Nuprl Lemma : Wmul-Wzero
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero,succ:A ⟶ 𝔹].
∀[z:W(A;a.B[a])]. ∀[w1:W(A;a.B[a])]. ((w1 * z) = z ∈ W(A;a.B[a])) supposing isZero(z)
supposing ∀a:A. ((↑(succ a)) ⇒ (Unit ⊆r B[a]))
Proof
Definitions occuring in Statement :
Wmul: (w1 * w2),
Wzero: isZero(w),
W: W(A;a.B[a]),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
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,
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),
Wmul: (w1 * w2),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
not: ¬A,
subtype_rel: A ⊆r B,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
true: True
Lemmas referenced :
W-induction,
Wzero_wf,
all_wf,
W_wf,
equal_wf,
Wmul_wf,
bool_wf,
eqtt_to_assert,
it_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
Wsup_wf,
squash_wf,
true_wf,
not_wf,
assert_wf,
subtype_rel_wf,
unit_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
functionEquality,
because_Cache,
hypothesis,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[zero,succ:A {}\mrightarrow{} \mBbbB{}].
\mforall{}[z:W(A;a.B[a])]. \mforall{}[w1:W(A;a.B[a])]. ((w1 * z) = z) supposing isZero(z)
supposing \mforall{}a:A. ((\muparrow{}(succ a)) {}\mRightarrow{} (Unit \msubseteq{}r B[a]))
Date html generated:
2017_04_14-AM-07_45_01
Last ObjectModification:
2017_02_27-PM-03_15_43
Theory : co-recursion
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