Nuprl Lemma : test-omega-auto-split
∀n:ℤ. (if 2 <z n then if 1 <z n then 1 else 2 fi if n ≤z 3 then 3 else 4 fi ~ if 2 <z n then 1 else 3 fi )
Proof
Definitions occuring in Statement :
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
all: ∀x:A. B[x],
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
le_int_wf,
assert_of_le_int,
le_wf,
intformle_wf,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation
Latex:
\mforall{}n:\mBbbZ{}
(if 2 <z n then if 1 <z n then 1 else 2 fi
if n \mleq{}z 3 then 3
else 4
fi \msim{} if 2 <z n then 1 else 3 fi )
Date html generated:
2018_05_21-PM-00_24_39
Last ObjectModification:
2018_05_19-AM-06_58_33
Theory : omega
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