Nuprl Lemma : triangular-num-add1

[n:ℕ]. (t(n 1) (t(n) 1) ∈ ℤ)


Proof




Definitions occuring in Statement :  triangular-num: t(n) nat: uall: [x:A]. B[x] add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] triangular-num: t(n) 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: ge: i ≥  decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top and: P ∧ Q squash: T nat_plus: + less_than: a < b less_than': less_than'(a;b) subtype_rel: A ⊆B iff: ⇐⇒ Q rev_implies:  Q subtract: m bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) ifthenelse: if then else fi  bfalse: ff bnot: ¬bb assert: b
Lemmas referenced :  nat_wf div_rem_sum subtype_base_sq int_subtype_base equal-wf-base true_wf nequal_wf nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf equal_wf rem_mul less_than_wf iff_weakening_equal rem_add1 eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int rem_base_case itermMultiply_wf int_term_value_mul_lemma decidable__lt intformless_wf int_formula_prop_less_lemma zero_ann eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int rem_bounds_1 decidable__equal_int intformeq_wf int_formula_prop_eq_lemma add-is-int-iff multiply-is-int-iff false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin multiplyEquality addEquality setElimination rename hypothesisEquality natural_numberEquality because_Cache dependent_set_memberEquality addLevel lambdaFormation instantiate cumulativity intEquality independent_isectElimination dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination voidElimination baseClosed unionElimination approximateComputation dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidEquality sqequalRule independent_pairFormation applyEquality imageElimination imageMemberEquality productElimination remainderEquality equalityElimination promote_hyp pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[n:\mBbbN{}].  (t(n  +  1)  =  (t(n)  +  n  +  1))



Date html generated: 2019_06_20-PM-02_38_05
Last ObjectModification: 2019_06_12-PM-00_26_34

Theory : num_thy_1


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