Nuprl Lemma : sum_functionality_wrt_sqequal
∀[n:ℕ]. ∀[f,g:Base]. Σ(f[x] | x < n) ~ Σ(g[x] | x < n) supposing ∀i:ℕn. (f[i] ~ g[i])
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],
all: ∀x:A. B[x],
natural_number: $n,
base: Base,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
so_apply: x[s],
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
lelt: i ≤ j < k,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
all_wf,
int_seg_wf,
sqequal-wf-base,
less_than_transitivity1,
less_than_irreflexivity,
base_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
decidable__lt,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sum-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
unionElimination,
productElimination,
equalityElimination,
lessCases,
imageMemberEquality,
imageElimination,
dependent_set_memberEquality,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:Base]. \mSigma{}(f[x] | x < n) \msim{} \mSigma{}(g[x] | x < n) supposing \mforall{}i:\mBbbN{}n. (f[i] \msim{} g[i])
Date html generated:
2017_04_14-AM-09_21_24
Last ObjectModification:
2017_02_27-PM-03_57_26
Theory : int_2
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