Nuprl Lemma : retraction-fun-path-before
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀L:T List. ∀x,y,a,b:T. a before b ∈ L ⇒ a = f+(b) supposing x=f*(y) via L))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
strict-fun-connected: y = f+(x),
fun-path: y=f*(x) via L,
l_before: x before y ∈ l,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
fun-path: y=f*(x) via L,
and: P ∧ Q,
not: ¬A,
false: False,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
strict-fun-connected: y = f+(x),
uiff: uiff(P;Q)
Lemmas referenced :
member-less_than,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
subtract_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
int_seg_wf,
fun-path-before,
l_before_wf,
fun-path_wf,
list_wf,
retraction_wf,
no_repeats_iff,
fun-path-no_repeats
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
axiomEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
addEquality,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
(retraction(T;f)
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}x,y,a,b:T. a before b \mmember{} L {}\mRightarrow{} a = f+(b) supposing x=f*(y) via L))
Date html generated:
2018_05_21-PM-07_47_40
Last ObjectModification:
2017_07_26-PM-05_25_36
Theory : general
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