Nuprl Lemma : retraction-fun-path-squash
∀[T:Type]
∀f:T ⟶ T. ∀h:T ⟶ ℕ.
((∀x:T. (↓((f x) = x ∈ T) ∨ h (f x) < h x))
⇒ (∀L:T List. ∀x,y:T. ↓(x = y ∈ T) ∨ h y < h x supposing y=f*(x) via L))
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
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],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
squash: ↓T,
nat: ℕ,
or: P ∨ Q,
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,
less_than': less_than'(a;b),
false: False,
uiff: uiff(P;Q),
guard: {T},
decidable: Dec(P),
not: ¬A,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list_induction,
all_wf,
isect_wf,
fun-path_wf,
squash_wf,
or_wf,
equal_wf,
less_than_wf,
list_wf,
nil_wf,
cons_wf,
nat_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
fun-path-cons,
decidable__lt,
length_wf,
iff_weakening_equal,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
rename,
dependent_functionElimination,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
universeEquality,
independent_isectElimination,
voidElimination,
voidEquality,
productElimination,
natural_numberEquality,
unionElimination,
inlFormation,
hyp_replacement,
Error :applyLambdaEquality,
inrFormation,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T. \mforall{}h:T {}\mrightarrow{} \mBbbN{}.
((\mforall{}x:T. (\mdownarrow{}((f x) = x) \mvee{} h (f x) < h x))
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}x,y:T. \mdownarrow{}(x = y) \mvee{} h y < h x supposing y=f*(x) via L))
Date html generated:
2016_10_25-AM-11_03_48
Last ObjectModification:
2016_07_12-AM-07_11_02
Theory : general
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