Nuprl Lemma : ni-min-nat
∀[n,m:ℕ]. (ni-min(n∞;m∞) = imin(n;m)∞ ∈ ℕ∞)
Proof
Definitions occuring in Statement :
ni-min: ni-min(f;g),
nat2inf: n∞,
nat-inf: ℕ∞,
imin: imin(a;b),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat-inf: ℕ∞,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat2inf: n∞,
ni-min: ni-min(f;g),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T
Lemmas referenced :
assert_wf,
ni-min_wf,
nat2inf_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
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,
all_wf,
nat_wf,
iff_imp_equal_bool,
band_wf,
lt_int_wf,
imin_wf,
less_than_wf,
iff_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_lt_int,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
imin_unfold,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionEquality,
functionExtensionality,
axiomEquality,
productEquality,
productElimination,
addLevel,
impliesFunctionality,
independent_functionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[n,m:\mBbbN{}]. (ni-min(n\minfty{};m\minfty{}) = imin(n;m)\minfty{})
Date html generated:
2017_10_01-AM-08_30_03
Last ObjectModification:
2017_07_26-PM-04_24_18
Theory : basic
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