Nuprl Lemma : Wleq-Wmul
∀[A:Type]. ∀[B:A ⟶ Type].
∀zero,succ:A ⟶ 𝔹.
((∀a:A. ((↑(succ a)) ⇒ (Unit ⊆r B[a])))
⇒ (∀a:A. (¬↑(zero a) ⇐⇒ B[a]))
⇒ (∀w3,w2,w1:W(A;a.B[a]). ((w1 ≤ w2) ⇒ ((w1 * w3) ≤ (w2 * w3)))))
Proof
Definitions occuring in Statement :
Wmul: (w1 * w2),
Wcmp: Wcmp(A;a.B[a];leq),
W: W(A;a.B[a]),
assert: ↑b,
btrue: tt,
bool: 𝔹,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
Wmul: (w1 * w2),
Wsup: Wsup(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
Wcmp: Wcmp(A;a.B[a];leq),
infix_ap: x f y
Lemmas referenced :
W-induction,
all_wf,
W_wf,
infix_ap_wf,
Wcmp_wf,
btrue_wf,
Wmul_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_wf,
not_wf,
assert_wf,
subtype_rel_wf,
unit_wf2,
Wcmp_transitivity,
Wadd_wf,
it_wf,
Wleq-Wadd,
Wleq-Wadd2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
hypothesis,
functionEquality,
instantiate,
universeEquality,
independent_isectElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
voidElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}zero,succ:A {}\mrightarrow{} \mBbbB{}.
((\mforall{}a:A. ((\muparrow{}(succ a)) {}\mRightarrow{} (Unit \msubseteq{}r B[a])))
{}\mRightarrow{} (\mforall{}a:A. (\mneg{}\muparrow{}(zero a) \mLeftarrow{}{}\mRightarrow{} B[a]))
{}\mRightarrow{} (\mforall{}w3,w2,w1:W(A;a.B[a]). ((w1 \mleq{} w2) {}\mRightarrow{} ((w1 * w3) \mleq{} (w2 * w3)))))
Date html generated:
2017_04_14-AM-07_45_05
Last ObjectModification:
2017_02_27-PM-03_16_20
Theory : co-recursion
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