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:
2019_06_20-PM-02_37_57
Last ObjectModification:
2019_06_12-PM-00_26_29
Theory : num_thy_1
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