hcju/isabelleps huggingface dataset:
independent_pdevs_pairwise_non_coll: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and y :: "real \<times> real" assumes "x \<in> set (independent_pdevs xs)"   and "y \<in> set (independent_pdevs xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> 0"   and "x \<noteq> y" shows "det3 0 x y \<noteq> 0"
M_in_subprob: shows "M1 \<in> space (subprob_algebra M1)"
CONSTRAINT_I: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "P x" shows "CONSTRAINT P x"
c1: assumes "c \<le> 0" shows "\<exists>w. \<forall>y<w. b * y + c < y\<^sup>2"
Abs_ge_self: fixes x :: "ant" shows "x \<le> Abs x"
UP_is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>) (\<star>) \<a> \<i> src trg UP cmp\<^sub>U\<^sub>P"
blocks_Nil: fixes e :: "expr" shows "blocks ([], [], [], e) = e"
sum_closure: fixes a :: "i"   and b :: "i" assumes "GM" shows "\<exists>z. \<forall>w. O w z = (O w a \<or> O w b)"
PTrans: shows "P \<longmapsto> \<alpha> \<prec> P'"
bot_ow_def: fixes U :: "'a set"   and bot :: "'a" shows "bot_ow U bot \<equiv> bot \<in> U"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_inv_4: fixes l :: "'d strand_label"   and a :: "('e, 'f) stateful_strand_step"   and k :: "'d strand_label"   and b :: "('e, 'f) stateful_strand_step" assumes "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, a) = (k, b)"   and "\<nexists>t. a = receive\<langle>t\<rangle> \<or> a = send\<langle>t\<rangle>" shows "b = a"
assms: shows "\<phi> i x\<^sub>0 \<down>"
Og_def: shows "Og = Ogood \<inter> {x \<in> space M. \<forall>n\<in>{1..Nf 1}. \<forall>l\<in>{1..n}. - (real K * real l) < u n x - u (n - l) x}"
Characters_def: fixes G :: "('c, 'd) monoid_scheme" shows "Characters G = \<lparr>carrier = characters G, mult = \<lambda>\<chi>\<^sub>1 \<chi>\<^sub>2 k. \<chi>\<^sub>1 k * \<chi>\<^sub>2 k, one = principal_char G\<rparr>"
adv_P1_def: fixes R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "OT_14.adv_P1 R1_12_0T n m1 m2 D \<equiv> \<bar>spmf (R1_14 R1_12_0T n m1 m2 \<bind> D) True - spmf (funct_OT_14 m1 m2 \<bind> (\<lambda>(o1, o2). S1_14 n m1 o1 \<bind> D)) True\<bar>"
IS_LEFT_UNIQUE_def: fixes R :: "('b \<times> 'd) set" shows "IS_LEFT_UNIQUE R \<equiv> single_valued (R\<inverse>)"
principal_pow_prime_condTr: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "prime_ideal R (R \<diamondsuit>\<^sub>p a \<^bsup>\<diamondsuit>R Suc n\<^esup>)" shows "n = 0"
unord: shows "Atom k \<^bold>\<or> disj_of_clause (removeAll (k\<^sup>+) F), \<^bold>\<not> (Atom k) \<^bold>\<or> disj_of_clause (removeAll (k\<inverse>) G), \<Gamma> \<Rightarrow>\<^sub>n"
ex_assn_move_out_1: fixes Q :: "'a \<Rightarrow> assn"   and R :: "assn" shows "(\<exists>\<^sub>Ax. Q x) * R = (\<exists>\<^sub>Ax. Q x * R)"
monad_altc_writerT': assumes "monad_altc Monad_Overloading.return Monad_Overloading.bind altc" shows "monad_altc Monad_Overloading.return Monad_Overloading.bind altc"
minus_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'b" assumes "bounded_bilinear prod" shows "prod (- a) b = - prod a b"
drop_rev: fixes n :: "nat"   and xs :: "'a list" shows "drop n (rev xs) = rev (take (length xs - n) xs)"
insert_absorb: fixes a :: "'e"   and A :: "'e set" assumes "a \<in> A" shows "insert a A = A"
PP_PPf: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (HBL1.PP var subst P (enc \<phi>)) = bprv (exi (Generic_Syntax.Variable var 11) (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf (Var (Generic_Syntax.Variable var 11)) (enc \<phi>)))"
inj_on_f_set_l: shows "inj_on (\<lambda>x. Operator (fst x) (index (\<Pi>\<^sub>\<O>) (snd x))) (set (concat (map (\<lambda>k. map (Pair k) (\<pi> ! k)) [0..<length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1])))"
