Nuprl Lemma : itop_split
∀[g:IMonoid]. ∀[a,b,c:ℤ].
(∀[E:{a..c-} ⟶ |g|]. (Π(*,e) a ≤ j < c. E[j] = (Π(*,e) a ≤ j < b. E[j] * Π(*,e) b ≤ j < c. E[j]) ∈ |g|)) supposing
((b ≤ c) and
(a ≤ b))
Proof
Definitions occuring in Statement :
itop: Π(op,id) lb ≤ i < ub. E[i],
imon: IMonoid,
grp_id: e,
grp_op: *,
grp_car: |g|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
imon: IMonoid,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
squash: ↓T,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
subtract: n - m
Lemmas referenced :
int_seg_wf,
grp_car_wf,
le_wf,
imon_wf,
int_le_to_int_upper,
isect_wf,
uall_wf,
equal_wf,
itop_wf,
grp_op_wf,
grp_id_wf,
infix_ap_wf,
int_upper_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__le,
int_upper_wf,
int_upper_ind,
subtract_wf,
itermSubtract_wf,
itermConstant_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
true_wf,
itop_unroll_base,
iff_weakening_equal,
mon_ident,
itop_unroll_lo,
subtract-add-cancel,
itop_unroll_hi,
mon_assoc,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-associates,
add-swap,
add-commutes,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionEquality,
extract_by_obid,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
intEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
instantiate,
lambdaFormation,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
cumulativity,
minusEquality
Latex:
\mforall{}[g:IMonoid]. \mforall{}[a,b,c:\mBbbZ{}].
(\mforall{}[E:\{a..c\msupminus{}\} {}\mrightarrow{} |g|]
(\mPi{}(*,e) a \mleq{} j < c. E[j] = (\mPi{}(*,e) a \mleq{} j < b. E[j] * \mPi{}(*,e) b \mleq{} j < c. E[j]))) supposing
((b \mleq{} c) and
(a \mleq{} b))
Date html generated:
2017_10_01-AM-08_15_55
Last ObjectModification:
2017_02_28-PM-02_01_38
Theory : groups_1
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