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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