Nuprl Lemma : sum-in-vs-shift
∀[k,n,m:ℤ]. ∀[f,vs:Top]. (Σ{f[i] | n≤i≤m} ~ Σ{f[i + k] | n - k≤i≤m - k})
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
subtract: n - m,
add: n + m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-bag-add: Σ{f[b] | b ∈ bs},
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-map: bag-map(f;bs),
squash: ↓T,
all: ∀x:A. B[x],
true: True,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than: a < b,
le: A ≤ B,
sq_type: SQType(T)
Lemmas referenced :
bag-summation-map,
istype-void,
from-upto_wf,
subtract_wf,
subtype_rel_list,
le_wf,
less_than_wf,
top_wf,
istype-le,
istype-less_than,
istype-top,
istype-int,
subtype_base_sq,
list_wf,
list_subtype_base,
set_subtype_base,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add-comm,
map_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
subtype_rel_self,
iff_weakening_equal,
from-upto-shift,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
subtype_rel_list_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
hypothesisEquality,
addEquality,
because_Cache,
closedConclusion,
natural_numberEquality,
applyEquality,
setEquality,
intEquality,
productEquality,
independent_isectElimination,
lambdaEquality_alt,
setIsType,
productIsType,
inhabitedIsType,
instantiate,
cumulativity,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
multiplyEquality,
minusEquality
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \mforall{}[f,vs:Top]. (\mSigma{}\{f[i] | n\mleq{}i\mleq{}m\} \msim{} \mSigma{}\{f[i + k] | n - k\mleq{}i\mleq{}m - k\})
Date html generated:
2019_10_31-AM-06_26_17
Last ObjectModification:
2019_08_08-PM-00_26_51
Theory : linear!algebra
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