hcju/isabelleps huggingface dataset:
flush_store_buffer_append: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and sb :: "'a memref list"   and sb' :: "'a memref list"   and is\<^sub>s\<^sub>b :: "instr list"   and "is" :: "instr list"   and p :: "'a"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p\<^sub>s\<^sub>b :: "'a"   and \<O>' :: "nat set"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "instrs (sb @ sb') @ is\<^sub>s\<^sub>b = is @ prog_instrs (sb @ sb')"   and "causal_program_history is\<^sub>s\<^sub>b (sb @ sb')"   and "ts ! i = (p, is, \<theta> |` (dom \<theta> - read_tmps (sb @ sb')), x, \<D>, \<O>, \<R>)"   and "p = hd_prog p\<^sub>s\<^sub>b (sb @ sb')"   and "last_prog p\<^sub>s\<^sub>b (sb @ sb') = p\<^sub>s\<^sub>b"   and "reads_consistent True \<O>' m sb"   and "history_consistent \<theta> p (sb @ sb')"   and "\<forall>sop\<in>write_sops sb. valid_sop sop"   and "distinct_read_tmps (sb @ sb')"   and "volatile_reads_consistent m sb" shows "\<exists>is'. instrs sb' @ is\<^sub>s\<^sub>b = is' @ prog_instrs sb' \<and> (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts[i := (last_prog (hd_prog p\<^sub>s\<^sub>b sb') sb, is', \<theta> |` (dom \<theta> - read_tmps sb'), x, \<D> \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}, acquired True sb \<O>, release sb (dom \<S>) \<R>)], flush sb m, share sb \<S>)"
lconf_hext: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "'a \<Rightarrow> val option"   and E :: "'a \<Rightarrow> ty option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,h \<turnstile> l (:\<le>)\<^sub>w E"   and "h \<unlhd> h'" shows "P,h' \<turnstile> l (:\<le>)\<^sub>w E"
femptyE: fixes a :: "'c"   and P :: "bool" assumes "a |\<in>| {||}" shows "P"
insert_subset_equality: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "a \<notin> A"   and "a \<notin> B"   and "insert a A = insert a B" shows "A = B"
z_facts_5: shows "z \<notin> Fvars \<phi>"
branchSubs: fixes subs :: "'a \<Rightarrow> 'a set"   and Gamma :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "branch subs Gamma f"   and "\<not> terminal subs (f n)" shows "f (Suc n) \<in> subs (f n)"
lm060: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum snd (summedBid bids ` a)"
graph_minus_aimpl: fixes E1 :: "('a \<times> 'a) set"   and E2 :: "('a \<times> 'a) set" shows "E1 - E2 = E_of_succ (\<lambda>x. E1 `` {x} - E2 `` {x})"
simps: fixes a1 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and a2 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.mm_equiv_abv a1 a2 = (\<exists>\<R> lc\<^sub>1 lc\<^sub>2. a1 = lc\<^sub>1 \<and> a2 = lc\<^sub>2 \<and> abs.strong_low_bisim_mm \<R> \<and> (lc\<^sub>1, lc\<^sub>2) \<in> \<R>)"
V0'_to_V0: fixes v :: "'v" assumes "v \<in> G'.V0" shows "fi v \<in> V0"
subst_freshen_conclusions: fixes pf :: "'form"   and s :: "'subst"   and a :: "nat" assumes "pf \<in> set conclusions" shows "subst s (freshen a pf) = pf"
echelon_form_intro: fixes A :: "(('b, 'c) vec, 'a) vec" assumes "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"   and "\<forall>i j. i < j \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ j $ n \<noteq> (0::'b))" shows "echelon_form A"
hyps_1: fixes n :: "'node" assumes "l = shortestPath g n"   and "\<And>ns. \<lbrakk>g \<turnstile> Entry g-ns\<rightarrow>n; EntryPath g ns\<rbrakk> \<Longrightarrow> thesis"   and "n \<in> set (\<alpha>n g)" shows "thesis"
Cons_1: fixes n :: "('a \<times> nat) list" assumes "monom_inv m"   and "monom_inv n" shows "(m = n) = (\<forall>x. sum_var_list m x = sum_var_list n x)"
minim_least: fixes r :: "('d \<times> 'd) set"   and B :: "'d set"   and b :: "'d" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "b \<in> B" shows "(minim r B, b) \<in> r"
