Nuprl Lemma : all-quotient-dependent
∀T:Type
(canonicalizable(T)
⇒ (∀S:T ⟶ ℙ. ∀E:t:T ⟶ S[t] ⟶ S[t] ⟶ ℙ.
((∀t:T. EquivRel(S t;a,b.E[t;a;b]))
⇒ (∀t:T. (x,y:S[t]//E[t;x;y]) ⇐⇒ f,g:∀t:T. S[t]//dep-fun-equiv(T;t,x,y.↓E[t;x;y];f;g)))))
Proof
Definitions occuring in Statement :
dep-fun-equiv: dep-fun-equiv(X;x,a,b.E[x; a; b];f;g),
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
canonicalizable: canonicalizable(T),
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
ext-eq: A ≡ B,
dep-fun-equiv: dep-fun-equiv(X;x,a,b.E[x; a; b];f;g),
true: True,
squash: ↓T,
guard: {T},
compose: f o g,
quotient: x,y:A//B[x; y],
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
isect2: T1 ⋂ T2,
exists: ∃x:A. B[x],
canonicalizable: canonicalizable(T),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
rev_implies: P ⇐ Q,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_apply: x[s1;s2;s3]
Lemmas referenced :
quotient-squash,
bool_wf,
iff_weakening_equal,
true_wf,
equal_wf,
subtype_rel-equal,
quotient-member-eq,
equal-wf-base,
isect2_subtype_rel2,
quotient-dep-function-subtype,
isect2_subtype_rel,
base_wf,
isect2_wf,
subtype_rel_dep_function,
canonicalizable_wf,
equiv_rel_wf,
subtype_rel_self,
equiv_rel_subtype,
equiv_rel_squash,
dep-fun-equiv-rel,
squash_wf,
dep-fun-equiv_wf,
quotient_wf,
all_wf
Rules used in proof :
hyp_replacement,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
instantiate,
productEquality,
pertypeElimination,
pointwiseFunctionalityForEquality,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
isect_memberEquality,
rename,
productElimination,
universeEquality,
functionEquality,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
hypothesis,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}T:Type
(canonicalizable(T)
{}\mRightarrow{} (\mforall{}S:T {}\mrightarrow{} \mBbbP{}. \mforall{}E:t:T {}\mrightarrow{} S[t] {}\mrightarrow{} S[t] {}\mrightarrow{} \mBbbP{}.
((\mforall{}t:T. EquivRel(S t;a,b.E[t;a;b]))
{}\mRightarrow{} (\mforall{}t:T. (x,y:S[t]//E[t;x;y]) \mLeftarrow{}{}\mRightarrow{} f,g:\mforall{}t:T. S[t]//dep-fun-equiv(T;t,x,y.\mdownarrow{}E[t;x;y];f;g)))))
Date html generated:
2017_09_29-PM-06_07_42
Last ObjectModification:
2017_09_07-PM-03_38_22
Theory : continuity
Home
Index