Nuprl Lemma : rel_plus-TI
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ((∀Q:T ⟶ ℙ. TI(T;x,y.R+ x y;x.Q[x])) ⇒ (∀Q:T ⟶ ℙ. TI(T;x,y.R[x;y];x.Q[x])))
Proof
Definitions occuring in Statement :
rel_plus: R+,
TI: 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,
TI: TI(T;x,y.R[x; y];t.Q[t]),
so_apply: x[s],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
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,
all_wf,
rel_plus_wf,
TI_wf,
rel-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
lemma_by_obid,
isectElimination,
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{}. TI(T;x,y.R\msupplus{} x y;x.Q[x])) {}\mRightarrow{} (\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. TI(T;x,y.R[x;y];x.Q[x])))
Date html generated:
2016_05_14-PM-03_54_08
Last ObjectModification:
2015_12_26-PM-06_57_22
Theory : relations2
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