Nuprl Lemma : AF-induction4
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
∀Q:T ⟶ ℙ. 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],
TI: 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 :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
not: ¬A,
false: False,
squash: ↓T,
true: True
Lemmas referenced :
rel_plus-TI,
AF-induction3,
rel_plus_wf,
rel_plus_trans,
exists_wf,
almost-full_wf,
all_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
independent_functionElimination,
isectElimination,
because_Cache,
hypothesis,
universeEquality,
independent_isectElimination,
functionEquality,
instantiate,
productEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation,
voidElimination,
addLevel,
hyp_replacement,
equalitySymmetry,
levelHypothesis,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
\mforall{}Q:T {}\mrightarrow{} \mBbbP{}. 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:
2016_10_21-AM-10_50_09
Last ObjectModification:
2016_07_12-AM-05_54_30
Theory : relations2
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