Nuprl Lemma : mon_itop_split_el
∀[g:IMonoid]. ∀[a,b,c:ℤ].
  (∀[E:{a..c-} ⟶ |g|]
     ((Π a ≤ j < c. E[j]) = ((Π a ≤ j < b. E[j]) * (E[b] * (Π b + 1 ≤ j < c. E[j]))) ∈ |g|)) supposing 
     (b < c and 
     (a ≤ b))
Proof
Definitions occuring in Statement : 
mon_itop: Π lb ≤ i < ub. E[i]
, 
imon: IMonoid
, 
grp_op: *
, 
grp_car: |g|
, 
int_seg: {i..j-}
, 
less_than: a < b
, 
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]
, 
add: n + m
, 
natural_number: $n
, 
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
, 
squash: ↓T
, 
all: ∀x:A. B[x]
, 
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
, 
and: P ∧ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
true: True
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
infix_ap: x f y
Lemmas referenced : 
int_seg_wf, 
grp_car_wf, 
less_than_wf, 
le_wf, 
imon_wf, 
equal_wf, 
squash_wf, 
true_wf, 
mon_itop_split, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
infix_ap_wf, 
grp_op_wf, 
mon_itop_wf, 
decidable__lt, 
lelt_wf, 
itermAdd_wf, 
itermConstant_wf, 
int_term_value_add_lemma, 
int_term_value_constant_lemma, 
iff_weakening_equal, 
mon_itop_unroll_lo
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
hypothesis, 
functionEquality, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
setElimination, 
rename, 
sqequalRule, 
isect_memberEquality, 
axiomEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
intEquality, 
applyEquality, 
lambdaEquality, 
imageElimination, 
universeEquality, 
independent_isectElimination, 
dependent_functionElimination, 
unionElimination, 
natural_numberEquality, 
dependent_pairFormation, 
int_eqEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
functionExtensionality, 
dependent_set_memberEquality, 
productElimination, 
addEquality, 
imageMemberEquality, 
baseClosed, 
independent_functionElimination
Latex:
\mforall{}[g:IMonoid].  \mforall{}[a,b,c:\mBbbZ{}].
    (\mforall{}[E:\{a..c\msupminus{}\}  {}\mrightarrow{}  |g|]
          ((\mPi{}  a  \mleq{}  j  <  c.  E[j])  =  ((\mPi{}  a  \mleq{}  j  <  b.  E[j])  *  (E[b]  *  (\mPi{}  b  +  1  \mleq{}  j  <  c.  E[j])))))  supposing 
          (b  <  c  and 
          (a  \mleq{}  b))
Date html generated:
2017_10_01-AM-08_16_17
Last ObjectModification:
2017_02_28-PM-02_01_17
Theory : groups_1
Home
Index