Nuprl Lemma : retract-compose_wf
∀[T,A:Type]. ∀[f:T ⟶ A]. ∀[h:Base].
(retract-compose(h;f) ∈ {g:T ⟶ A| g = f ∈ (T ⟶ A)} ) supposing (((T ⊆r Base) ∧ (A ⊆r Base) ∧ retract(T;h)) and value\000C-type(T))
Proof
Definitions occuring in Statement :
retract-compose: retract-compose(h;f),
retract: retract(T;f),
value-type: value-type(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
retract-compose: retract-compose(h;f),
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
has-value: (a)↓,
retract: retract(T;f)
Lemmas referenced :
equal_wf,
subtype_rel_wf,
base_wf,
retract_wf,
value-type_wf,
subtype_base_sq,
value-type-has-value
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalRule,
axiomEquality,
productEquality,
independent_isectElimination,
isect_memberEquality,
because_Cache,
universeEquality,
instantiate,
dependent_functionElimination,
independent_functionElimination,
callbyvalueReduce
Latex:
\mforall{}[T,A:Type]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[h:Base].
(retract-compose(h;f) \mmember{} \{g:T {}\mrightarrow{} A| g = f\} ) supposing (((T \msubseteq{}r Base) \mwedge{} (A \msubseteq{}r Base) \mwedge{} retract(T;h)) \000Cand value-type(T))
Date html generated:
2017_04_17-AM-09_15_47
Last ObjectModification:
2017_02_27-PM-05_21_13
Theory : decidable!equality
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