Nuprl Lemma : Wleq-Wadd3
∀[A:Type]. ∀[B:A ⟶ Type]. ∀zero:A ⟶ 𝔹. ((∀a:A. (¬↑(zero a) ⇐⇒ B[a])) ⇒ (∀w1,w2:W(A;a.B[a]). (w1 ≤ (w1 + w2))))
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],
subtype_rel: A ⊆r B,
prop: ℙ,
Wsup: Wsup(a;b),
Wcmp: Wcmp(A;a.B[a];leq),
infix_ap: x f y,
ifthenelse: if b then t else f fi ,
btrue: tt,
Wadd: (w1 + w2),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q
Lemmas referenced :
W-induction,
all_wf,
W_wf,
infix_ap_wf,
Wcmp_wf,
btrue_wf,
Wadd_wf,
bfalse_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_weakening
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,
instantiate,
universeEquality,
independent_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
voidElimination,
functionEquality,
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{}w1,w2:W(A;a.B[a]). (w1 \mleq{} (w1 + w2))))
Date html generated:
2017_04_14-AM-07_44_30
Last ObjectModification:
2017_02_27-PM-03_15_47
Theory : co-recursion
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