Nuprl Lemma : fun_exp_add1
∀[f,x:Top]. ∀[n:ℕ]. (f (f^n x) ~ f^n + 1 x)
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fun_exp: f^n,
nat: ℕ,
top: Top,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
prop: ℙ,
compose: f o g,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
primrec-unroll,
nat_wf,
top_wf,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
sq_stable__le,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
bnot_wf,
not_wf,
add-subtract-cancel,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalAxiom,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
intEquality,
independent_functionElimination,
imageMemberEquality,
imageElimination,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
minusEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination
Latex:
\mforall{}[f,x:Top]. \mforall{}[n:\mBbbN{}]. (f (f\^{}n x) \msim{} f\^{}n + 1 x)
Date html generated:
2017_04_14-AM-07_34_48
Last ObjectModification:
2017_02_27-PM-03_07_40
Theory : fun_1
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