mat_unitary_mat: fixes M :: "complex mat" assumes "unitary M" shows "inverts_mat M M\<^sup>\<dagger>"
cinner_commute': fixes y :: "'a"   and x :: "'a" shows "cnj (y \<bullet>\<^sub>C x) = x \<bullet>\<^sub>C y"
sinks_aux_single_dom: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and u :: "'a"   and xs :: "'b list" shows "sinks_aux I D {u} xs = insert u (sinks I D u xs)"
assms_2: shows "\<zeta> \<notin> path_image p"
hs: shows "distinct [Xh d1h1, Xh d1h2, Xh d1h3]"
inc_Arity_def: shows "inc_Arity \<equiv> map_fun Rep_Arity to_Arity Suc"
option_while'_inj_step: fixes C :: "'d \<Rightarrow> bool"   and s :: "'d"   and B :: "'d \<Rightarrow> 'd option"   and s' :: "'d"   and t :: "'d option"   and t' :: "'d option" assumes "C s"   and "B s = Some s'"   and "(Some s, t) \<in> option_while' C B"   and "(Some s', t') \<in> option_while' C B" shows "t = t'"
add_minus_zero_rel: fixes a :: "('a \<times> 'a) set" assumes "cq.valid_frac a" shows "cq.add_rel a (cq.uminus_rel a) = cq.zero_rel"
shiftl_mask_is_0: fixes x :: "'b word"   and n :: "nat" shows "(x << n) AND mask n = 0"
bprv_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>" shows "prv \<phi>"
VV_impl1: fixes v :: "'a"   and p :: "Player" assumes "v \<in> G'.VV p" shows "v \<notin> G'.VV p**"
Block_self: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> Partition" shows "Block a = A"
assms_1: shows "g \<in> R \<rightarrow>\<^sub>M stock_measure t"
assms_1: shows "lower_on \<sigma> \<eta> (vars_of_cl (cl_ecl C))"
summable_0_powser: fixes f :: "nat \<Rightarrow> 'a" shows "summable (\<lambda>n. f n * (0::'a) ^ n)"
assms_3: shows "split_i_j_x i j x = (a, b)"
bexp_vars_pred_False: shows "bexp_vars pred_False = {}"
new_tv_Cons: fixes n :: "nat"   and x :: "'a"   and l :: "'a list" shows "new_tv n (x # l) = (new_tv n x \<and> new_tv n l)"
continuous_map_eq_image_closure_subset_alt: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (\<forall>S\<subseteq>topspace X. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
sp_append_sub: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v3 :: "'d \<times> nat" assumes "SubRel.subpath g v1 es v2 subs"   and "(v2, v3) \<in> subs" shows "SubRel.subpath g v1 es v3 subs"
partn_lst_VWF_nontriv: fixes \<beta> :: "V"   and \<alpha> :: "V list"   and n :: "nat"   and l :: "nat" assumes "partn_lst_VWF \<beta> \<alpha> n"   and "l = length \<alpha>"   and "Ord \<beta>"   and "0 < l" obtains i :: "nat" where "i < l" and "\<alpha> ! i \<le> \<beta>"
infiter_unfold: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> = c ; c\<^sup>\<infinity>"
resCap_cf_def: fixes cf :: "'b \<Rightarrow> 'a"   and p :: "'b list" shows "resCap_cf cf p \<equiv> Min {cf e |e. e \<in> set p}"
trans_equiv_less: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b = Equiv"   and "compare cmp b c = Less" shows "compare cmp a c = Less"
xvecFreshP: shows "(p \<bullet> xvec) \<sharp>* P"
rep_eq: fixes x :: "'a pmf" shows "pmf x = (\<lambda>x. measure_pmf.prob x {x})"
pt_name2: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'a" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
lprefix_lhdD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lprefix xs ys"   and "\<not> lnull xs" shows "lhd xs = lhd ys"
alg_dvd_0_right: fixes x :: "'a" shows "x alg_dvd (0::'a)"
execd_preserve_deadlocked: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "preserve_deadlocked JVM_final (mexecd P) convert_RA (correct_jvm_state \<Phi>)"
add_col_sub_row_def: fixes a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "add_col_sub_row a k l A = addrow (- a) k l (addcol a l k A)"
lin_subspace: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "Subspace.subspace (lin x) V"
add_eq_0_iff2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (a = uminus b)"
