Nuprl Lemma : sg-reachable_wf
∀[g:SimpleGame]. ∀[x,y:Pos(g)]. (sg-reachable(g;x;y) ∈ ℙ)
Proof
Definitions occuring in Statement :
sg-reachable: sg-reachable(g;x;y),
sg-pos: Pos(g),
simple-game: SimpleGame,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T
Definitions unfolded in proof :
less_than: a < b,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
so_apply: x[s],
squash: ↓T,
sq_stable: SqStable(P),
guard: {T},
nat: ℕ,
true: True,
top: Top,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
sg-reachable: sg-reachable(g;x;y),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_properties,
nat_plus_properties,
omega-shadow,
minus-zero,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
simple-game_wf,
nat_plus_subtype_nat,
le-add-cancel2,
int_subtype_base,
set_subtype_base,
le_weakening2,
sg-legal2_wf,
nat_plus_wf,
sq_stable__le,
multiply_nat_wf,
add_nat_wf,
mul-associates,
le_wf,
mul_bounds_1a,
sg-legal1_wf,
squash_wf,
nat_wf,
all_wf,
le-add-cancel-alt,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
lelt_wf,
false_wf,
seq-item_wf,
equal_wf,
seq-len_wf,
less_than_wf,
sg-pos_wf,
sequence_wf,
exists_wf
Rules used in proof :
axiomEquality,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
imageElimination,
baseClosed,
imageMemberEquality,
rename,
setElimination,
multiplyEquality,
functionEquality,
intEquality,
minusEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
lambdaFormation,
independent_pairFormation,
dependent_set_memberEquality,
applyEquality,
because_Cache,
natural_numberEquality,
productEquality,
lambdaEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[g:SimpleGame]. \mforall{}[x,y:Pos(g)]. (sg-reachable(g;x;y) \mmember{} \mBbbP{})
Date html generated:
2018_07_25-PM-01_33_45
Last ObjectModification:
2018_06_18-PM-05_03_16
Theory : co-recursion
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