Nuprl Lemma : member_upto
∀n,i:ℕ. ((i ∈ upto(n)) ⇐⇒ i < n)
Proof
Definitions occuring in Statement :
upto: upto(n),
l_member: (x ∈ l),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
uimplies: b supposing a,
prop: ℙ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q
Lemmas referenced :
length_upto,
select_upto,
nat_properties,
decidable__lt,
select_wf,
int_seg_wf,
upto_wf,
le_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
l_member_wf,
nat_wf,
subtype_rel_list,
int_seg_subtype_nat,
false_wf,
decidable__equal_int,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
less_than_wf,
length_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
because_Cache,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
productEquality
Latex:
\mforall{}n,i:\mBbbN{}. ((i \mmember{} upto(n)) \mLeftarrow{}{}\mRightarrow{} i < n)
Date html generated:
2017_04_17-AM-07_57_58
Last ObjectModification:
2017_02_27-PM-04_29_40
Theory : list_1
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