Nuprl Lemma : list_extensionality
∀[T:Type]. ∀[a,b:T List].
(a = b ∈ (T List)) supposing ((∀i:ℕ. (i < ||a|| ⇒ (a[i] = b[i] ∈ T))) and (||a|| = ||b|| ∈ ℤ))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
all: ∀x:A. B[x],
guard: {T},
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
false: False,
subtract: n - m,
sq_type: SQType(T),
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
not: ¬A,
cons: [a / b],
less_than: a < b,
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P)
Lemmas referenced :
list_wf,
le_weakening,
length_wf,
less_than_transitivity1,
sq_stable__le,
select_wf,
equal_wf,
less_than_wf,
nat_wf,
all_wf,
list_induction,
equal-wf-base-T,
nil_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
equal-wf-base,
length_of_cons_lemma,
non_neg_length,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
cons_wf,
less_than_irreflexivity,
equal-wf-T-base,
add-commutes,
subtract_wf,
minus-add,
add-associates,
minus-one-mul,
zero-add,
add-swap,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
one-mul,
subtype_base_sq,
minus-zero,
nat_properties,
and_wf,
true_wf,
squash_wf,
nat_plus_wf,
add_nat_plus,
false_wf,
le-add-cancel2,
add_functionality_wrt_le,
minus-one-mul-top,
condition-implies-le,
le_antisymmetry_iff,
not-equal-2,
decidable__int_equal,
less-iff-le,
not-lt-2,
decidable__lt,
le-add-cancel,
not-le-2,
decidable__le,
iff_weakening_equal,
select_cons_tl
Rules used in proof :
axiomEquality,
isect_memberEquality,
isect_memberFormation,
universeEquality,
intEquality,
dependent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
hypothesisEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
because_Cache,
rename,
setElimination,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaFormation,
voidEquality,
voidElimination,
dependent_pairFormation,
sqequalIntensionalEquality,
applyEquality,
productElimination,
promote_hyp,
addEquality,
minusEquality,
multiplyEquality,
instantiate,
hyp_replacement,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[a,b:T List].
(a = b) supposing ((\mforall{}i:\mBbbN{}. (i < ||a|| {}\mRightarrow{} (a[i] = b[i]))) and (||a|| = ||b||))
Date html generated:
2019_06_20-PM-00_41_09
Last ObjectModification:
2018_08_06-PM-02_09_12
Theory : list_0
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