Nuprl Lemma : fun_exp_add1_sub
∀[T:Type]. ∀[f:T ⟶ T]. ∀[x:T]. ∀[n:ℕ+]. ((f (f^n - 1 x)) = (f^n x) ∈ T)
Proof
Definitions occuring in Statement :
fun_exp: f^n,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T
Lemmas referenced :
nat_plus_wf,
fun_exp_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
false_wf,
decidable__le,
subtract_wf,
fun_exp_add1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
applyEquality,
lambdaEquality,
intEquality,
minusEquality,
because_Cache,
imageElimination,
imageMemberEquality,
baseClosed,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[x:T]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((f (f\^{}n - 1 x)) = (f\^{}n x))
Date html generated:
2016_05_13-PM-04_06_56
Last ObjectModification:
2016_01_14-PM-07_46_13
Theory : fun_1
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