Nuprl Lemma : triangular-num-add1
∀[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:
2019_06_20-PM-02_38_05
Last ObjectModification:
2019_06_12-PM-00_26_34
Theory : num_thy_1
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