Nuprl Lemma : sum_difference
∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ]. ∀[d:ℤ]. Σ(f[x] | x < n) = (Σ(g[x] | x < n) + d) ∈ ℤ supposing Σ(f[x] - g[x] | x < n) = d ∈ ℤ
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
equal-wf-T-base,
sum_wf,
subtract_wf,
int_seg_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
sum_linear,
subtype_rel_self,
iff_weakening_equal,
sum_functionality,
int_seg_properties,
nat_properties,
decidable__equal_int,
lelt_wf,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
independent_functionElimination,
Error :universeIsType,
sqequalRule,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
because_Cache,
Error :inhabitedIsType,
functionEquality,
Error :functionIsType,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
addEquality,
lambdaFormation,
unionElimination,
dependent_set_memberEquality,
independent_pairFormation,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[d:\mBbbZ{}].
\mSigma{}(f[x] | x < n) = (\mSigma{}(g[x] | x < n) + d) supposing \mSigma{}(f[x] - g[x] | x < n) = d
Date html generated:
2019_06_20-PM-01_18_08
Last ObjectModification:
2018_09_26-PM-02_37_38
Theory : int_2
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