Nuprl Lemma : Wleq-Wadd
∀[A:Type]. ∀[B:A ⟶ Type]. ∀zero:A ⟶ 𝔹. ∀w3,w2,w1:W(A;a.B[a]). ((w1 ≤ w2) ⇒ ((w1 + w3) ≤ (w2 + w3)))
Proof
Definitions occuring in Statement :
Wadd: (w1 + w2),
Wcmp: Wcmp(A;a.B[a];leq),
W: W(A;a.B[a]),
btrue: tt,
bool: 𝔹,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
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,
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,
Wadd_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-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
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}zero:A {}\mrightarrow{} \mBbbB{}. \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_44_25
Last ObjectModification:
2017_02_27-PM-03_15_07
Theory : co-recursion
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