Nuprl Lemma : sqeq-copath2
∀[a,b,n,m:Top]. (if n + 1=m then a else b ~ if n=m - 1 then a else b)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x]
,
top: Top
,
int_eq: if a=b then c else d
,
subtract: n - m
,
add: n + m
,
natural_number: $n
,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
true: True
,
less_than': less_than'(a;b)
,
le: A ≤ B
,
top: Top
,
subtype_rel: A ⊆r B
,
subtract: n - m
,
nequal: a ≠ b ∈ T
,
not: ¬A
,
false: False
,
assert: ↑b
,
ifthenelse: if b then t else f fi
,
bnot: ¬bb
,
guard: {T}
,
sq_type: SQType(T)
,
or: P ∨ Q
,
prop: ℙ
,
exists: ∃x:A. B[x]
,
bfalse: ff
,
uiff: uiff(P;Q)
,
btrue: tt
,
it: ⋅
,
unit: Unit
,
bool: 𝔹
,
implies: P
⇒ Q
,
all: ∀x:A. B[x]
,
uimplies: b supposing a
,
and: P ∧ Q
,
has-value: (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,
le_antisymmetry_iff,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
base_wf,
add-associates,
condition-implies-le,
not-equal-2,
add-is-int-iff,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
subtract_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
int-value-type,
value-type-has-value,
exception-not-bottom,
bottom_diverge,
exception-not-value_1
Rules used in proof :
isect_memberEquality,
axiomSqEquality,
isect_memberFormation,
because_Cache,
sqleReflexivity,
hypothesis,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
sqequalRule,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
divergentSqle,
thin,
sqleRule,
cut,
sqequalSqle,
natural_numberEquality,
voidEquality,
lambdaEquality,
applyEquality,
minusEquality,
addEquality,
int_eqReduceFalseSq,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
int_eqReduceTrueSq,
equalityElimination,
unionElimination,
lambdaFormation,
callbyvalueAdd,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
intEquality,
productElimination,
callbyvalueIntEq,
exceptionSqequal,
axiomSqleEquality,
int_eqExceptionCases,
exceptionInteq,
addExceptionCases
Latex:
\mforall{}[a,b,n,m:Top]. (if n + 1=m then a else b \msim{} if n=m - 1 then a else b)
Date html generated:
2019_06_20-PM-01_11_59
Last ObjectModification:
2019_01_02-PM-01_35_34
Theory : co-recursion-2
Home
Index