piecewise_continuous_onE: fixes a :: "'c"   and b :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "piecewise_continuous_on a b I f" obtains l :: "'c \<Rightarrow> 'd" and u :: "'c \<Rightarrow> 'd" where "finite I" and "continuous_on ({a..b} - I) f" and "\<And>i. \<lbrakk>i \<in> I; a < i; i \<le> b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> l i) (at_left i)" and "\<And>i. \<lbrakk>i \<in> I; a \<le> i; i < b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> u i) (at_right i)"
norm_assertion_simps_1: fixes a :: "assn" shows "emp * a = a"
os_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet A B X Y" shows "A \<noteq> B \<and> A \<noteq> X \<and> A \<noteq> Y \<and> B \<noteq> X \<and> B \<noteq> Y"
lincomb_closed: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "S \<subseteq> carrier_vec (n + 1)" shows "next_dim.lincomb a S \<in> carrier_vec (n + 1)"
hm_size_def: fixes ht :: "('c, 'd) hashtable" shows "hm_size ht \<equiv> return (the_size ht)"
assms_1: shows "2 * k + 1 < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
index_minus_vec_2: fixes v\<^sub>1 :: "'c vec"   and v\<^sub>2 :: "'c vec" shows "dim_vec (v\<^sub>1 - v\<^sub>2) = dim_vec v\<^sub>2"
simps_1: fixes v :: "'v" shows "adj_path v [] = (v \<in> V)"
Diff_subset_conv: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
fundamental_theorem_of_calculus: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b})" shows "(f' has_integral f b - f a) {a..b}"
uses'_code_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and n :: "'node" shows "uninst.uses'_code u p g next n \<equiv> uninst.substNext_code p g next ` u g n"
ne_INF_commute: fixes U\<^sub>2 :: "'c set"   and U\<^sub>3 :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'al"   and B :: "'d set"   and A :: "'c set" assumes "U\<^sub>2 \<noteq> {}"   and "U\<^sub>3 \<noteq> {}"   and "\<forall>x\<in>U\<^sub>2. \<forall>y\<in>U\<^sub>3. f x y \<in> U"   and "B \<subseteq> U\<^sub>3"   and "A \<subseteq> U\<^sub>2" shows "\<Sqinter>\<^sub>o\<^sub>w ((\<lambda>i. \<Sqinter>\<^sub>o\<^sub>w (f i ` B)) ` A) = \<Sqinter>\<^sub>o\<^sub>w ((\<lambda>j. \<Sqinter>\<^sub>o\<^sub>w ((\<lambda>i. f i j) ` A)) ` B)"
iter_induct_nil: fixes c :: "'a::refinement_lattice"   and x :: "'a::refinement_lattice" assumes "nil \<sqinter> (c::'a::refinement_lattice) ; (x::'a::refinement_lattice) \<sqsubseteq> x" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> (x::'a::refinement_lattice)"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs (CFG_node n) = CFG.valid_node sourcenode targetnode valid_edge n"
div_condn_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and n :: "nat"   and g :: "'e"   and f :: "'e" shows "div_condn R S X n g f = (f \<in> carrier R \<and> n = deg_n R S X f \<longrightarrow> (\<exists>q. q \<in> carrier R \<and> (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> q \<cdot>\<^sub>r\<^bsub>R\<^esub> g = \<zero>\<^bsub>R\<^esub> \<or> deg_n R S X (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> q \<cdot>\<^sub>r\<^bsub>R\<^esub> g) < deg_n R S X g)))"
real_cond_exp_cmult: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f" shows "AE x in M. real_cond_exp M Invariants (\<lambda>x. c * f x) x = c * real_cond_exp M Invariants f x"
1: shows "make_vert_equiv k = mat1 (2 ^ k)"
fps_mult_right_fps_X_plus_1_nth: fixes a :: "'a fps"   and n :: "nat" shows "fps_nth (a * (1 + fps_X)) n = (if n = 0 then fps_nth a n else fps_nth a n + fps_nth a (n - 1))"
h'_f: fixes x :: "nat" shows "h' x sums f' x"
fresh_some: fixes a :: "'d"   and x :: "'e" shows "a \<sharp> Some x = a \<sharp> x"
