Nuprl Lemma : same-final-iterate-one-one
∀[A:Type]
∀f:A ⟶ (A + Top)
(SWellFounded(p-graph(A;f) y x)
⇒ ∀x,y:A.
∃n:ℕ. ((p-graph(A;f^n) x y) ∨ (p-graph(A;f^n) y x)) supposing final-iterate(f;x) = final-iterate(f;y) ∈ A
supposing p-inject(A;A;f))
Proof
Definitions occuring in Statement :
final-iterate: final-iterate(f;x),
p-graph: p-graph(A;f),
p-inject: p-inject(A;B;f),
p-fun-exp: f^n,
strongwellfounded: SWellFounded(R[x; y]),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
p-graph: p-graph(A;f),
member: t ∈ T,
exists: ∃x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
le: A ≤ B,
subtract: n - m,
sq_type: SQType(T),
p-inject: p-inject(A;B;f)
Lemmas referenced :
final-iterate-property,
decidable__le,
final-iterate_wf,
p-inject_wf,
strongwellfounded_wf,
p-graph_wf2,
subtype_rel_self,
istype-top,
istype-universe,
subtract_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-assert,
can-apply_wf,
p-fun-exp_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
do-apply_wf,
can-apply-fun-exp,
trivial-int-eq1,
can-apply-fun-exp-add,
p-fun-exp-injection,
minus-one-mul,
add-commutes,
add-associates,
add-mul-special,
zero-mul,
zero-add,
subtype_base_sq,
int_subtype_base,
equal_wf,
assert_wf,
add-swap,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
because_Cache,
productElimination,
setElimination,
rename,
unionElimination,
equalityIstype,
independent_isectElimination,
universeIsType,
lambdaEquality_alt,
applyEquality,
instantiate,
inhabitedIsType,
functionIsType,
unionIsType,
universeEquality,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
inlFormation_alt,
productIsType,
unionEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
intEquality,
applyLambdaEquality,
promote_hyp,
hyp_replacement,
functionEquality,
inrFormation_alt
Latex:
\mforall{}[A:Type]
\mforall{}f:A {}\mrightarrow{} (A + Top)
(SWellFounded(p-graph(A;f) y x)
{}\mRightarrow{} \mforall{}x,y:A.
\mexists{}n:\mBbbN{}. ((p-graph(A;f\^{}n) x y) \mvee{} (p-graph(A;f\^{}n) y x))
supposing final-iterate(f;x) = final-iterate(f;y)
supposing p-inject(A;A;f))
Date html generated:
2020_05_20-AM-08_09_14
Last ObjectModification:
2020_01_28-PM-04_51_04
Theory : general
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