Nuprl Lemma : Wmul-Wadd
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[zero,succ:A ⟶ 𝔹].
∀[w3,w2,w1:W(A;a.B[a])]. ((w1 * (w2 + w3)) = ((w1 * w2) + (w1 * 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),
Wadd: (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,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
Wadd: (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 ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
Wmul: (w1 * w2),
not: ¬A,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
W-induction,
all_wf,
W_wf,
equal_wf,
Wmul_wf,
Wadd_wf,
assert_wf,
decidable__assert,
subtype_rel_wf,
unit_wf2,
not_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
it_wf,
iff_weakening_equal,
Wadd-assoc,
Wsup_wf,
squash_wf,
true_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,
hypothesis,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
functionEquality,
isect_memberEquality,
axiomEquality,
productEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
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) + (w1 * 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_56
Last ObjectModification:
2017_02_27-PM-03_15_56
Theory : co-recursion
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