Nuprl Lemma : not-quotient-function-subtype
¬(∀[X,A:Type]. ∀[E:A ⟶ A ⟶ ℙ].
(EquivRel(A;a,b.E[a;b]) ⇒ ((X ⟶ (a,b:A//E[a;b])) ⊆r (f,g:X ⟶ A//fun-equiv(X;a,b.↓E[a;b];f;g)))))
Proof
Definitions occuring in Statement :
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
quotient: x,y:A//B[x; y],
false: False,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
true: True,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
istype-universe,
equiv_rel_wf,
subtype_rel_wf,
quotient_wf,
fun-equiv_wf,
base_wf,
true_wf,
istype-base,
equiv_rel_true,
squash_wf,
quotient-member-eq,
subtype_base_sq,
subtype_rel_self,
int_subtype_base,
fun-equiv-rel,
equiv_rel_squash,
quotient-squash
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalRule,
Error :isectIsType,
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
hypothesis,
Error :inhabitedIsType,
hypothesisEquality,
Error :functionIsType,
Error :universeIsType,
because_Cache,
Error :lambdaEquality_alt,
applyEquality,
functionEquality,
independent_isectElimination,
independent_functionElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
pertypeElimination,
promote_hyp,
productElimination,
Error :productIsType,
Error :equalityIstype,
sqequalBase,
dependent_functionElimination,
natural_numberEquality,
cumulativity,
intEquality,
voidElimination
Latex:
\mneg{}(\mforall{}[X,A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;a,b.E[a;b])
{}\mRightarrow{} ((X {}\mrightarrow{} (a,b:A//E[a;b])) \msubseteq{}r (f,g:X {}\mrightarrow{} A//fun-equiv(X;a,b.\mdownarrow{}E[a;b];f;g)))))
Date html generated:
2019_06_20-PM-00_32_59
Last ObjectModification:
2018_11_26-AM-00_13_31
Theory : quot_1
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