Nuprl Lemma : transitive-reflexive-closure-map
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].
∀f:A ⟶ A. ((∀x,y:A. ((R x y) ⇒ (R (f x) (f y)))) ⇒ (∀x,y:A. ((R^* x y) ⇒ (R^* (f x) (f y)))))
Proof
Definitions occuring in Statement :
transitive-reflexive-closure: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
transitive-reflexive-closure: R^*,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
and_wf,
equal_wf,
transitive-closure_wf,
transitive-reflexive-closure_wf,
all_wf,
transitive-closure-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
inlFormation,
cut,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
independent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
applyLambdaEquality,
setElimination,
rename,
productElimination,
applyEquality,
cumulativity,
functionExtensionality,
inrFormation,
lambdaEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}f:A {}\mrightarrow{} A
((\mforall{}x,y:A. ((R x y) {}\mRightarrow{} (R (f x) (f y)))) {}\mRightarrow{} (\mforall{}x,y:A. ((R\^{}* x y) {}\mRightarrow{} (R\^{}* (f x) (f y)))))
Date html generated:
2017_01_19-PM-02_17_49
Last ObjectModification:
2017_01_14-PM-06_52_49
Theory : relations2
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