Nuprl Lemma : fun-path-fixedpoint
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L:T List]. ∀[x,y,z:T]. (y = z ∈ T) supposing (((f y) = y ∈ T) and (y ∈ L) and z=f*(x) via L)
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
fun-path: y=f*(x) via L,
select: L[n],
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,
iff: P ⇐⇒ Q,
or: P ∨ Q,
cons: [a / b],
not: ¬A,
uiff: uiff(P;Q),
guard: {T},
nat_plus: ℕ+,
true: True,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list_induction,
uall_wf,
isect_wf,
fun-path_wf,
l_member_wf,
equal_wf,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
nil_wf,
less_than_wf,
equal-wf-T-base,
all_wf,
int_seg_wf,
equal-wf-base-T,
not_wf,
equal-wf-base,
cons_member,
reduce_hd_cons_lemma,
cons_wf,
list-cases,
product_subtype_list,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
fun-path-cons,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
independent_functionElimination,
lambdaFormation,
rename,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
baseClosed,
independent_isectElimination,
voidElimination,
voidEquality,
productElimination,
imageElimination,
productEquality,
natural_numberEquality,
minusEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
applyLambdaEquality,
setElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[L:T List]. \mforall{}[x,y,z:T].
(y = z) supposing (((f y) = y) and (y \mmember{} L) and z=f*(x) via L)
Date html generated:
2018_05_21-PM-07_43_26
Last ObjectModification:
2018_05_19-PM-04_48_52
Theory : general
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