Nuprl Lemma : iseg_product-split
∀[i,j,k:ℤ]. (iseg_product(i;j) ~ iseg_product(i;k) * iseg_product(k + 1;j)) supposing (k < j and (i ≤ k) and (1 ≤ i))
Proof
Definitions occuring in Statement :
iseg_product: iseg_product(i;j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iseg_product: iseg_product(i;j),
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,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
ge: i ≥ j ,
subtract: n - m
Lemmas referenced :
mul-commutes,
zero-add,
add-zero,
zero-mul,
add-commutes,
add-mul-special,
add-swap,
minus-one-mul,
minus-minus,
minus-add,
add-associates,
nat_properties,
combinations_wf,
false_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
multiply-is-int-iff,
decidable__equal_int,
int_term_value_mul_lemma,
itermMultiply_wf,
combinations_wf_int,
less_than_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
itermSubtract_wf,
itermAdd_wf,
subtract_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
combinations-split,
int_subtype_base,
set_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
hypothesis,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
hypothesisEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
multiplyEquality,
minusEquality,
pointwiseFunctionality,
rename,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
applyEquality,
setElimination,
setEquality
Latex:
\mforall{}[i,j,k:\mBbbZ{}].
(iseg\_product(i;j) \msim{} iseg\_product(i;k) * iseg\_product(k + 1;j)) supposing
(k < j and
(i \mleq{} k) and
(1 \mleq{} i))
Date html generated:
2016_05_15-PM-06_01_48
Last ObjectModification:
2016_01_16-PM-00_41_32
Theory : general
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