hcju/isabelleps huggingface dataset:
steps_rel: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" assumes "(x, y) \<in> rel" shows "(x, y) \<in> steps rel n"
comp_def: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
distributed_distr_eq_density: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal" assumes "distributed M N X f" shows "distr M N X = density N f"
square_free_m_factor_1: fixes f :: "int poly"   and g :: "int poly" assumes "square_free_m (f * g)" shows "square_free_m f"
PolysD_2: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "p \<in> P[X]" shows "indets p \<subseteq> X"
diff_eq_diff_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less a b = less c d"
unifiers_occur_left_is_Fun: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" assumes "x \<in> vars_term t"   and "is_Fun t" shows "unifiers (insert (Var x, t) E) = {}"
recurrence_time_restricts_compact_flow: fixes T :: "real"   and x :: "'a" assumes "T \<in> existence_ivl0 x"   and "0 < T"   and "flow0 x T = x" shows "range (flow0 x) = flow0 x ` {0..T}"
wbalanced_balance_tree: fixes t :: "'b tree" shows "wbalanced (balance_tree t)"
strict_iff_order: fixes x :: "'o"   and a :: "'a"   and b :: "'a" shows "lt x a b = (le x a b \<and> a \<noteq> b)"
assms_3: shows "g \<noteq> \<zero>\<^bsub>P\<^esub>"
simps: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and r' :: "C" shows "(f|\<^bsub>r\<^esub>)\<cdot>r' = f\<cdot>(r \<sqinter> r')"
trss_mon_w_no_ctx: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "mon_w fg w \<inter> mon_c fg c = {}"
keys_monom_mult_subset: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "keys (local.punit.monom_mult c t p) \<subseteq> local.punit.splus t ` keys p"
incoming_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat" shows "Graph.incoming c v \<equiv> {(u, v) |u. (u, v) \<in> Graph.E c}"
rule0: fixes a :: "'c" shows "a \<equiv> a + (0::'c)"
is_lasso_pre_def: fixes L :: "'v lasso" shows "is_lasso_pre L \<equiv> lasso_v0 L \<in> V0 \<and> path E (lasso_v0 L) (lasso_reach L) (lasso_va L) \<and> path E (lasso_va L) (lasso_cycle L) (lasso_va L)"
Skip: fixes G :: "'b triple set"   and P :: "'b assn" shows "G|-{P}. SKIP .{P}"
one_closed: fixes R :: "'a set" assumes "Ring1 R" shows "(1::'a) \<in> R"
null_if_pos_func_has_zero_int: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and A :: "'b set" assumes "integrable M f"   and "A \<in> sets M"   and "AE x\<in>A in M. 0 < f x"   and "set_lebesgue_integral M A f = 0" shows "A \<in> null_sets M"
smult_assoc_simp: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x) = a \<otimes> b \<odot>\<^bsub>M\<^esub> x"
conv_bot: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv bot = bot"
lens_indep_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" shows "X \<bowtie> Y \<equiv> (\<forall>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v) \<and> (\<forall>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>) \<and> (\<forall>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>)"
Scoped_Graph_def: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option" shows "Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps \<equiv> Port_Graph nodes inPorts outPorts vertices nodeOf edges \<and> Port_Graph_Signature_Scoped inPorts outPorts hyps"
covector_inf_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "times top x = x"   and "times top y = y" shows "times top (inf x y) = inf x y"
balanced: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "int (card ps) = 2" shows "int (\<B> index ps) = \<Lambda>"
map_id: fixes t :: "'h + 'i" shows "map_sum id id t = t"
cinner_adj_left: fixes G :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and x :: "'b"   and y :: "'a" shows "(G* *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)"
sort_map: shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x) (map fst (alls B xs))"
hassoc_map_assn_Cons: fixes m :: "'a \<Rightarrow> 'b option"   and p :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" shows "hassoc_map_assn m p \<Longrightarrow>\<^sub>A hassoc_map_assn (m(k \<mapsto> v)) ((k, v) # p) * true"
inner_add_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (plus x y) z = inner x z + inner y z"
