Nuprl Lemma : between-fun-connected
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀y,x:T. (y is f*(x) ⇒ f x is f*(y) ⇒ ((y = x ∈ T) ∨ (y = (f x) ∈ T)))))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-connected: y is f*(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
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],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s1;s2],
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
retraction: retraction(T;f),
exists: ∃x:A. B[x],
fun-connected: y is f*(x),
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
fun-connected-induction,
fun-connected_wf,
or_wf,
equal_wf,
not_wf,
retraction_wf,
fun-connected_transitivity,
fun-connected_weakening,
strict-fun-connected-step,
squash_wf,
true_wf,
iff_weakening_equal,
and_wf,
retraction-fun-path,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
inlFormation,
because_Cache,
axiomEquality,
rename,
voidElimination,
universeEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
unionElimination,
inrFormation,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T. (retraction(T;f) {}\mRightarrow{} (\mforall{}y,x:T. (y is f*(x) {}\mRightarrow{} f x is f*(y) {}\mRightarrow{} ((y = x) \mvee{} (y = (f x))))))
Date html generated:
2018_05_21-PM-07_48_32
Last ObjectModification:
2017_07_26-PM-05_26_18
Theory : general
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