Nuprl Lemma : decomp-map-if-has-value
∀[t,f:Base]. if ispair(t) then t ~ <fst(t), snd(t)> else t ~ Ax fi supposing (map(f;t))↓
Proof
Definitions occuring in Statement :
map: map(f;as),
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
ispair: if z is a pair then a otherwise b,
pair: <a, b>,
base: Base,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
map: map(f;as),
list_ind: list_ind,
has-value: (a)↓,
prop: ℙ,
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
or: P ∨ Q,
pi1: fst(t),
pi2: snd(t),
btrue: tt,
bfalse: ff,
not: ¬A,
false: False
Lemmas referenced :
has-value-if-has-value-map,
ispair-bool-if-has-value,
has-value_wf_base,
base_wf,
equal_wf,
bool_wf,
has-value-implies-dec-ispair,
has-value-implies-dec-isaxiom,
bottom_diverge
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
callbyvalueCallbyvalue,
callbyvalueReduce,
because_Cache,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
baseClosed,
instantiate,
isect_memberFormation,
lambdaFormation,
unionElimination,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination
Latex:
\mforall{}[t,f:Base]. if ispair(t) then t \msim{} <fst(t), snd(t)> else t \msim{} Ax fi supposing (map(f;t))\mdownarrow{}
Date html generated:
2018_05_21-PM-10_19_16
Last ObjectModification:
2017_07_26-PM-06_36_54
Theory : eval!all
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