Nuprl Lemma : Wleq-Wadd2
∀[A:Type]. ∀[B:A ⟶ Type].
∀zero:A ⟶ 𝔹. ((∀a:A. (¬↑(zero a)
⇐⇒ B[a]))
⇒ (∀w3,w2,w1:W(A;a.B[a]). ((w2 ≤ w3)
⇒ ((w1 + w2) ≤ (w1 + w3)))))
Proof
Definitions occuring in Statement :
Wadd: (w1 + w2)
,
Wcmp: Wcmp(A;a.B[a];leq)
,
W: W(A;a.B[a])
,
assert: ↑b
,
btrue: tt
,
bool: 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
,
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
,
Wadd: (w1 + w2)
,
Wsup: Wsup(a;b)
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
uimplies: b supposing a
,
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
,
infix_ap: x f y
,
not: ¬A
,
Wcmp: Wcmp(A;a.B[a];leq)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
Lemmas referenced :
W-induction,
all_wf,
W_wf,
infix_ap_wf,
Wcmp_wf,
btrue_wf,
Wadd_wf,
Wsup_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,
Wleq_weakening2,
Wleq-Wadd3,
bool_cases,
assert_of_bnot
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_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
voidElimination,
rename
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}zero:A {}\mrightarrow{} \mBbbB{}
((\mforall{}a:A. (\mneg{}\muparrow{}(zero a) \mLeftarrow{}{}\mRightarrow{} B[a]))
{}\mRightarrow{} (\mforall{}w3,w2,w1:W(A;a.B[a]). ((w2 \mleq{} w3) {}\mRightarrow{} ((w1 + w2) \mleq{} (w1 + w3)))))
Date html generated:
2017_04_14-AM-07_44_35
Last ObjectModification:
2017_02_27-PM-03_15_59
Theory : co-recursion
Home
Index