vimage_UNIV: fixes f :: "'h \<Rightarrow> 'i" shows "f -` UNIV = UNIV"
forward_terminating_path_end_points_3: fixes x :: "'a" assumes "forward_terminating_path x" shows "start_points x \<le> x\<^sup>+ ; end_points x"
dagger_ext: fixes x :: "'a" shows "x \<le> x\<^sup>\<dagger>"
tabulate_aux1: fixes cnfi :: "(lit \<times> lit) list"   and M :: "(lit, lit list) mapping" shows "E_of_succ (\<lambda>v. set (\<alpha>sl (fold ins_edges cnfi M) v)) = imp_graph (cnf_\<alpha> cnfi) \<union> E_of_succ ((set \<circ>\<circ> \<alpha>sl) M)"
gHom_def: fixes G :: "('c, 'e) Group_scheme"   and F :: "('d, 'f) Group_scheme" shows "gHom G F = {f. (f \<in> extensional (carrier G) \<and> f \<in> carrier G \<rightarrow> carrier F) \<and> (\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. f (x \<cdot>\<^bsub>G\<^esub> y) = f x \<cdot>\<^bsub>F\<^esub> f y)}"
add_diff_cancel_right': fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and a :: "'f"   and b :: "'f" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
formulaTrueAppend: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list"   and valuation :: "Literal list" shows "formulaTrue (formula1 @ formula2) valuation = (formulaTrue formula1 valuation \<and> formulaTrue formula2 valuation)"
covering_space_locally_compact_eq: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "covering_space C p S" shows "locally compact S = locally compact C"
prime_imp_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p n" shows "algebraic_semidom.coprime times one p n"
def: shows "usubstr \<sigma> (A @ B, C) \<noteq> None"
no_overlaps_lroft_s3_hlp: fixes amr :: "(16 word \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and ifs :: "char list list" assumes "distinct (map fst amr)"   and "distinct ifs" shows "no_overlaps OF_match_fields_unsafe (map (split3 OFEntry) (lr_of_tran_s3 ifs amr))"
real_vector_rpr_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" shows "real_vector_rpr carrier relation \<equiv> rational_preference carrier relation"
rij_def: shows "rij = Cn 1 r [r_const i, r_const j, recf.Id 1 0]"
homotopic_loops_imp_subset: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_loops s p q" shows "path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
eventually_at_left_field: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "eventually P (at_left x) = (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
landau_product_preprocess_53: fixes a :: "'a" shows "a + (0::'a) = a"
finite_char_def: fixes C :: "'a set set" shows "finite_char C \<equiv> \<forall>S. (S \<in> C) = (\<forall>S'. finite S' \<longrightarrow> S' \<subseteq> S \<longrightarrow> S' \<in> C)"
mrConcEnd: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x (b # p) = \<lfloor>a\<rfloor>"   and "a \<noteq> b" shows "applied_rule_rev Cp x p = \<lfloor>a\<rfloor>"
wf_imp_irrefl: fixes r :: "('c \<times> 'c) set" assumes "wf r" shows "irrefl r"
eq_infE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.eq_inf ind w\<^sub>1 w\<^sub>2" obtains "traces.le_inf ind w\<^sub>1 w\<^sub>2" and "traces.le_inf ind w\<^sub>2 w\<^sub>1"
DERIV_deriv_iff_field_differentiable: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "(f has_field_derivative deriv f x) (at x) = (f field_differentiable at x)"
indep_vars_iff_distr_eq_PiM: fixes I :: "'b set"   and X :: "'b \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> 'c"   and M' :: "'b \<Rightarrow> 'c measure" assumes "I \<noteq> {}"   and "\<And>i. M'.random_variable (M' i) (X i)" shows "M'.indep_vars M' X I = (distr M'__ (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr M'__ (M' i) (X i)))"
