Nuprl Lemma : before-upto
∀n:ℕ. ∀x,y:ℕn. (x before y ∈ upto(n) ⇐⇒ x < y)
Proof
Definitions occuring in Statement :
upto: upto(n),
l_before: x before y ∈ l,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_apply: x[s],
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
increasing: increasing(f;k),
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
eq_int: (i =z j)
Lemmas referenced :
length_of_cons_lemma,
length_of_nil_lemma,
exists_wf,
int_seg_wf,
length_wf,
upto_wf,
increasing_wf,
false_wf,
le_wf,
all_wf,
equal_wf,
select_wf,
cons_wf,
nil_wf,
int_seg_properties,
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,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
non_neg_length,
lelt_wf,
length_wf_nat,
length_upto,
less_than_wf,
nat_wf,
select_upto,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
zero-add,
ifthenelse_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_seg_subtype,
int_seg_cases
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaFormation,
independent_pairFormation,
productElimination,
isectElimination,
functionEquality,
natural_numberEquality,
because_Cache,
setElimination,
rename,
lambdaEquality,
productEquality,
dependent_set_memberEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
equalityElimination,
promote_hyp,
hypothesis_subsumption
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbN{}n. (x before y \mmember{} upto(n) \mLeftarrow{}{}\mRightarrow{} x < y)
Date html generated:
2017_04_17-AM-07_58_22
Last ObjectModification:
2017_02_27-PM-04_29_51
Theory : list_1
Home
Index