ide_char: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VV.ide a = (Span.VV.arr a \<and> Span.VxV.ide a)"
mq: shows "mq < 2 ^ (LENGTH('a) - s1)"
OV_inv2D: fixes x :: "'a obsv_state_scheme"   and r :: "nat" assumes "x \<in> OV_inv2"   and "next_round x \<le> r" shows "obsv_state.obs x r = Map.empty"
subprob_emeasure_le_1: fixes p :: "'b spmf"   and X :: "'b set" shows "emeasure (measure_spmf p) X \<le> 1"
assms: shows "Fun g S \<in> timpl_closure (Fun f T) TI"
prems_3: shows "g \<in> insert x F \<rightarrow> carrier G"
mk_root_args: fixes t :: "'a" assumes "t \<in> trees" shows "mk (fst (root t)) (args t) = t"
balanced_prob0_jozsa_algo: assumes "prob0_fst_qubits n jozsa_algo = 0" shows "is_balanced"
pa_invar_def: fixes H1 :: "('a, 'c, 'd) hashedTa_scheme"   and H2 :: "('b, 'c, 'e) hashedTa_scheme" shows "pa_invar H1 H2 \<equiv> pa_invar_add \<inter> {s. pa_\<alpha> s \<in> frp_invar (hta_\<alpha> H1) (hta_\<alpha> H2)}"
simps_20: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (Minus e1 e2) (Plus v va) = False"
simps_12: fixes v :: "nat" shows "generalize_instr (ICJump v) = ICJump v"
OR3: fixes evs3 :: "event list"   and KAB :: "nat"   and M :: "nat"   and A :: "agent"   and B :: "agent"   and NA :: "nat"   and NB :: "nat" assumes "evs3 \<in> orb"   and "Key KAB \<notin> used evs3"   and "Gets Server \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Nonce M, Agent A, Agent B\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs3" shows "Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>Crypt (shrK A) \<lbrace>Nonce NA, Key KAB\<rbrace>, Nonce NB, Key KAB\<rbrace>\<rbrace> # evs3 \<in> orb"
assms: fixes f :: "'a" assumes "f \<in> space\<^sub>N N1" shows "eNorm N2 f \<le> ennreal C * eNorm N1 f"
u_A: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "u x = 0"
inverse_le_iff_le: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less zero b" shows "less_eq (inverse a) (inverse b) = less_eq b a"
order_refl: fixes x :: "nat list" shows "x \<le>\<^sub>v x"
C: shows "Min (offs_set_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (x # xs) index key mi ma i) \<in> offs_set_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (x # xs) index key mi ma i"
equality: fixes r :: "('c, 'd) carrier_scheme"   and r' :: "('c, 'd) carrier_scheme" assumes "carrier r = carrier r'"   and "carrier.more r = carrier.more r'" shows "r = r'"
prime_power_cancel2: fixes p :: "'b"   and m :: "'b"   and k :: "nat"   and m' :: "'b"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "\<not> p dvd m"   and "\<not> p dvd m'" obtains "m = m'" and "k = k'"
prod_ps: shows "n = (\<Prod>m\<in>#mset ps. m)"
linorder_not_less: fixes x :: "'r"   and y :: "'r" shows "(\<not> x < y) = (y \<le> x)"
index_less_size: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "index xs x < length xs"
safety_unless: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "safety P"   and "safety Q" shows "safety (P \<W> Q)"
finite: shows "finite (range (\<lambda>i. SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))))"
ceiling_unique: fixes z :: "int"   and x :: "'a" assumes "of_int z - (1::'a) < x"   and "x \<le> of_int z" shows "\<lceil>x\<rceil> = z"
refl: fixes a :: "'b DBMEntry" shows "a \<preceq> a"
mult_closed: fixes r :: "'r"   and s :: "'r" assumes "r \<in> R"   and "s \<in> R" shows "r * s \<in> R"
finite_reach: shows "finite (reach \<Sigma> \<delta> q\<^sub>0)"
assms: shows "run (f \<bind> g) \<sigma> \<sigma>'' r"
assms_2: shows "[b] \<preceq>\<^sub>F w"
