Nuprl Lemma : mu-unroll
∀[f:Top]. (mu(f) ~ if f 0 then 0 else mu(λi.(f (i + 1))) + 1 fi )
Proof
Definitions occuring in Statement :
mu: mu(f),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
lambda: λx.A[x],
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
mu-ge: mu-ge(f;n),
mu: mu(f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
bottom: ⊥,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
compose: f o g,
subtract: n - m,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
strict4: strict4(F),
has-value: (a)↓,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B
Lemmas referenced :
istype-top,
base_wf,
istype-sqequal,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
fun_exp_unroll,
istype-le,
strictness-apply,
bottom-sqle,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lifting-strict-ifthenelse,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
lifting-strict-callbyvalue,
int_subtype_base,
strictness-add-left,
exception-not-value
Rules used in proof :
extract_by_obid,
axiomSqEquality,
hypothesis,
sqleReflexivity,
callbyvalueReduce,
sqequalRule,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :universeIsType,
Error :lambdaFormation_alt,
thin,
Error :dependent_pairFormation_alt,
baseClosed,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
promote_hyp,
sqequalSqle,
fixpointLeast,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomSqleEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :dependent_set_memberEquality_alt,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
instantiate,
cumulativity,
baseApply,
closedConclusion,
callbyvalueAdd,
intEquality,
addExceptionCases,
exceptionSqequal,
Error :inrFormation_alt,
imageMemberEquality,
imageElimination,
Error :inlFormation_alt,
sqleRule,
divergentSqle,
applyEquality,
callbyvalueCallbyvalue,
callbyvalueExceptionCases,
addEquality
Latex:
\mforall{}[f:Top]. (mu(f) \msim{} if f 0 then 0 else mu(\mlambda{}i.(f (i + 1))) + 1 fi )
Date html generated:
2019_06_20-PM-01_17_18
Last ObjectModification:
2019_05_01-PM-05_04_08
Theory : int_2
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