progress_time_conv: fixes j :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<sigma>' :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula" assumes "\<forall>i<j. \<tau> \<sigma> i = \<tau> \<sigma>' i" shows "Monitor.progress \<sigma> P \<phi> j = Monitor.progress \<sigma>' P \<phi> j"
countable_lfp: fixes F :: "'d set \<Rightarrow> 'd set" assumes "\<And>Y. countable Y \<Longrightarrow> countable (F Y)"   and "sup_continuous F" shows "countable (lfp F)"
domIff: fixes a :: "'a"   and m :: "'a \<Rightarrow> 'b option" shows "(a \<in> dom m) = (m a \<noteq> \<bottom>)"
add_right_cancel: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "(b + a = c + a) = (b = c)"
\<section>: shows "continuous_on {0..1} (\<lambda>t. if t \<le> 1 / 2 then 2 *\<^sub>R t else 1)"
e_type_local: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and n :: "nat"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'" shows "\<exists>tls. i < length (s_inst \<S>) \<and> length tls = n \<and> \<S>\<bullet>(s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ map typeof vs, return := Some tls\<rparr> \<turnstile> es : [] _> tls \<and> ts' = ts @ tls"
cr_ACI_rexp_def: shows "cr_ACI_rexp \<equiv> \<lambda>x. (=) (ACI_class x)"
axioms_1: fixes G :: "('e, 'f) monoid_scheme" assumes "comm_group G" shows "Group.comm_monoid G"
listset_length: fixes xs :: "'b list"   and Xs :: "'b set list" assumes "xs \<in> listset Xs" shows "length xs = length Xs"
replace3: fixes f' :: "'c"   and ls :: "'c list"   and f :: "'c"   and fs :: "'c list" assumes "f' \<in> set ls"   and "f \<in> set fs" shows "f \<in> set (replace f' fs ls)"
nextElem_last: fixes xs :: "'a list"   and c :: "'a" assumes "distinct xs" shows "nextElem xs c (last xs) = c"
residuated_inf: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "residual f (x \<sqinter> y) = residual f x \<sqinter> residual f y"
upper_asymptotic_density_zero_Union: fixes A :: "nat \<Rightarrow> nat set" assumes "\<And>n. upper_asymptotic_density (A n) = 0" shows "\<exists>B. upper_asymptotic_density B = 0 \<and> (\<forall>n. \<exists>N. A n \<inter> {N..} \<subseteq> B)"
dim_zero_eq: fixes A :: "('b, 'c) vec set" assumes "vec.dim A = 0" shows "A = {} \<or> A = {0}"
cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa \<Colon> ys"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Ok x2 = Ok y2) = (x2 = y2)"
FCIA1: shows "FCIA \<rho>1 \<Gamma>1 \<V>1 Tr\<^bsub>ES1\<^esub>"
numeral_powr_numeral_real: fixes m :: "num"   and n :: "num" shows "numeral m powr numeral n = numeral m ^ numeral n"
degree0_coeffs: fixes p :: "'c poly" assumes "degree p = 0" shows "\<exists>a. p = [:a:]"
un_def: shows "un \<equiv> hom_induced (int n) (usphere n) (equator n) (nsphere n) (equator n) id zn"
ts_bool_True_False: fixes x :: "nat \<Rightarrow> bool list"   and i :: "nat" assumes "ts x" shows "x i = [True] \<or> x i = [False]"
ds_eq: shows "{d \<in> ds \<union> {0..<dm}. d < length (start dm)} = {0..<dm}"
assms_1: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "P c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "P c1"
eventually_mono: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "eventually P F"   and "\<And>x. P x \<Longrightarrow> Q x" shows "eventually Q F"
asubg_test: fixes A :: "('d, 'e) aGroup_scheme"   and H :: "'d set" assumes "aGroup A"   and "H \<subseteq> carrier A"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b \<in> H" shows "A +> H"
run_fail_state: fixes fail :: "'d"   and s :: "'e" shows "run_state (fail_state fail) s = fail"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "U.SAT \<Phi> \<equiv> \<forall>\<xi>. U.wtE \<xi> \<longrightarrow> U.satPB \<xi> \<Phi>"
ChamberComplexMorphism_axioms: shows "ChamberComplexMorphism X X g"
xor_def: fixes x :: "'b"   and y :: "'b" shows "x XOR y = x AND NOT y OR NOT x AND y"
