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: uall: [x:A]. B[x] remainder: rem m divide: n ÷ m multiply: m add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: int_nzero: -o true: True nequal: a ≠ b ∈  not: ¬A implies:  Q uimplies: supposing a sq_type: SQType(T) all: x:A. B[x] guard: {T} false: False prop: nat_plus: + 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 ⊆B iff: ⇐⇒ Q rev_implies:  Q decidable: Dec(P) or: P ∨ Q top: Top ge: i ≥  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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