Nuprl Lemma : list_extensionality_iff
∀[T:Type]. ∀[a,b:T List]. (a = b ∈ (T List) ⇐⇒ (||a|| = ||b|| ∈ ℤ) ∧ (∀i:ℕ||a||. (a[i] = b[i] ∈ T)))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
true: True,
all: ∀x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
less_than: a < b,
nat: ℕ,
le: A ≤ B,
ge: i ≥ j ,
subtype_rel: A ⊆r B
Lemmas referenced :
length_wf,
int_seg_wf,
equal_wf,
list_wf,
all_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
less_than_wf,
list_extensionality,
squash_wf,
true_wf,
lelt_wf,
nat_properties,
iff_weakening_equal,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
cumulativity,
productElimination,
productEquality,
setElimination,
rename,
independent_isectElimination,
equalityTransitivity,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[a,b:T List]. (a = b \mLeftarrow{}{}\mRightarrow{} (||a|| = ||b||) \mwedge{} (\mforall{}i:\mBbbN{}||a||. (a[i] = b[i])))
Date html generated:
2017_04_14-AM-09_24_54
Last ObjectModification:
2017_02_27-PM-03_59_23
Theory : list_1
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