`∀[n:ℕ]. (t(n + 1) = (t(n) + n + 1) ∈ ℤ)`

Proof

Definitions occuring in Statement :  triangular-num: `t(n)` nat: `ℕ` uall: `∀[x:A]. B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` triangular-num: `t(n)` 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` ge: `i ≥ j ` 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: `ℕ+` less_than: `a < b` less_than': `less_than'(a;b)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f 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: 2018_05_21-PM-07_53_36
Last ObjectModification: 2018_05_19-PM-04_51_55

Theory : general

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