Nuprl Lemma : double_sum_difference
∀[n,m:ℕ]. ∀[f,g:ℕn ⟶ ℕm ⟶ ℤ]. ∀[d:ℤ].
  sum(f[x;y] | x < n; y < m) = (sum(g[x;y] | x < n; y < m) + d) ∈ ℤ 
  supposing sum(f[x;y] - g[x;y] | x < n; y < m) = d ∈ ℤ
Proof
Definitions occuring in Statement : 
double_sum: sum(f[x; y] | x < n; y < m)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s1;s2]
, 
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
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
nat: ℕ
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
double_sum: sum(f[x; y] | x < n; y < m)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
int_seg: {i..j-}
, 
ge: i ≥ j 
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
false: False
, 
uiff: uiff(P;Q)
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
Lemmas referenced : 
equal-wf-T-base, 
double_sum_wf, 
subtract_wf, 
int_seg_wf, 
int_subtype_base, 
nat_wf, 
sum_difference, 
sum_wf, 
subtype_base_sq, 
sum_functionality, 
int_seg_properties, 
nat_properties, 
decidable__equal_int, 
add-is-int-iff, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformeq_wf, 
itermSubtract_wf, 
itermVar_wf, 
itermAdd_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_subtract_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_formula_prop_wf, 
false_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
introduction, 
cut, 
hypothesis, 
Error :universeIsType, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
intEquality, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
natural_numberEquality, 
setElimination, 
rename, 
isect_memberEquality, 
axiomEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
Error :inhabitedIsType, 
functionEquality, 
Error :functionIsType, 
independent_isectElimination, 
instantiate, 
cumulativity, 
dependent_functionElimination, 
independent_functionElimination, 
lambdaFormation, 
productElimination, 
unionElimination, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
baseClosed, 
approximateComputation, 
dependent_pairFormation, 
int_eqEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation
Latex:
\mforall{}[n,m:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}m  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[d:\mBbbZ{}].
    sum(f[x;y]  |  x  <  n;  y  <  m)  =  (sum(g[x;y]  |  x  <  n;  y  <  m)  +  d) 
    supposing  sum(f[x;y]  -  g[x;y]  |  x  <  n;  y  <  m)  =  d
Date html generated:
2019_06_20-PM-02_29_44
Last ObjectModification:
2018_09_26-PM-06_05_08
Theory : num_thy_1
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