Nuprl Lemma : Wzero-unique
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero:A ⟶ 𝔹].
∀[z1,z2:W(A;a.B[a])]. z1 = z2 ∈ W(A;a.B[a]) supposing isZero(z1) ∧ isZero(z2)
supposing (∀a1,a2:A. ((↑(zero a1)) ⇒ (↑(zero a2)) ⇒ (a1 = a2 ∈ A))) ∧ (∀a:A. ((¬B[a]) ⇒ (↑(zero a))))
Proof
Definitions occuring in Statement :
Wzero: isZero(w),
W: W(A;a.B[a]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
Wsup: Wsup(a;b),
Wzero: isZero(w),
pi1: fst(t),
squash: ↓T,
not: ¬A,
false: False,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
W-induction,
all_wf,
W_wf,
Wzero_wf,
equal_wf,
Wsup_wf,
squash_wf,
true_wf,
not_wf,
assert_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
hypothesis,
functionEquality,
independent_functionElimination,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[zero:A {}\mrightarrow{} \mBbbB{}].
\mforall{}[z1,z2:W(A;a.B[a])]. z1 = z2 supposing isZero(z1) \mwedge{} isZero(z2)
supposing (\mforall{}a1,a2:A. ((\muparrow{}(zero a1)) {}\mRightarrow{} (\muparrow{}(zero a2)) {}\mRightarrow{} (a1 = a2)))
\mwedge{} (\mforall{}a:A. ((\mneg{}B[a]) {}\mRightarrow{} (\muparrow{}(zero a))))
Date html generated:
2017_04_14-AM-07_44_43
Last ObjectModification:
2017_02_27-PM-03_15_45
Theory : co-recursion
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