### 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

Home Index