Nuprl Lemma : decidable__fun-connected
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) ⇒ (∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀x,y:T. Dec(x is f*(y))))
Proof
Definitions occuring in Statement :
retraction: retraction(T;f),
fun-connected: y is f*(x),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
retraction: retraction(T;f),
exists: ∃x:A. B[x],
nat: ℕ,
guard: {T},
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q)
Lemmas referenced :
all_wf,
decidable_wf,
equal_wf,
retraction_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
less_than_wf,
nat_wf,
subtract_wf,
fun-connected_wf,
set_wf,
primrec-wf2,
fun-connected_weakening_eq,
not_wf,
fun-connected-fixedpoint,
decidable__lt,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
fun-connected-step,
fun-connected_transitivity,
fun-connected-step-back,
add_nat_wf,
false_wf,
le_wf,
decidable__le,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionExtensionality,
applyEquality,
functionEquality,
universeEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
unionElimination,
inlFormation,
inrFormation,
independent_functionElimination,
imageElimination,
dependent_set_memberEquality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. (retraction(T;f) {}\mRightarrow{} (\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}x,y:T. Dec(x is f*(y))))
Date html generated:
2018_05_21-PM-07_48_21
Last ObjectModification:
2017_07_26-PM-05_26_08
Theory : general
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