Nuprl Lemma : sq-stable-coW-game-legal1
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w,w':coW(A;a.B[a])]. ∀p,q:Pos(coW-game(a.B[a];w;w')). SqStable(Legal1(p;q))
Proof
Definitions occuring in Statement :
coW-game: coW-game(a.B[a];w;w'),
coW: coW(A;a.B[a]),
sg-legal1: Legal1(x;y),
sg-pos: Pos(g),
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
coW-game: coW-game(a.B[a];w;w'),
sg-legal1: Legal1(x;y),
pi2: snd(t),
pi1: fst(t),
sg-pos: Pos(g),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
subtract: n - m,
top: Top,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False
Lemmas referenced :
squash_wf,
sg-legal1_wf,
coW-game_wf,
sg-pos_wf,
coW_wf,
decidable__int_equal,
copath-length_wf,
nat_wf,
sq_stable__copathAgree,
equal_wf,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
copathAgree_wf,
copath_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
instantiate,
cumulativity,
functionEquality,
universeEquality,
dependent_functionElimination,
setElimination,
rename,
addEquality,
because_Cache,
natural_numberEquality,
unionElimination,
imageElimination,
inlFormation,
independent_pairFormation,
independent_functionElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
intEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
productEquality,
inrFormation
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w,w':coW(A;a.B[a])].
\mforall{}p,q:Pos(coW-game(a.B[a];w;w')). SqStable(Legal1(p;q))
Date html generated:
2019_06_20-PM-00_57_46
Last ObjectModification:
2019_01_02-PM-01_34_27
Theory : co-recursion-2
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