Nuprl Lemma : Wmul-assoc
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero,succ:A ⟶ 𝔹]. ∀[w3,w2,w1:W(A;a.B[a])].
(w1 * (w2 * w3)) = ((w1 * w2) * w3) ∈ W(A;a.B[a])
supposing ∀a:A. (((↑(succ a))
⇒ (Unit ⊆r B[a])) ∧ ((↑(zero a))
⇒ (¬B[a])))
Proof
Definitions occuring in Statement :
Wmul: (w1 * w2)
,
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]
,
not: ¬A
,
implies: P
⇒ Q
,
and: 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]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
and: P ∧ Q
,
prop: ℙ
,
Wmul: (w1 * w2)
,
Wsup: Wsup(a;b)
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
ifthenelse: if b then t else f fi
,
squash: ↓T
,
subtype_rel: A ⊆r B
,
true: True
,
guard: {T}
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
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-induction,
all_wf,
equal_wf,
Wmul_wf,
assert_wf,
bool_wf,
eqtt_to_assert,
squash_wf,
true_wf,
Wmul-Wadd,
it_wf,
Wadd_wf,
iff_weakening_equal,
W_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtype_rel_wf,
unit_wf2,
not_wf,
Wsup_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
independent_isectElimination,
lambdaFormation,
hypothesis,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
functionEquality,
productEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[zero,succ:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[w3,w2,w1:W(A;a.B[a])].
(w1 * (w2 * w3)) = ((w1 * w2) * w3)
supposing \mforall{}a:A. (((\muparrow{}(succ a)) {}\mRightarrow{} (Unit \msubseteq{}r B[a])) \mwedge{} ((\muparrow{}(zero a)) {}\mRightarrow{} (\mneg{}B[a])))
Date html generated:
2017_04_14-AM-07_44_59
Last ObjectModification:
2017_02_27-PM-03_15_51
Theory : co-recursion
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