Nuprl Lemma : nat-pred-as-sub
∀n:ℕ. (0 < n ⇒ (n-1 = (n - 1) ∈ ℕ))
Proof
Definitions occuring in Statement :
nat-pred: n-1,
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
int_upper: {i...},
nequal: a ≠ b ∈ T ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
less_than': less_than'(a;b),
le: A ≤ B,
prop: ℙ,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat-pred: n-1,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
nat_wf,
less_than_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
int_upper_properties,
zero-add,
nequal-le-implies,
int_upper_subtype_nat,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_wf,
false_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
subtract_wf,
decidable__equal_int,
nat_properties,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf
Rules used in proof :
hypothesis_subsumption,
int_eqReduceFalseSq,
independent_functionElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_set_memberEquality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
dependent_functionElimination,
hypothesisEquality,
int_eqReduceTrueSq,
sqequalRule,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. (0 < n {}\mRightarrow{} (n-1 = (n - 1)))
Date html generated:
2017_04_21-AM-11_21_48
Last ObjectModification:
2017_04_20-PM-03_50_09
Theory : continuity
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