Nuprl Lemma : fun-connected_transitivity
∀[T:Type]. ∀f:T ⟶ T. ∀x,y,z:T. (y is f*(x) ⇒ z is f*(y) ⇒ z is f*(x))
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fun-connected: y is f*(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
cons: [a / b],
fun-path: y=f*(x) via L,
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
subtract: n - m,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
not: ¬A,
uiff: uiff(P;Q),
guard: {T},
cand: A c∧ B,
true: True
Lemmas referenced :
exists_wf,
list_wf,
fun-path_wf,
list-cases,
product_subtype_list,
length_of_nil_lemma,
stuck-spread,
base_wf,
last_lemma,
null_nil_lemma,
length_of_cons_lemma,
reduce_hd_cons_lemma,
null_cons_lemma,
false_wf,
append_wf,
cons_wf,
fun-path-append,
and_wf,
equal_wf,
nil_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
dependent_pairFormation,
addLevel,
levelHypothesis,
equalityTransitivity,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
setEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}x,y,z:T. (y is f*(x) {}\mRightarrow{} z is f*(y) {}\mRightarrow{} z is f*(x))
Date html generated:
2016_10_25-AM-11_02_59
Last ObjectModification:
2016_07_12-AM-07_10_33
Theory : general
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