prems_1: shows "ps \<in> snd ` set (e # pth)"
Lxy_in: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "Lxy [x] S = [x]"
intros: fixes p :: "'a \<times> 'd" shows "fst p \<in> Basic_BNFs.fsts p"
IH: shows "(\<Prod>x\<in>A. (1::'a) + (1::'a) / x) = (\<Sum>x\<in>Pow A. (1::'a) / \<Prod>x)"
rel_G_pos_distr_cond_eq: fixes tytok :: "('g \<times> 'h \<times> 'i \<times> 'j \<times> 'k \<times> 'l \<times> 'm) itself" shows "rel_G_pos_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
contf': shows "continuous_on (S \<union> \<Union> G) f"
rel_bset_aux_infinite: fixes a :: "'c set['d]"   and b :: "'e set['d]"   and R :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "((\<forall>t\<in>set_bset a. \<exists>u\<in>set_bset b. R t u) \<and> (\<forall>u\<in>set_bset b. \<exists>t\<in>set_bset a. R t u)) = ((BNF_Def.Grp {a. set_bset a \<subseteq> {(a, b). R a b}} (map_bset fst))\<inverse>\<inverse> OO BNF_Def.Grp {a. set_bset a \<subseteq> {(a, b). R a b}} (map_bset snd)) a b"
msg_nStore: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and i :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "inf_disj n nSend"   and "i < n"   and "\<forall>i<n. msg (Suc 0) (nReturn i)"   and "Cable n nSend recv" shows "msg (Suc 0) (nStore i)"
tarjan_disc_aux: fixes x :: "\<lparr>sccs :: 'v set set, lowlink :: 'v \<Rightarrow> nat option, tj_stack :: 'v list\<rparr>"   and s' :: "'v tarjan_state"   and v :: "'v"   and s :: "'v tarjan_state"   and w :: "'v" assumes "\<And>w. \<lbrakk>Tarjan_invar G (s'\<lparr>state.more := x\<rparr>); w \<in> dom (discovered (s'\<lparr>state.more := x\<rparr>)); w \<noteq> v\<rbrakk> \<Longrightarrow> lowlink_set (s'\<lparr>state.more := x\<rparr>) w \<subseteq> lowlink_set s w \<union> {v}"   and "tree_edges s' = tree_edges s \<or> (\<exists>u. u \<noteq> v \<and> tree_edges s' = tree_edges s \<union> {(u, v)})"   and "Tarjan_invar G s"   and "discovered s' = discovered s(v \<mapsto> counter s)"   and "finished s' = finished s"   and "lowlink s' = lowlink s"   and "cross_edges s' = cross_edges s"   and "back_edges s' = back_edges s"   and "v \<notin> dom (discovered s)"   and "\<And>w. w \<in> dom (discovered s) \<Longrightarrow> \<zeta> s w = Min (\<delta> s ` lowlink_set s w)"   and "\<And>\<Psi>. tarjan_disc v s' \<le>\<^sub>n SPEC \<Psi> \<Longrightarrow> \<Psi> x"   and "DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>)"   and "w \<in> dom (discovered (s'\<lparr>state.more := x\<rparr>))" shows "\<zeta> (s'\<lparr>state.more := x\<rparr>) w = Min (\<delta> (s'\<lparr>state.more := x\<rparr>) ` lowlink_set (s'\<lparr>state.more := x\<rparr>) w)"
G_soundness: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "ModelIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl intT intF intP" shows "CM.Model (ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl) (ProblemIkTpartG.GE_wtPsym wtPsym prot protFw) (ProblemIkTpartG.GE_arOf arOf) (ProblemIkTpartG.GE_resOf resOf) (ProblemIkTpartG.GE_parOf parOf) (ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> prot protFw protCl) intT (ModelIkTpartG.GE_intF intT intF) (ModelIkTpartG.GE_intP intP)"
S2safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S2 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
simps_5: fixes dims :: "nat list"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com" shows "state_sig.well_com dims (While M S) = (measurement (state_sig.d dims) 2 M \<and> state_sig.well_com dims S)"
the_eqI: fixes P :: "'a Predicate.pred"   and x :: "'a" assumes "(THE x. pred.eval P x) = x" shows "Predicate.the P = x"
coeff_reflect_poly: fixes p :: "'a poly"   and n :: "nat" shows "coeff (reflect_poly p) n = (if degree p < n then 0::'a else coeff p (degree p - n))"