Nuprl Lemma : rel_plus-uniform-TI
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ((∀Q:T ⟶ ℙ. uniform-TI(T;x,y.R+ x y;x.Q[x]))
⇒ (∀Q:T ⟶ ℙ. uniform-TI(T;x,y.R[x;y];x.Q[x])))
Proof
Definitions occuring in Statement :
rel_plus: R+
,
uniform-TI: uniform-TI(T;x,y.R[x; y];t.Q[t])
,
prop: ℙ
,
so_apply: x[s1;s2]
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
apply: f a
,
function: x:A ⟶ B[x]
,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
uniform-TI: uniform-TI(T;x,y.R[x; y];t.Q[t])
,
uall: ∀[x:A]. B[x]
,
so_apply: x[s]
,
member: t ∈ T
,
prop: ℙ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s1;s2]
,
subtype_rel: A ⊆r B
,
so_lambda: λ2x y.t[x; y]
,
infix_ap: x f y
Lemmas referenced :
set_wf,
uall_wf,
rel_plus_wf,
all_wf,
uniform-TI_wf,
rel-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
lemma_by_obid,
lambdaEquality,
applyEquality,
universeEquality,
setEquality,
setElimination,
rename,
functionEquality,
because_Cache,
cumulativity,
instantiate,
dependent_set_memberEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
((\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. uniform-TI(T;x,y.R\msupplus{} x y;x.Q[x])) {}\mRightarrow{} (\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. uniform-TI(T;x,y.R[x;y];x.Q[x])))
Date html generated:
2016_05_14-PM-03_54_10
Last ObjectModification:
2015_12_26-PM-06_57_26
Theory : relations2
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