Nuprl Lemma : sum-map-append1
∀[f:Top]. ∀[L:Top List]. ∀[x:Top]. (Σf[x] for x ∈ L @ [x] ~ Σf[x] for x ∈ L + f[x])
Proof
Definitions occuring in Statement :
sum-map: Σf[x] for x ∈ L,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sum-map: Σf[x] for x ∈ L,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uimplies: b supposing a,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
nat_plus: ℕ+,
uiff: uiff(P;Q),
nat: ℕ,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
le: A ≤ B,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
select: L[n],
cons: [a / b]
Lemmas referenced :
sum-unroll,
decidable__lt,
length_wf,
top_wf,
append_wf,
cons_wf,
nil_wf,
less_than_wf,
length-append,
subtract_wf,
subtype_base_sq,
int_subtype_base,
length_of_cons_lemma,
length_of_nil_lemma,
non_neg_length,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
false_wf,
equal_wf,
list_wf,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma,
ge_wf,
le_wf,
decidable__le,
nat_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
select-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
because_Cache,
lessCases,
sqequalAxiom,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaFormation,
imageElimination,
productElimination,
independent_functionElimination,
addEquality,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
intWeakElimination,
equalityElimination
Latex:
\mforall{}[f:Top]. \mforall{}[L:Top List]. \mforall{}[x:Top]. (\mSigma{}f[x] for x \mmember{} L @ [x] \msim{} \mSigma{}f[x] for x \mmember{} L + f[x])
Date html generated:
2018_05_21-PM-08_29_06
Last ObjectModification:
2017_07_26-PM-05_56_13
Theory : general
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