Nuprl Lemma : coW-pos-agree_transitivity
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w,w':coW(A;a.B[a])].
∀p,q,r:Pos(coW-game(a.B[a];w;w')).
(coW-pos-agree(a.B[a];w;w';p;q) ⇒ coW-pos-agree(a.B[a];w;w';q;r) ⇒ coW-pos-agree(a.B[a];w;w';p;r))
Proof
Definitions occuring in Statement :
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
sg-pos: Pos(g),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
sq_stable: SqStable(P),
member: t ∈ T,
le: A ≤ B,
cand: A c∧ B,
and: P ∧ Q,
pi1: fst(t),
sg-pos: Pos(g),
coW-game: coW-game(a.B[a];w;w'),
coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
copathAgree_transitivity,
coW_wf,
coW-game_wf,
sg-pos_wf,
coW-pos-agree_wf,
le_transitivity,
sq_stable__le
Rules used in proof :
dependent_functionElimination,
universeEquality,
functionEquality,
cumulativity,
instantiate,
applyEquality,
lambdaEquality,
independent_pairFormation,
imageElimination,
baseClosed,
hypothesisEquality,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
cut,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w,w':coW(A;a.B[a])].
\mforall{}p,q,r:Pos(coW-game(a.B[a];w;w')).
(coW-pos-agree(a.B[a];w;w';p;q)
{}\mRightarrow{} coW-pos-agree(a.B[a];w;w';q;r)
{}\mRightarrow{} coW-pos-agree(a.B[a];w;w';p;r))
Date html generated:
2018_07_25-PM-01_43_25
Last ObjectModification:
2018_06_20-PM-02_47_33
Theory : co-recursion
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