Nuprl Lemma : strat2play-invariant
∀g:SimpleGame. ∀n:ℕ. ∀s:win2strat(g;n). ∀moves:strat2play(g;n;s). ∀i:ℕ(2 * n) + 1.
((((i mod 2) = 0 ∈ ℤ) ⇒ (↓Legal1(moves[i];moves[i + 1])))
∧ (((i mod 2) = 1 ∈ ℤ) ⇒ (↓Legal2(moves[i];moves[i + 1]))))
Proof
Definitions occuring in Statement :
strat2play: strat2play(g;n;s),
win2strat: win2strat(g;n),
play-item: moves[i],
sg-legal2: Legal2(x;y),
sg-legal1: Legal1(x;y),
simple-game: SimpleGame,
modulus: a mod n,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
strat2play-invariant-1,
int_seg_wf,
strat2play_wf,
win2strat_wf,
istype-nat,
simple-game_wf,
div_rem_sum,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
rem_bounds_1,
int_seg_subtype_nat,
istype-false,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
div_bounds_1,
int_seg_properties,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
istype-le,
modulus-is-rem,
set_subtype_base,
lelt_wf,
decidable__equal_int,
decidable__le
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
Error :universeIsType,
isectElimination,
natural_numberEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :equalityIstype,
baseClosed,
sqequalBase,
applyEquality,
sqequalRule,
independent_pairFormation,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
because_Cache,
divideEquality,
closedConclusion,
Error :inhabitedIsType,
imageElimination,
int_eqEquality,
Error :productIsType,
baseApply,
imageMemberEquality
Latex:
\mforall{}g:SimpleGame. \mforall{}n:\mBbbN{}. \mforall{}s:win2strat(g;n). \mforall{}moves:strat2play(g;n;s). \mforall{}i:\mBbbN{}(2 * n) + 1.
((((i mod 2) = 0) {}\mRightarrow{} (\mdownarrow{}Legal1(moves[i];moves[i + 1])))
\mwedge{} (((i mod 2) = 1) {}\mRightarrow{} (\mdownarrow{}Legal2(moves[i];moves[i + 1]))))
Date html generated:
2019_06_20-PM-00_52_25
Last ObjectModification:
2019_01_02-PM-03_30_36
Theory : co-recursion-2
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