Nuprl Lemma : test-omega2
∀p0,p1,q0,q1,r0,r1,t0,t1:ℤ.
((((p0 * q1) + (q0 * r1) + (r0 * p1)) - (p1 * q0) + (q1 * r0) + (r1 * p0))
= ((((t0 * q1) + (q0 * r1) + (r0 * t1)) - (t1 * q0) + (q1 * r0) + (r1 * t0))
+ (((p0 * t1) + (t0 * r1) + (r0 * p1)) - (p1 * t0) + (t1 * r0) + (r1 * p0))
+ (((p0 * q1) + (q0 * t1) + (t0 * p1)) - (p1 * q0) + (q1 * t0) + (t1 * p0)))
∈ ℤ)
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ
Lemmas referenced :
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesis,
unionElimination,
isectElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
hypothesisEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule
Latex:
\mforall{}p0,p1,q0,q1,r0,r1,t0,t1:\mBbbZ{}.
((((p0 * q1) + (q0 * r1) + (r0 * p1)) - (p1 * q0) + (q1 * r0) + (r1 * p0))
= ((((t0 * q1) + (q0 * r1) + (r0 * t1)) - (t1 * q0) + (q1 * r0) + (r1 * t0))
+ (((p0 * t1) + (t0 * r1) + (r0 * p1)) - (p1 * t0) + (t1 * r0) + (r1 * p0))
+ (((p0 * q1) + (q0 * t1) + (t0 * p1)) - (p1 * q0) + (q1 * t0) + (t1 * p0))))
Date html generated:
2017_09_29-PM-05_56_34
Last ObjectModification:
2017_05_30-PM-11_58_44
Theory : omega
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