Nuprl Lemma : sqeq-copath1
∀[a,b,n:Top]. (λm.if (m) < (n - 1) then a[m] else b[m] ~ λm.if (m + 1) < (n) then a[m] else b[m])
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
less: if (a) < (b) then c else d,
lambda: λx.A[x],
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
has-value: (a)↓,
le: A ≤ B,
subtype_rel: A ⊆r B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
prop: ℙ,
false: False,
not: ¬A,
squash: ↓T,
true: True,
top: Top,
less_than': less_than'(a;b),
less_than: a < b,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtract: n - m,
uimplies: b supposing a,
is-exception: is-exception(t)
Lemmas referenced :
top_wf,
is-exception_wf,
has-value_wf_base,
minus-minus,
le-add-cancel,
zero-add,
add_functionality_wrt_le,
add-commutes,
minus-add,
less-iff-le,
minus-one-mul-top,
add-swap,
minus-one-mul,
base_wf,
add-associates,
condition-implies-le,
not-lt-2,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
less_than_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
int-value-type,
value-type-has-value,
exception-not-bottom,
bottom_diverge,
exception-not-value_1,
add-is-int-iff,
subtract_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
axiomSqEquality,
sqleReflexivity,
hypothesis,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
divergentSqle,
thin,
sqleRule,
sqequalSqle,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
callbyvalueLess,
minusEquality,
lambdaEquality,
applyEquality,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
addEquality,
independent_functionElimination,
imageElimination,
imageMemberEquality,
natural_numberEquality,
voidEquality,
voidElimination,
independent_pairFormation,
lessCases,
equalityElimination,
unionElimination,
lambdaFormation,
callbyvalueAdd,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
intEquality,
exceptionSqequal,
axiomSqleEquality,
lessExceptionCases,
exceptionLess,
addExceptionCases
Latex:
\mforall{}[a,b,n:Top].
(\mlambda{}m.if (m) < (n - 1) then a[m] else b[m] \msim{} \mlambda{}m.if (m + 1) < (n) then a[m] else b[m])
Date html generated:
2019_06_20-PM-01_11_52
Last ObjectModification:
2019_01_02-PM-01_35_32
Theory : co-recursion-2
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