Nuprl Lemma : fun-connected-tree
∀[T:Type]
∀f:T ⟶ T
∀x,y:T. (x is f*(y) ⇒ (∀z:T. (x is f*(z) ⇒ (z is f*(y) ∨ y is f*(z)))))
supposing ∀a,b:T. (((f a) = (f b) ∈ T) ⇒ (¬((f a) = a ∈ T)) ⇒ (¬((f b) = b ∈ T)) ⇒ (a = b ∈ T))
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
guard: {T},
or: P ∨ Q,
not: ¬A,
false: False,
fun-connected: y is f*(x),
exists: ∃x:A. B[x],
cons: [a / b],
fun-path: y=f*(x) via L,
select: L[n],
nil: [],
it: ⋅,
top: Top,
subtract: n - m,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
decidable: Dec(P),
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
equal_wf,
fun-connected-induction,
all_wf,
fun-connected_wf,
or_wf,
list-cases,
product_subtype_list,
length_of_nil_lemma,
stuck-spread,
base_wf,
fun-path-cons,
decidable__lt,
length_wf,
and_wf,
squash_wf,
true_wf,
fun-path_wf,
fun-connected_transitivity,
fun-connected_weakening,
strict-fun-connected-step,
iff_weakening_equal,
strict-fun-connected_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
functionExtensionality,
rename,
functionEquality,
independent_functionElimination,
inrFormation,
because_Cache,
voidElimination,
universeEquality,
productElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidEquality,
imageElimination,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
instantiate,
imageMemberEquality,
dependent_pairFormation,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
\mforall{}x,y:T. (x is f*(y) {}\mRightarrow{} (\mforall{}z:T. (x is f*(z) {}\mRightarrow{} (z is f*(y) \mvee{} y is f*(z)))))
supposing \mforall{}a,b:T. (((f a) = (f b)) {}\mRightarrow{} (\mneg{}((f a) = a)) {}\mRightarrow{} (\mneg{}((f b) = b)) {}\mRightarrow{} (a = b))
Date html generated:
2018_05_21-PM-07_45_51
Last ObjectModification:
2017_07_26-PM-05_23_23
Theory : general
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