Nuprl Lemma : fun_exp_unroll
∀[n:ℕ]. ∀[f:Top]. (f^n ~ if (n =z 0) then λx.x else f o f^n - 1 fi )
Proof
Definitions occuring in Statement :
fun_exp: f^n,
compose: f o g,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
lambda: λx.A[x],
subtract: n - m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
fun_exp: f^n,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
subtract: n - m
Lemmas referenced :
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eq_int_wf,
bool_wf,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-equal-2,
sq_stable__le,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less-iff-le,
le-add-cancel2,
not_functionality_wrt_uiff,
assert_wf,
less_than_wf,
not-lt-2,
le_antisymmetry_iff,
top_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
applyEquality,
lambdaEquality,
intEquality,
minusEquality,
axiomSqEquality,
Error :universeIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:Top]. (f\^{}n \msim{} if (n =\msubz{} 0) then \mlambda{}x.x else f o f\^{}n - 1 fi )
Date html generated:
2019_06_20-PM-00_26_40
Last ObjectModification:
2018_09_26-PM-00_12_07
Theory : fun_1
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