Nuprl Lemma : Wadd-Wzero
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero:A ⟶ 𝔹].
∀[z:W(A;a.B[a])]
∀[w1:W(A;a.B[a])]. (((w1 + z) = w1 ∈ W(A;a.B[a])) ∧ ((z + w1) = w1 ∈ W(A;a.B[a]))) supposing isZero(z)
supposing (∀a:A. (↑(zero a) ⇐⇒ ¬B[a])) ∧ (∀a1,a2:A. ((↑(zero a1)) ⇒ (↑(zero a2)) ⇒ (a1 = a2 ∈ A)))
Proof
Definitions occuring in Statement :
Wadd: (w1 + w2),
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],
iff: P ⇐⇒ Q,
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,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
Wsup: Wsup(a;b),
Wzero: isZero(w),
pi1: fst(t),
Wadd: (w1 + w2),
squash: ↓T,
true: True,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
W_wf,
Wzero_wf,
all_wf,
iff_wf,
assert_wf,
not_wf,
equal_wf,
bool_wf,
W-induction,
Wadd_wf,
squash_wf,
true_wf,
ite_rw_true,
Wsup_wf,
iff_weakening_equal,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
Wzero-unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
sqequalRule,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productEquality,
functionEquality,
universeEquality,
independent_functionElimination,
lambdaFormation,
imageElimination,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[zero:A {}\mrightarrow{} \mBbbB{}].
\mforall{}[z:W(A;a.B[a])]. \mforall{}[w1:W(A;a.B[a])]. (((w1 + z) = w1) \mwedge{} ((z + w1) = w1)) supposing isZero(z)
supposing (\mforall{}a:A. (\muparrow{}(zero a) \mLeftarrow{}{}\mRightarrow{} \mneg{}B[a])) \mwedge{} (\mforall{}a1,a2:A. ((\muparrow{}(zero a1)) {}\mRightarrow{} (\muparrow{}(zero a2)) {}\mRightarrow{} (a1 = a2)))
Date html generated:
2017_04_14-AM-07_44_48
Last ObjectModification:
2017_02_27-PM-03_15_49
Theory : co-recursion
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