Nuprl Lemma : fun-connected-step
∀[T:Type]. ∀f:T ⟶ T. ∀x:T. (Dec((f x) = x ∈ T) ⇒ f x is f*(x))
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: 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,
fun-connected: y is f*(x),
decidable: Dec(P),
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
exists: ∃x:A. B[x],
fun-path: y=f*(x) via L,
top: Top,
subtract: n - m,
last: last(L),
select: L[n],
cons: [a / b],
and: P ∧ Q,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
sq_type: SQType(T)
Lemmas referenced :
decidable_wf,
equal_wf,
cons_wf,
nil_wf,
length_of_cons_lemma,
length_of_nil_lemma,
reduce_hd_cons_lemma,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
select_wf,
int_seg_wf,
fun-path_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
dependent_pairFormation,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
setElimination,
rename,
productElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
instantiate,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}x:T. (Dec((f x) = x) {}\mRightarrow{} f x is f*(x))
Date html generated:
2018_05_21-PM-07_44_53
Last ObjectModification:
2017_07_26-PM-05_22_25
Theory : general
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