Nuprl Lemma : itop_unroll_hi
∀[g:IMonoid]. ∀[i,j:ℤ].
∀[E:{i..j-} ⟶ |g|]. (Π(*,e) i ≤ k < j. E[k] = (Π(*,e) i ≤ k < j - 1. E[k] * E[j - 1]) ∈ |g|) supposing i < j
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-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: 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,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff,
guard: {T}
Lemmas referenced :
int_seg_wf,
grp_car_wf,
less_than_wf,
imon_wf,
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_lt_int,
infix_ap_wf,
grp_op_wf,
itop_wf,
grp_id_wf,
subtract_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_int_wf,
le_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf
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,
lambdaFormation,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
lambdaEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[g:IMonoid]. \mforall{}[i,j:\mBbbZ{}].
\mforall{}[E:\{i..j\msupminus{}\} {}\mrightarrow{} |g|]. (\mPi{}(*,e) i \mleq{} k < j. E[k] = (\mPi{}(*,e) i \mleq{} k < j - 1. E[k] * E[j - 1]))
supposing i < j
Date html generated:
2017_10_01-AM-08_15_37
Last ObjectModification:
2017_02_28-PM-02_00_18
Theory : groups_1
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