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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