Hermite_of_upt_row_i_in_range: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat" assumes "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "res_function res"   and "mod_type_class.to_nat i < k"   and "k \<le> nrows A" shows "Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range ass"
assms: shows "selected_edge h j g \<le> - forest_components f"
Cd_Above_dX: fixes X :: "'a set"   and d :: "'b" assumes "ManiPref'.dX X d \<subseteq> Field (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d)" shows "ManiPref'.Cd d X = Above (TruePref.Pd_shuffle_to_top ds' (ManiPref.cop ds) d) (ManiPref'.dX X d) \<inter> X"
conv_compl_aux: fixes x :: "'a" shows "(- x)\<^sup>\<smile> = (- x)\<^sup>\<smile> + - x\<^sup>\<smile>"
typeI_and_typII_one_chains_have_gen_common_subdiv: shows "common_sudiv_exists one_chain_typeI one_chain_typeII"
transpose_rank: fixes A :: "'a mat" shows "vec_space.rank (dim_row A) A = vec_space.rank (dim_col A) A\<^sup>T"
fresh_2: fixes a :: "atom"   and tm1 :: "tm"   and tm2 :: "tm" shows "a \<sharp> tm1 EQ tm2 = (a \<sharp> tm1 \<and> a \<sharp> tm2)"
"1_2": shows "subst_domain \<theta> \<inter> set (bvars\<^sub>s\<^sub>t\<^sub>p a) = {}"
ordered_sel_resolvent_is_ordered_resolvent: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and Sel :: "'at set" assumes "ordered_resolvent P1 P2 C"   and "selected_part Sel P1 = {}"   and "selected_part Sel P2 = {}" shows "ordered_sel_resolvent Sel P1 P2 C"
EM_remainder_converges_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'a"   and a :: "int" shows "EM_remainder_converges n f a = (\<exists>L. ((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) \<longlongrightarrow> L) at_top)"
last_message_hold_length: fixes xs :: "'b message_af list" shows "length (last_message_hold xs) = length xs"
project_LeadsETo_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> extend_set h A LeadsTo[givenBy f] extend_set h B" shows "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> A \<longmapsto>w B"
multiset_def: shows "multiset = {f. finite {x. 0 < f x}}"
iLastWeak_True: fixes t0 :: "nat"   and I :: "nat set" shows "\<ominus>\<^sub>W t t0 I. True"
assms: shows "f \<in> Invariants \<rightarrow>\<^sub>M N"
sat_def: fixes S :: "'b formula set" shows "sat S \<equiv> \<exists>\<A>. \<forall>F\<in>S. \<A> \<Turnstile> F"
length_generate_PS: fixes emBits :: "nat" shows "length (generate_PS emBits 160) = roundup emBits 8 * 8 - sLen - 160 - 16"
Key_in_parts_respond: fixes K :: "nat"   and RB :: "msg"   and PB :: "msg"   and K' :: "nat"   and evs :: "event list" assumes "Key K \<in> parts {RB}"   and "(PB, RB, K') \<in> respond evs" shows "Key K \<notin> used evs"
IH: fixes p1 :: "'c"   and p2 :: "'d" assumes "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path A (ws || r1s) p1 \<and> path B (ws || r2s) p2"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path AB (ws || r1s || r2s) (p1, p2)"
csupport_on_def: fixes X :: "'c set"   and S :: "'c \<Rightarrow> 'd" shows "csupport_on X S = closure (support_on X S)"
fromQ_inverse: fixes x :: "'a['b, 'c]" shows "toQ (fromQ x) = x"
2: fixes i :: "nat"   and j :: "nat" assumes "i < length (seq' j)" shows "seq i = rev (seq' j) ! i"
IH: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat"   and e :: "cexpr" shows "cexpr_sem \<sigma> (cexpr_comp_aux x e e') = cexpr_sem (\<sigma>(x := cexpr_sem \<sigma> e)) e'"
add_mult_distrib_mat: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and C :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr n"   and "C \<in> carrier_mat n nc" shows "(A + B) * C = A * C + B * C"
NF_not_suc: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set" assumes "(x, y) \<in> r\<^sup>*"   and "x \<in> NF r" shows "x = y"
