Nuprl Lemma : intlex-aux-antisym
∀[l1:ℤ List]. ∀[l2:{as:ℤ List| ||as|| = ||l1|| ∈ ℤ} ].
(l1 = l2 ∈ (ℤ List)) supposing (intlex-aux(l2;l1) = tt and intlex-aux(l1;l2) = tt)
Proof
Definitions occuring in Statement :
intlex-aux: intlex-aux(l1;l2),
length: ||as||,
list: T List,
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
or: P ∨ Q,
cons: [a / b],
top: Top,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
sq_type: SQType(T),
nat: ℕ,
intlex-aux: intlex-aux(l1;l2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
isl: isl(x),
bool: 𝔹,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
list_wf,
equal-wf-base,
isect_wf,
bool_wf,
list_subtype_base,
int_subtype_base,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
length_wf_nat,
nat_wf,
subtype_rel-equal,
base_wf,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
equal_wf,
set_subtype_base,
set_wf,
add-associates,
subtract_wf,
minus-zero,
add-swap,
subtype_base_sq,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul,
nat_properties,
spread_cons_lemma,
decidable__lt,
top_wf,
less_than_wf,
less_than_transitivity2,
less_than_irreflexivity,
less_than_transitivity1,
le_weakening,
bfalse_wf,
and_wf,
isl_wf,
unit_wf2,
btrue_neq_bfalse,
decidable__equal_int,
false_wf,
not-equal-2,
not-lt-2,
cons_wf,
squash_wf,
true_wf,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
setEquality,
intEquality,
hypothesis,
lambdaFormation,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
sqequalIntensionalEquality,
addEquality,
natural_numberEquality,
minusEquality,
axiomEquality,
instantiate,
cumulativity,
multiplyEquality,
lessCases,
sqequalAxiom,
independent_pairFormation,
imageMemberEquality,
imageElimination,
int_eqReduceFalseSq,
dependent_set_memberEquality,
applyLambdaEquality,
int_eqReduceTrueSq,
universeEquality
Latex:
\mforall{}[l1:\mBbbZ{} List]. \mforall{}[l2:\{as:\mBbbZ{} List| ||as|| = ||l1||\} ].
(l1 = l2) supposing (intlex-aux(l2;l1) = tt and intlex-aux(l1;l2) = tt)
Date html generated:
2017_09_29-PM-05_48_46
Last ObjectModification:
2017_07_26-PM-01_37_15
Theory : list_0
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