Nuprl Lemma : AF-uniform-induction4-ext
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
∀Q:T ⟶ ℙ. uniform-TI(T;x,y.R[x;y];t.Q[t])
supposing ∃R':T ⟶ T ⟶ ℙ. (AFx,y:T.R'[x;y] ∧ (∀x,y:T. (R+[x;y] ⇒ (¬R'[x;y]))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
almost-full: AFx,y:T.R[x; y],
uniform-TI: uniform-TI(T;x,y.R[x; y];t.Q[t]),
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
AF-uniform-induction4,
rel_plus-uniform-TI,
AF-uniform-induction3,
AF-uniform-induction2,
AF-uniform-induction
Lemmas referenced :
AF-uniform-induction4,
rel_plus-uniform-TI,
AF-uniform-induction3,
AF-uniform-induction2,
AF-uniform-induction
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. uniform-TI(T;x,y.R[x;y];t.Q[t])
supposing \mexists{}R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (AFx,y:T.R'[x;y] \mwedge{} (\mforall{}x,y:T. (R\msupplus{}[x;y] {}\mRightarrow{} (\mneg{}R'[x;y]))))
Date html generated:
2018_05_21-PM-00_52_15
Last ObjectModification:
2018_05_19-AM-06_40_45
Theory : relations2
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