Nuprl Lemma : sub-bag-rep
∀[T:Type]. ∀x:T. ∀n,m:ℕ. uiff(sub-bag(T;bag-rep(n;x);bag-rep(m;x));n ≤ m)
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-rep: bag-rep(n;x),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtract: n - m
Lemmas referenced :
less_than'_wf,
sub-bag_wf,
bag-rep_wf,
list-subtype-bag,
le_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
bag-size_wf,
bag_wf,
iff_weakening_equal,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_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_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
bag-size-rep,
bag-size-append,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
bag-append_wf,
bag-rep-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
because_Cache,
independent_isectElimination,
universeEquality,
imageElimination,
intEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
voidEquality,
unionElimination,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}n,m:\mBbbN{}. uiff(sub-bag(T;bag-rep(n;x);bag-rep(m;x));n \mleq{} m)
Date html generated:
2017_10_01-AM-08_52_59
Last ObjectModification:
2017_07_26-PM-04_34_23
Theory : bags
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