foo_def: fixes f2v :: "real"   and f2 :: "real" shows "foo f2v f2 = (case if f2v = 0 then [- f2] else [] of [] \<Rightarrow> None | a # x \<Rightarrow> Map.empty x)"
uppert_to_jb: fixes n :: "nat"   and A :: "'b mat" assumes "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.uppert A"   and "A \<in> carrier_mat n n" shows "inv_upto n ??.Jordan_Normal_Form_Existence.jb A 1"
interrupt_actions_ok'_nthD: fixes "is" :: "'b set"   and ias :: "'b interrupt_action list"   and n :: "nat" assumes "interrupt_actions_ok' is ias"   and "n < length ias" shows "interrupt_action_ok' (redT_updIs is (take n ias)) (ias ! n)"
cop_stable_no_blocking_on: fixes ds :: "'b set" shows "stable_no_blocking_on ds (cop ds)"
infsetsum_Diff: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "infsetsum f (B - A) = infsetsum f B - infsetsum f A"
src_cod: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "src\<^sub>D (D.cod \<mu>) = src\<^sub>D \<mu>"
distinct_15: fixes x3 :: "bool"   and x4 :: "32 word" shows "Bool x3 \<noteq> Intg x4"
secure_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" shows "secure P I D \<equiv> \<forall>xs y ys Y zs Z. (y # ys, Y) \<in> futures P xs \<and> (zs, Z) \<in> futures P xs \<longrightarrow> (ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> futures P xs \<and> (y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> futures P xs"
size_multiset_eq: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset" shows "size_multiset f M = (\<Sum>x\<in>set_mset M. count M x * Suc (f x))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_new_element_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
enum_def: fixes i :: "nat"   and j :: "nat" shows "enum i j = (if j \<in> upd ` {..<i} then Suc (base j) else base j)"
asymp_equiv_altdef: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<sim>[F] g) = ((\<lambda>x. f x - g x) \<in> o[F](g))"
simps_1: fixes uu :: "nat" shows "max_depth (Atom uu) = 0"
b: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> b) = (\<exists>t. meets p t \<and> meets t q)"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set (map fst pairs)"   and "y \<in> set (map fst pairs)"   and "P x"   and "P y" shows "x = y"
abs_0: fixes abs :: "'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs zero = zero"
converse_inv_image: fixes R :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
fst_scaleR: fixes r :: "real"   and A :: "'a \<times> 'b" shows "fst (r *\<^sub>R A) = r *\<^sub>R fst A"
semiattractive_axioms: shows "semiattractive A (\<lambda>x y. r y x)"
Diff_insert2: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A\<setminus>insert a B = A\<setminus>{a}\<setminus>B"
scaleQ_one: fixes x :: "'e['f, 'g]" shows "(1::'e) *\<^sub>Q x = x"
llist_all2_LCons_LCons: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and x22 :: "'c llist"   and y21 :: "'d"   and y22 :: "'d llist" shows "llist_all2 R (LCons x21 x22) (LCons y21 y22) = (R x21 y21 \<and> llist_all2 R x22 y22)"
indep_subset_carrier: fixes X :: "'a set" assumes "indep_in \<E> X" shows "X \<subseteq> \<E>"
assms_2: shows "\<alpha> \<in> elts (ordertype M r)"
Literal_eq_iff: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and s :: "String.literal"   and c0 :: "bool"   and c1 :: "bool"   and c2 :: "bool"   and c3 :: "bool"   and c4 :: "bool"   and c5 :: "bool"   and c6 :: "bool"   and t :: "String.literal" shows "(String.Literal b0 b1 b2 b3 b4 b5 b6 s = String.Literal c0 c1 c2 c3 c4 c5 c6 t) = (b0 = c0 \<and> b1 = c1 \<and> b2 = c2 \<and> b3 = c3 \<and> b4 = c4 \<and> b5 = c5 \<and> b6 = c6 \<and> s = t)"
BEq_def: fixes a :: "aexp"   and b :: "aexp" shows "BEq a b \<equiv> And (Less a (Plus b (N 1))) (Less b (Plus a (N 1)))"