Nuprl Lemma : upto_equal_nil
∀[n:ℕ]. uiff(upto(n) = [] ∈ (ℤ List);n = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
upto: upto(n),
nil: [],
list: T List,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
prop: ℙ,
label: ...$L... t,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
sq_type: SQType(T),
all: ∀x:A. B[x]
Lemmas referenced :
upto_is_nil,
subtype_base_sq,
list_wf,
int_seg_wf,
list_subtype_base,
set_subtype_base,
lelt_wf,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
sqequal-nil,
nil_wf,
iff_weakening_equal,
equal-wf-T-base,
upto_wf,
subtype_rel_list,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
instantiate,
cumulativity,
natural_numberEquality,
setElimination,
rename,
hypothesis,
because_Cache,
sqequalRule,
intEquality,
lambdaEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[n:\mBbbN{}]. uiff(upto(n) = [];n = 0)
Date html generated:
2017_04_17-AM-07_57_26
Last ObjectModification:
2017_02_27-PM-04_29_04
Theory : list_1
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