Nuprl Lemma : sqeq-copath3
∀[a,b,c,f:Top]. ∀[n,m:ℤ].
(λx.if (x) < (m) then f if x=n then a[x] else b[x] else c[x] ~ λx.if (x) < (m)
then if x=n then f a[x] else (f b[x])
else c[x])
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x]
,
top: Top
,
so_apply: x[s]
,
less: if (a) < (b) then c else d
,
int_eq: if a=b then c else d
,
apply: f a
,
lambda: λx.A[x]
,
int: ℤ
,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
subtype_rel: A ⊆r B
,
has-value: (a)↓
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
uimplies: b supposing a
,
less_than: a < b
,
less_than': less_than'(a;b)
,
top: Top
,
true: True
,
squash: ↓T
,
not: ¬A
,
false: False
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
guard: {T}
,
bnot: ¬bb
,
ifthenelse: if b then t else f fi
,
assert: ↑b
,
nequal: a ≠ b ∈ T
,
rev_implies: P
⇐ Q
,
iff: P
⇐⇒ Q
,
prop: ℙ
Lemmas referenced :
has-value_wf_base,
int_subtype_base,
is-exception_wf,
istype-int,
istype-top,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-void,
eq_int_wf,
assert_of_eq_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
exception-not-value_1,
value-type-has-value,
int-value-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalSqle,
sqleRule,
thin,
divergentSqle,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqleReflexivity,
axiomSqEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
callbyvalueLess,
productElimination,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lessCases,
independent_pairFormation,
voidElimination,
natural_numberEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
int_eqReduceTrueSq,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
int_eqReduceFalseSq,
Error :equalityIsType1,
Error :universeIsType,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal,
intEquality
Latex:
\mforall{}[a,b,c,f:Top]. \mforall{}[n,m:\mBbbZ{}].
(\mlambda{}x.if (x) < (m) then f if x=n then a[x] else b[x] else c[x] \msim{} \mlambda{}x.if (x) < (m)
then if x=n
then f a[x]
else (f b[x])
else c[x])
Date html generated:
2019_06_20-PM-01_12_05
Last ObjectModification:
2019_01_02-PM-01_35_36
Theory : co-recursion-2
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