CatExpCatAx: fixes A :: "('g, 'h, 'i) Category_scheme"   and B :: "('j, 'k, 'l) Category_scheme" assumes "Category A"   and "Category B" shows "Category_axioms (CatExp' A B)"
alternating_list_SucSuc_ConsCons: fixes k :: "nat"   and s :: "'a"   and t :: "'a" shows "alternating_list (Suc (Suc k)) s t = s # t # alternating_list k s t"
inv_inv: fixes x :: "'c" assumes "x \<in> carrier H" shows "inv\<^bsub>H\<^esub> (inv\<^bsub>H\<^esub> x) = x"
apply_opers_idx_elems: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and xs :: "('a \<times> 'b \<times> bool) list" assumes "es prefix of i"   and "apply_operations es = Some xs" shows "element_ids xs = set (indices es)"
one_real_alg_def: shows "1 \<equiv> of_rat_real_alg 1"
zero_power2: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero 2 = zero"
assms_2: shows "\<forall>x\<in>set (xs @ ys). \<not> z \<prec> x"
one: fixes R :: "('d \<Rightarrow> real) set"   and S :: "'d set"   and U :: "'d set"   and t0 :: "'d"   and t1 :: "'d" assumes "function_ring_on R S"   and "open U"   and "t0 \<in> S"   and "t0 \<in> U"   and "t1 \<in> S - U" shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and> (\<forall>e>0. \<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>t\<in>S \<inter> V. f t < e) \<and> (\<forall>t\<in>S - U. 1 - e < f t))"
cart_prod_snd: fixes x :: "'e \<times> 'f"   and A :: "'e set"   and B :: "'f set" assumes "x \<in> A \<times> B" shows "snd x \<in> B"
iNextStrong_True: fixes t0 :: "nat"   and I :: "nat set" shows "(\<circle>\<^sub>S t t0 I. True) = (I \<down>> t0 \<noteq> {} \<and> t0 \<in> I)"
uminus_numeral_One: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "uminus (numeral.numeral one plus num.One) = uminus one"
that: fixes g :: "'a \<Rightarrow> 'b"   and k :: "nat \<Rightarrow> nat" assumes "continuous_on S g"   and "strict_mono k"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. N \<le> n \<and> x \<in> S \<longrightarrow> norm (\<F> (k n) x - g x) < e" shows "thesis"
stable_no_blocking_on_def: fixes ds :: "'d set"   and X :: "'x set" shows "stable_no_blocking_on ds X = (\<forall>h X'. \<not> blocking_on ds X h X')"
B23: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x \<le> y"   and "y \<cdot> y = y" shows "x\<^sup>\<star> \<le> y"
cases: fixes x :: "('b \<times> 'c) tree" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> 'c) tree" and a :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (a, uu_), r\<rangle>"
inv\<^sub>1E_1: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P\<^sub>1 :: "'b set set"   and P\<^sub>2 :: "'b set set"   and B\<^sub>1 :: "'b set"   and B\<^sub>2 :: "'b set"   and V :: "'b set" assumes "BinPacking U w c S L"   and "BinPacking.inv\<^sub>1 U w c P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "BinPacking.bp U w c (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V})"
Suc: shows "card {a \<in> {a. odd a}. a < 2 * n} = n"
mod_pred: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "(n - Suc 0) mod m = (if n mod m = 0 then m - Suc 0 else n mod m - Suc 0)"
prod_mset_prime_factorization: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "\<Prod>\<^sub># (prime_factorization x) = normalize x"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "less_option x xa" obtains ao :: "'a option" and bo :: "'a option" where "x = ao" and "xa = bo" and "less_eq_option ao bo \<and> ao \<noteq> bo"
that: fixes us :: "'b list"   and vs :: "'b list" assumes "xs = us @ vs"   and "map f us = ys"   and "map f vs = zs" shows "thesis"
is_OK_returns_heap_I: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and h' :: "'x" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
subalgebra_axioms_def: fixes K :: "'c set"   and V :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subalgebra_axioms K V R \<equiv> \<forall>k v. k \<in> K \<longrightarrow> v \<in> V \<longrightarrow> k \<otimes>\<^bsub>R\<^esub> v \<in> V"
Eq\<^sub>EE: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "[x \<^bold>=\<^sub>E y in v]" shows "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
eval_inl_correct: fixes xs :: "dynamic list"   and opinl :: "opinl" assumes "length xs = ar (deinl opinl)" shows "eval_inl opinl xs = eval (deinl opinl) xs"
le_cases: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" obtains "a \<preceq>\<^bsub>D\<^esub> b"   | "b \<preceq>\<^bsub>D\<^esub> a"
