Nuprl Lemma : fpf-conversion-test
4 : 2 ⊕ 6 : 2 ⊕ 7 : 5 = <[4; 6; 7], λx.if x ∈b [4; 6] then 2 else 5 fi > ∈ i:ℤ fp-> ℤ
Proof
Definitions occuring in Statement :
fpf-single: x : v
,
fpf-join: f ⊕ g
,
fpf: a:A fp-> B[a]
,
deq-member: x ∈b L
,
cons: [a / b]
,
nil: []
,
int-deq: IntDeq
,
ifthenelse: if b then t else f fi
,
lambda: λx.A[x]
,
pair: <a, b>
,
natural_number: $n
,
int: ℤ
,
equal: s = t ∈ T
Definitions unfolded in proof :
cons: [a / b]
,
nil: []
,
it: ⋅
,
ifthenelse: if b then t else f fi
,
deq-member: x ∈b L
,
reduce: reduce(f;k;as)
,
list_ind: list_ind,
bor: p ∨bq
,
int-deq: IntDeq
,
eq_int: (i =z j)
,
uall: ∀[x:A]. B[x]
,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
,
member: t ∈ T
,
so_apply: x[s1;s2;s3;s4]
,
so_lambda: λ2x.t[x]
,
top: Top
,
so_apply: x[s]
,
uimplies: b supposing a
,
strict4: strict4(F)
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
has-value: (a)↓
,
prop: ℙ
,
guard: {T}
,
or: P ∨ Q
,
squash: ↓T
,
btrue: tt
,
bfalse: ff
,
fpf-join: f ⊕ g
,
append: as @ bs
,
pi1: fst(t)
,
fpf-single: x : v
,
filter: filter(P;l)
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
bnot: ¬bb
,
fpf-dom: x ∈ dom(f)
,
fpf-cap: f(x)?z
,
fpf-ap: f(x)
,
pi2: snd(t)
Lemmas referenced :
lifting-strict-int_eq,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
lifting-strict-spread,
lifting-strict-callbyvalue,
lifting-strict-ispair,
lifting-strict-decide,
lifting-strict-isaxiom,
fpf-join_wf,
fpf-single_wf,
int-deq_wf
Rules used in proof :
comment,
sqequalSubstitution,
sqequalRule,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesis,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation,
instantiate,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueExceptionCases,
callbyvalueApply,
applyExceptionCases,
intEquality,
lambdaEquality,
natural_numberEquality
Latex:
4 : 2 \moplus{} 6 : 2 \moplus{} 7 : 5 = <[4; 6; 7], \mlambda{}x.if x \mmember{}\msubb{} [4; 6] then 2 else 5 fi >
Date html generated:
2018_05_21-PM-09_25_16
Last ObjectModification:
2018_02_09-AM-10_21_00
Theory : finite!partial!functions
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