Nuprl Lemma : rel_plus-weak-TI
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. (∀Q:T ⟶ ℙ. weak-TI(T;x,y.R[x;y];x.Q[x]) ⇐⇒ ∀Q:T ⟶ ℙ. weak-TI(T;x,y.R+ x y;x.Q[x]))
Proof
Definitions occuring in Statement :
rel_plus: R+,
weak-TI: weak-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],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
weak-TI: weak-TI(T;x,y.R[x; y];t.Q[t]),
cand: A c∧ B,
guard: {T},
infix_ap: x f y,
exists: ∃x:A. B[x],
or: P ∨ Q
Lemmas referenced :
all_wf,
weak-TI_wf,
rel_plus_wf,
rel_plus_iff2,
rel-star-iff-rel-plus,
and_wf,
equal_wf,
rel-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
dependent_functionElimination,
productEquality,
independent_functionElimination,
productElimination,
unionElimination,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. weak-TI(T;x,y.R[x;y];x.Q[x]) \mLeftarrow{}{}\mRightarrow{} \mforall{}Q:T {}\mrightarrow{} \mBbbP{}. weak-TI(T;x,y.R\msupplus{} x y;x.Q[x]))
Date html generated:
2016_10_21-AM-10_50_06
Last ObjectModification:
2016_07_12-AM-05_54_17
Theory : relations2
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