Nuprl Lemma : tuple-equiv-is-equiv
∀L:(X:Type × (X ⟶ X ⟶ ℙ)) List
((∀i:ℕ||L||. let X,E = L[i] in EquivRel(X;x,y.E x y))
⇒ EquivRel(tuple-type(map(λp.(fst(p));L));t,t'.tuple-equiv(L) t t'))
Proof
Definitions occuring in Statement :
tuple-equiv: tuple-equiv(L),
tuple-type: tuple-type(L),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
int_seg: {i..j-},
prop: ℙ,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
tuple-equiv: tuple-equiv(L),
let: let,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
subtype_rel: A ⊆r B,
trans: Trans(T;x,y.E[x; y]),
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi2: snd(t),
pi1: fst(t),
less_than': less_than'(a;b),
label: ...$L... t,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
tuple-type_wf,
map_wf,
pi1_wf,
istype-universe,
tuple-equiv_wf,
subtype_rel_self,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
equiv_rel_wf,
list_wf,
select-tuple_wf,
int_seg_subtype_nat,
istype-false,
map-length,
equal_wf,
squash_wf,
true_wf,
length-map-sq,
subtype_rel_list,
top_wf,
iff_weakening_equal,
select-map,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
productEquality,
universeEquality,
functionEquality,
cumulativity,
hypothesisEquality,
Error :lambdaEquality_alt,
hypothesis,
Error :productIsType,
Error :functionIsType,
Error :inhabitedIsType,
applyEquality,
because_Cache,
natural_numberEquality,
closedConclusion,
setElimination,
rename,
independent_isectElimination,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
intEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}L:(X:Type \mtimes{} (X {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{})) List
((\mforall{}i:\mBbbN{}||L||. let X,E = L[i] in EquivRel(X;x,y.E x y))
{}\mRightarrow{} EquivRel(tuple-type(map(\mlambda{}p.(fst(p));L));t,t'.tuple-equiv(L) t t'))
Date html generated:
2019_06_20-PM-02_16_41
Last ObjectModification:
2019_03_18-PM-04_19_42
Theory : tuples
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