Nuprl Lemma : sum-map-cons
∀[T:Type]. ∀[f:T ⟶ ℤ]. ∀[L:T List]. ∀[x:T].  (Σf[x] for x ∈ [x / L] ~ f[x] + Σf[x] for x ∈ L)
Proof
Definitions occuring in Statement : 
sum-map: Σf[x] for x ∈ L
, 
cons: [a / b]
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
function: x:A ⟶ B[x]
, 
add: n + m
, 
int: ℤ
, 
universe: Type
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
top: Top
, 
prop: ℙ
, 
guard: {T}
, 
sq_type: SQType(T)
, 
sum-map: Σf[x] for x ∈ L
, 
squash: ↓T
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
true: True
, 
and: P ∧ Q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
not: ¬A
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
le: A ≤ B
, 
nat: ℕ
, 
subtract: n - m
, 
append: as @ bs
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
so_apply: x[s1;s2;s3]
Lemmas referenced : 
subtype_base_sq, 
int_subtype_base, 
last_induction, 
all_wf, 
equal_wf, 
sum-map_wf, 
cons_wf, 
list_wf, 
sum_map_nil_lemma, 
length_of_cons_lemma, 
length_of_nil_lemma, 
squash_wf, 
true_wf, 
sum_split_first, 
less_than_wf, 
select_wf, 
nil_wf, 
int_seg_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
length-singleton, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
int_seg_wf, 
iff_weakening_equal, 
add_functionality_wrt_eq, 
false_wf, 
sum_wf, 
subtract_wf, 
le_wf, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
empty_support, 
select-cons-hd, 
list_ind_cons_lemma, 
subtype_rel_list, 
top_wf, 
decidable__equal_int, 
intformeq_wf, 
itermAdd_wf, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
sum-map-append1
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
instantiate, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
cumulativity, 
intEquality, 
independent_isectElimination, 
hypothesis, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
addEquality, 
independent_functionElimination, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
lambdaFormation, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
sqequalAxiom, 
functionEquality, 
universeEquality, 
imageElimination, 
dependent_set_memberEquality, 
natural_numberEquality, 
independent_pairFormation, 
imageMemberEquality, 
baseClosed, 
setElimination, 
rename, 
productElimination, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
computeAll
Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[L:T  List].  \mforall{}[x:T].    (\mSigma{}f[x]  for  x  \mmember{}  [x  /  L]  \msim{}  f[x]  +  \mSigma{}f[x]  for  x  \mmember{}  L)
Date html generated:
2018_05_21-PM-08_29_19
Last ObjectModification:
2017_07_26-PM-05_56_25
Theory : general
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