### 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

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