Nuprl Lemma : triangular-num-alt
∀[n:ℕ]. (t(n) = (((n ÷ 2) + (n rem 2)) * ((2 * (n ÷ 2)) + 1)) ∈ ℤ)
Proof
Definitions occuring in Statement : 
triangular-num: t(n)
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
remainder: n rem m
, 
divide: n ÷ m
, 
multiply: n * m
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
int_nzero: ℤ-o
, 
true: True
, 
nequal: a ≠ b ∈ T 
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
uimplies: b supposing a
, 
sq_type: SQType(T)
, 
all: ∀x:A. B[x]
, 
guard: {T}
, 
false: False
, 
prop: ℙ
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
and: P ∧ Q
, 
triangular-num: t(n)
, 
le: A ≤ B
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
top: Top
, 
ge: i ≥ j 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
div_rem_sum, 
subtype_base_sq, 
int_subtype_base, 
equal-wf-base, 
true_wf, 
nequal_wf, 
rem_bounds_1, 
less_than_wf, 
nat_wf, 
divide_wf, 
lelt_wf, 
int_seg_wf, 
equal_wf, 
squash_wf, 
add_functionality_wrt_eq, 
iff_weakening_equal, 
decidable__equal_int, 
mul-distributes, 
mul-distributes-right, 
mul-associates, 
add-associates, 
mul-swap, 
mul-commutes, 
zero-mul, 
zero-add, 
add-zero, 
one-mul, 
add-commutes, 
div-cancel, 
int_seg_properties, 
nat_properties, 
satisfiable-full-omega-tt, 
intformnot_wf, 
intformeq_wf, 
itermAdd_wf, 
itermMultiply_wf, 
itermConstant_wf, 
itermVar_wf, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
int_term_value_mul_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
set_subtype_base, 
intformand_wf, 
intformless_wf, 
intformle_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_le_lemma, 
decidable__le, 
decidable__lt, 
add-swap, 
add-mul-special
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
dependent_set_memberEquality, 
natural_numberEquality, 
addLevel, 
lambdaFormation, 
instantiate, 
cumulativity, 
intEquality, 
independent_isectElimination, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
voidElimination, 
baseClosed, 
sqequalRule, 
independent_pairFormation, 
imageMemberEquality, 
productElimination, 
because_Cache, 
multiplyEquality, 
addEquality, 
divideEquality, 
applyEquality, 
lambdaEquality, 
imageElimination, 
universeEquality, 
unionElimination, 
isect_memberEquality, 
voidEquality, 
dependent_pairFormation, 
int_eqEquality, 
computeAll
Latex:
\mforall{}[n:\mBbbN{}].  (t(n)  =  (((n  \mdiv{}  2)  +  (n  rem  2))  *  ((2  *  (n  \mdiv{}  2))  +  1)))
Date html generated:
2018_05_21-PM-07_53_23
Last ObjectModification:
2017_07_26-PM-05_30_54
Theory : general
Home
Index