Nuprl Lemma : bag-rep-add
∀x:Top. ∀n,m:ℕ. (bag-rep(n + m;x) ~ bag-rep(n;x) + bag-rep(m;x))
Proof
Definitions occuring in Statement :
bag-rep: bag-rep(n;x),
bag-append: as + bs,
nat: ℕ,
top: Top,
all: ∀x:A. B[x],
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
top: Top,
nat: ℕ,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
prop: ℙ,
bag-rep: bag-rep(n;x),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
bag-append: as + bs,
empty-bag: {},
cons-bag: x.b,
primrec: primrec(n;b;c),
append: as @ bs,
nil: [],
cons: [a / b],
list_ind: list_ind,
bottom: ⊥
Lemmas referenced :
nat_wf,
top_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
primrec-unroll,
empty_bag_append_lemma,
zero-add,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
cons_bag_append_lemma,
general_arith_equation1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
addEquality,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
lambdaFormation,
isectElimination,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}x:Top. \mforall{}n,m:\mBbbN{}. (bag-rep(n + m;x) \msim{} bag-rep(n;x) + bag-rep(m;x))
Date html generated:
2017_10_01-AM-08_52_03
Last ObjectModification:
2017_07_26-PM-04_33_46
Theory : bags
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