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
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