fls_divide_base_factor_numer: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor f / g = fls_shift (fls_subdegree f) (f / g)"
D: fixes x :: "real" assumes "x \<in> {1 / 2..1}" shows "x powr (a - 1) \<le> D"
left_add_mult_distrib: fixes i :: "nat"   and u :: "nat"   and j :: "nat"   and k :: "nat" shows "i * u + (j * u + k) = (i + j) * u + k"
is_expansion_const: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and c :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "length basis = expansion_level TYPE('a)" shows "is_expansion (const_expansion c) basis"
add_less_same_cancel2: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a + b \<subset># b) = (a \<subset># {#})"
sq_norm_gso_le_f: fixes i :: "nat" assumes "i < length fs" shows "\<parallel>gso i\<parallel>\<^sup>2 \<le> \<parallel>fs ! i\<parallel>\<^sup>2"
Cons_5: shows "set accu \<subseteq> set (edgesL G)"
cut_greater_Min_greater: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>> t \<noteq> {}" shows "t < iMin (I \<down>> t)"
hext_trans: fixes h :: "'heap"   and h' :: "'heap"   and h'' :: "'heap" assumes "h \<unlhd> h'"   and "h' \<unlhd> h''" shows "h \<unlhd> h''"
idmap_funcs: fixes A :: "'c set" shows "idmap A \<in> A \<rightarrow> A"
redT_updLns_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and ln :: "'a \<Rightarrow>f nat"   and las :: "'a lock_actions" shows "redT_updLns ls t ln las = (\<lambda>(l, n, la). upd_threadRs n l t la) \<circ>$ ($ls, ($ln, las$)$)"
map_add_le_mapI: fixes f :: "'d \<Rightarrow> 'e option"   and h :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "f \<subseteq>\<^sub>m h"   and "g \<subseteq>\<^sub>m h" shows "f ++ g \<subseteq>\<^sub>m h"
invertible_qmpt_def: shows "I.invertible_qmpt \<equiv> bij Tinv \<and> random_variable M (inv Tinv)"
singular_subdivision_simplicial_simplex: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "singular_subdivision p c = simplicial_subdivision p c"
word_plus_mcs_4: fixes v :: "'b word"   and x :: "'b word"   and w :: "'b word" assumes "v + x \<le> w + x"   and "x \<le> v + x" shows "v \<le> w"
acts_MapOps_lookup_def: shows "acts_MapOps_lookup \<equiv> \<lambda>m (cec, aec). Option.bind (lookup_trie m (map fst (toList cec))) (\<lambda>m. Option.bind (lookup_trie m (map snd (toList cec))) (\<lambda>m. Option.bind (lookup_trie m (map fst (toList aec))) (\<lambda>m. lookup_trie m (map snd (toList aec)))))"
iso_iff: shows "((\<lambda>e. \<lambda>i\<in>I. (if i \<in> I then J i else (\<lambda>_. undefined)) (e i)) \<in> Group.iso (DirProds Gs I) (DirProds (f \<circ> Gs) I)) = (carrier (DirProds (f \<circ> Gs) I) \<subseteq> (\<lambda>e. \<lambda>i\<in>I. (if i \<in> I then J i else (\<lambda>_. undefined)) (e i)) ` carrier (DirProds Gs I) \<and> (\<forall>x\<in>carrier (DirProds Gs I). (\<lambda>i\<in>I. (if i \<in> I then J i else (\<lambda>_. undefined)) (x i)) = \<one>\<^bsub>DirProds (f \<circ> Gs) I\<^esub> \<longrightarrow> x = \<one>\<^bsub>DirProds Gs I\<^esub>))"
length_codomain_block: fixes ls :: "brick list" shows "length (blockmat ls) = 2 ^ nat (codomain_block ls)"
divides_one: fixes a :: "'a" assumes "a \<in> carrier R" shows "a divides \<one> = (a \<in> Units R)"
finite_neq_0_inv': fixes h :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b" assumes "finite {x. h (f x) (g x) \<noteq> (0::'b)}"   and "finite {x. f x \<noteq> (0::'c)}"   and "\<And>x. h (0::'c) x = x" shows "finite {x. g x \<noteq> (0::'b)}"
uabc_uhalt: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp)" shows "{\<lambda>nl. nl = [code tp, bl2wc (<lm>)]} F_aprog \<up>"
exec: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Normal t"
cont: shows "continuous_on {real_of_int a..real_of_int b} f"
siso_cont: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd"   and i :: "nat"   and s :: "'b" assumes "PL_Indis cval indis"   and "PL_Indis.siso aval tval cval indis c"   and "i < PL.brn c" shows "PL_Indis.siso aval tval cval indis (PL.cont aval tval cval c s i)"
