Nuprl Lemma : int-prod-isolate
∀[n:ℕ]. ∀[m:ℕn]. ∀[f:ℕn ⟶ ℤ]. (Π(f[x] | x < n) = (Π(if (x =z m) then 1 else f[x] fi | x < n) * f[m]) ∈ ℤ)
Proof
Definitions occuring in Statement :
int-prod: Π(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
le: A ≤ B,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
sq_type: SQType(T),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
nat: ℕ,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
false_wf,
add-member-int_seg1,
le_wf,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
itermSubtract_wf,
intformle_wf,
decidable__le,
int_seg_properties,
subtract_wf,
int_seg_wf,
eq_int_wf,
ifthenelse_wf,
int_subtype_base,
subtype_base_sq,
lelt_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
int-prod-split,
iff_weakening_equal,
btrue_wf,
eq_int_eq_true,
true_wf,
squash_wf,
zero-add,
int-prod-single,
int_formula_prop_eq_lemma,
intformeq_wf,
int_seg_subtype_nat,
int-prod_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
decidable__equal_int
Rules used in proof :
axiomEquality,
functionEquality,
promote_hyp,
equalityElimination,
lambdaFormation,
functionExtensionality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
because_Cache,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
addEquality,
dependent_functionElimination,
hypothesis,
independent_pairFormation,
productElimination,
dependent_set_memberEquality,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[m:\mBbbN{}n]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(\mPi{}(f[x] | x < n) = (\mPi{}(if (x =\msubz{} m) then 1 else f[x] fi | x < n) * f[m]))
Date html generated:
2018_05_21-PM-00_29_36
Last ObjectModification:
2017_12_10-PM-11_43_03
Theory : int_2
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