Nuprl Lemma : bigger-int-property
∀[L:ℤ List]. ∀[n:ℤ]. (∀x∈L.x < bigger-int(n;L))
Proof
Definitions occuring in Statement :
bigger-int: bigger-int(n;L),
l_all: (∀x∈L.P[x]),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
guard: {T},
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B,
bigger-int: bigger-int(n;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
sq_type: SQType(T),
bnot: ¬bb
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,
member-less_than,
select_wf,
int_seg_properties,
length_wf,
bigger-int_wf,
int_seg_wf,
le_wf,
list_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
decidable__assert,
null_wf,
list-cases,
l_all_nil,
nil_wf,
length_of_nil_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
list_accum_append,
subtype_rel_list,
top_wf,
append_wf,
cons_wf,
last_wf,
length-append,
length_of_cons_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
length_wf_nat,
l_all_wf,
l_member_wf,
equal_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
value-type-has-value,
int-value-type,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_all_single,
l_all_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
productElimination,
imageElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
baseClosed,
impliesFunctionality,
productEquality,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
setEquality,
callbyvalueReduce,
equalityElimination,
instantiate,
cumulativity,
addLevel,
levelHypothesis,
andLevelFunctionality
Latex:
\mforall{}[L:\mBbbZ{} List]. \mforall{}[n:\mBbbZ{}]. (\mforall{}x\mmember{}L.x < bigger-int(n;L))
Date html generated:
2017_04_17-AM-07_48_49
Last ObjectModification:
2017_02_27-PM-04_23_02
Theory : list_1
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