Nuprl Lemma : sbhomout_wf
∀[a,b,c,d:ℕ]. (0 < a + b ⇒ 0 < c + d ⇒ (∀[L:ℕ2 List]. (sbhomout(a;b;c;d;L) ∈ ℕ2 List)))
Proof
Definitions occuring in Statement :
sbhomout: sbhomout(a;b;c;d;L),
list: T List,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
sbhomout: sbhomout(a;b;c;d;L),
nil: [],
it: ⋅,
nat_plus: ℕ+,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
has-value: (a)↓,
true: True,
mtge1: mtge1(a;b;c;d),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
nat_wf,
equal-wf-T-base,
colength_wf_list,
int_seg_wf,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
itermAdd_wf,
int_term_value_add_lemma,
sbcode_wf,
add-is-int-iff,
set_subtype_base,
int_subtype_base,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
decidable__equal_int,
int_seg_properties,
mtge1_wf,
bool_wf,
eqtt_to_assert,
value-type-has-value,
int-value-type,
cons_wf,
false_wf,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
assert_wf,
bor_wf,
band_wf,
le_int_wf,
lt_int_wf,
or_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
assert_of_le_int,
assert_of_lt_int,
intformor_wf,
int_formula_prop_or_lemma,
decidable__lt,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
applyEquality,
because_Cache,
unionElimination,
isect_memberFormation,
dependent_set_memberEquality,
productElimination,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
instantiate,
cumulativity,
imageElimination,
equalityElimination,
callbyvalueReduce,
imageMemberEquality,
productEquality,
orFunctionality,
multiplyEquality
Latex:
\mforall{}[a,b,c,d:\mBbbN{}]. (0 < a + b {}\mRightarrow{} 0 < c + d {}\mRightarrow{} (\mforall{}[L:\mBbbN{}2 List]. (sbhomout(a;b;c;d;L) \mmember{} \mBbbN{}2 List)))
Date html generated:
2018_05_21-PM-11_41_10
Last ObjectModification:
2017_07_26-PM-06_42_51
Theory : rationals
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