ bag(T))
∧ ((#(as) = 1 ∈ ℤ) ⇒ (#(bs) = 1 ∈ ℤ) ⇒ (only(as) = only(bs) ∈ T)))
supposing (#(as) ≤ 1) ∧ (#(bs) ≤ 1)
Proof
Definitions occuring in Statement :
bag-only: only(bs),
bag-size: #(bs),
empty-bag: {},
bag: bag(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
cand: A c∧ B
Lemmas referenced :
equal-wf-T-base,
bag_wf,
and_wf,
equal_wf,
bag-only_wf,
bag-size_wf,
decidable__lt,
iff_wf,
le_wf,
decidable__le,
nat_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
empty-bag_wf,
bag-size-zero,
bag_size_empty_lemma,
less_than_wf,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
single-bag_wf,
bag-size-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
hypothesis,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
baseClosed,
because_Cache,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
independent_isectElimination,
hyp_replacement,
intEquality,
applyEquality,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
natural_numberEquality,
unionElimination,
productEquality,
functionEquality,
isect_memberEquality,
imageElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:bag(T)].
uiff(as = bs;(as = \{\} \mLeftarrow{}{}\mRightarrow{} bs = \{\}) \mwedge{} ((\#(as) = 1) {}\mRightarrow{} (\#(bs) = 1) {}\mRightarrow{} (only(as) = only(bs))))
supposing (\#(as) \mleq{} 1) \mwedge{} (\#(bs) \mleq{} 1)
Date html generated:
2017_10_01-AM-08_52_42
Last ObjectModification:
2017_07_26-PM-04_34_11
Theory : bags
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