Nuprl Lemma : sum_switch
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[i:ℕn - 1]. (Σ(f[(i, i + 1) x] | x < n) = Σ(f[x] | x < n) ∈ ℤ)
Proof
Definitions occuring in Statement :
flip: (i, j),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
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,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
flip: (i, j),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lt_int: i <z j,
subtract: n - m,
true: True
Lemmas referenced :
int_seg_wf,
subtract_wf,
flip_wf,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_properties,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
subtype_base_sq,
int_subtype_base,
sum_split,
lelt_wf,
false_wf,
int_seg_subtype_nat,
sum_functionality,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
not_wf,
bnot_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
equal_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
primrec1_lemma,
squash_wf,
true_wf,
istype-universe,
add-comm,
subtype_rel_self,
iff_weakening_equal,
zero-add,
add-commutes,
sum-as-primrec,
primrec-unroll,
add-member-int_seg1,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesis,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
Error :isect_memberEquality_alt,
hypothesisEquality,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
voidElimination,
Error :productIsType,
addEquality,
imageElimination,
instantiate,
cumulativity,
intEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
lambdaEquality,
lambdaFormation,
promote_hyp,
equalityElimination,
baseClosed,
impliesFunctionality,
Error :lambdaFormation_alt,
Error :equalityIstype,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[i:\mBbbN{}n - 1]. (\mSigma{}(f[(i, i + 1) x] | x < n) = \mSigma{}(f[x] | x < n))
Date html generated:
2019_06_20-PM-02_29_56
Last ObjectModification:
2019_02_06-PM-03_51_25
Theory : num_thy_1
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