Nuprl Lemma : from-upto-member-nat
∀n,m,k:ℕ. uiff((k ∈ [n, m));(n ≤ k) ∧ k < m)
Proof
Definitions occuring in Statement :
from-upto: [n, m),
l_member: (x ∈ l),
nat: ℕ,
less_than: a < b,
uiff: uiff(P;Q),
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
l_member: (x ∈ l),
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
less_than'_wf,
member-less_than,
l_member_wf,
nat_wf,
from-upto_wf,
subtype_rel_list,
le_wf,
less_than_wf,
subtype_rel_sets,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length-from-upto,
lt_int_wf,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
select-from-upto,
lelt_wf,
subtract_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__lt,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
equal_wf,
bool_cases_sqequal,
assert-bnot,
decidable__equal_int,
length_wf,
select_wf,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
applyEquality,
setEquality,
intEquality,
productEquality,
because_Cache,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
instantiate,
cumulativity,
independent_functionElimination,
impliesFunctionality,
dependent_set_memberEquality,
applyLambdaEquality,
equalityElimination,
promote_hyp,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}n,m,k:\mBbbN{}. uiff((k \mmember{} [n, m));(n \mleq{} k) \mwedge{} k < m)
Date html generated:
2017_04_17-AM-07_55_39
Last ObjectModification:
2017_02_27-PM-04_28_41
Theory : list_1
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