Nuprl Lemma : injective-quotient-inject
∀[T,S:Type]. ∀[f:T ⟶ S]. Inj(T//x.f[x];S;λx.f[x])
Proof
Definitions occuring in Statement :
injective-quotient: T//x.f[x],
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
injective-quotient: T//x.f[x],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
injective-quotient-typing,
istype-universe,
injective-quotient_wf,
quotient-member-eq,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
Error :equalityIstype,
applyEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
Error :lambdaEquality_alt,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
productElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
Error :productIsType,
sqequalBase,
independent_pairFormation
Latex:
\mforall{}[T,S:Type]. \mforall{}[f:T {}\mrightarrow{} S]. Inj(T//x.f[x];S;\mlambda{}x.f[x])
Date html generated:
2019_06_20-PM-00_33_07
Last ObjectModification:
2018_12_19-PM-05_30_19
Theory : quot_1
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