cases: fixes a1 :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and a2 :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) llist" assumes "mthr.Runs a1 a2" obtains s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" where "a1 = sa__" and "a2 = LNil" and "\<And>tl s'. \<not> redT sa__ tl s'"   | s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" and tls :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) llist" and tl :: "'thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action" where "a1 = s" and "a2 = LCons tl tls" and "redT s tl s'" and "mthr.Runs s' tls"
idem: fixes a :: "'t" shows "ord_term_lin.max a a = a"
star_inductl_one: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "less_eq_prod (sup_prod one_prod (times_prod' x y)) y" shows "less_eq_prod (prod.qstar x) y"
fps_nonzeroI: fixes f :: "'a fps"   and n :: "nat" assumes "fps_nth f n \<noteq> (0::'a)" shows "f \<noteq> 0"
assms: shows "\<forall>x. count M x \<le> 1"
prj_simps_2: shows "C.seq \<mu>_\<nu>\<pi>.prj\<^sub>0 \<p>\<^sub>1[\<nu>\<pi>.leg0 \<cdot> \<mu>_\<nu>\<pi>.prj\<^sub>0, \<rho>.leg1]"
prems_3: shows "preds (slice_kinds S (a # as)) s"
rel_set_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
card_0_eq: fixes A :: "'d set" assumes "finite A" shows "(card A = 0) = (A = {})"
map_ran_conv_map: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "('a \<times> 'c) list" shows "map_ran f xs = map (\<lambda>(k, v). (k, f k v)) xs"
fold_A_morph_A_A: shows "ChamberComplexMorphism A A fold_A"
bind_resT_simps_1: fixes f :: "'g \<rightarrow> 'e\<cdot>'f resT" shows "\<bottom> \<bind> f = \<bottom>"
simps_2: shows "\<^bold>\<lfloor>\<^bold>\<I>\<^bold>\<rfloor> = \<^bold>\<I>"
g_set_xyy_loc_def: fixes ops0 :: "('a, 'b, 'c) set_ops_scheme"   and ops1 :: "('d, 'e, 'f) set_ops_scheme"   and ops2 :: "('d, 'g, 'h) set_ops_scheme" shows "g_set_xyy_loc ops0 ops1 ops2 \<equiv> StdSet ops0 \<and> g_set_xx_loc ops1 ops2"
lincomb_clean: fixes A :: "'a vec set"   and Z :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "A \<subseteq> carrier_vec n"   and "Z \<subseteq> carrier_vec n"   and "finite A"   and "finite Z"   and "a \<in> A \<rightarrow> UNIV"   and "a \<in> Z \<rightarrow> {0::'a}" shows "lincomb a (A \<union> Z) = lincomb a A"
inter_guards_terminates: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile>c1 \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
mod_ring_1__integer: shows "mod_ring_rel_integer 1 1"
path_iff_aux_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and star :: "'b \<Rightarrow> 'b"   and r :: "'b"   and x :: "'b" assumes "class.relation_algebra_rtc minus uminus inf less_eq less sup bot top composition converse unit star"   and "relation_algebra.is_inj less_eq composition converse unit r \<and> relation_algebra.is_sur less_eq composition converse unit r" shows "less_eq (composition r x) (sup (star x) (star (converse x))) = less_eq (converse x) (composition (sup (star x) (star (converse x))) r)"
isps: shows "product_prob_space (\<lambda>i. measure_pmf (bernoulli_pmf p))"
eq: shows "topspace X - {x \<in> topspace X. f x \<in> T} = {x \<in> topspace X. f x \<in> topspace Y - T}"
Cons_1: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "\<And>u. u \<in> set vs \<Longrightarrow> u \<prec>\<^sub>t v"   and "u \<in> set (fst (sym_preproc_addnew gs vs fs v))" shows "u \<prec>\<^sub>t v"
lift: fixes e1 :: "nat rexp"   and e2 :: "nat rexp"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" assumes "lang e1 = lang e2" shows "lang (subst e1 \<sigma>) = lang (subst e2 \<sigma>)"
m_comm: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> y = y \<otimes>\<^bsub>P\<^esub> x"
elims: fixes x :: "Vars.var list"   and xa :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and y :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "to_gc_conf x xa = y" obtains r :: "Vars.var list" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "x = r" and "xa = (\<Gamma>, e, S)" and "y = (restrictA (- set r) \<Gamma>, e, restr_stack (- set r) S @ map Dummy (rev r))"
set_assert_option_conv: fixes b :: "bool" shows "set_option (assert_option b) = (if b then {()} else {})"
extremum_uniqueI: fixes a :: "'b" assumes "a \<le> bot" shows "a = bot"
distinct_32: fixes x51 :: "('c, 'd) Term.term"   and x52 :: "('c, 'd) Term.term"   and x41 :: "('c, 'd) Term.term"   and x42 :: "('c, 'd) Term.term" shows "delete\<langle>x51,x52\<rangle> \<noteq> insert\<langle>x41,x42\<rangle>"
GSMP_subterms: fixes M :: "('fun, 'var) Term.term set" shows "subterms\<^sub>s\<^sub>e\<^sub>t (GSMP M) = GSMP M"