hcju/isabelleps huggingface dataset:
push_bit_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n (plus a b) = plus (push_bit n a) (push_bit n b)"
graphE: fixes x :: "'c"   and y :: "real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "(x, y) \<in> graph F f" obtains "x \<in> F" and "y = f x"
assms_1: shows "wa_refine WC WB \<alpha>1"
update_convs_1: fixes arrays' :: "(typerep \<Rightarrow> nat \<Rightarrow> nat list) \<Rightarrow> typerep \<Rightarrow> nat \<Rightarrow> nat list"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'a" shows "arrays_update arrays' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays' arrays, refs = refs, lim = lim, \<dots> = more\<rparr>"
disc_3: fixes x11 :: "String.literal"   and x12 :: "typ list" shows "\<not> is_Tv (Ty x11 x12)"
iteratei_rule_insert_P: assumes "set_iterator_genord it S0 (\<lambda>_ _. True)"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {x}. True; \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> I (insert x S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
trivial_limit_at_top_linorder: shows "at_top \<noteq> bot"
injo: shows "inj_on (inv_into X f) (f ` X)"
unit_prod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one"   and "dvd.dvd times b one" shows "dvd.dvd times (times a b) one"
ro: shows "unowned_shared (snd (snd (c n))) (fst (c n))"
tends: shows "AE x in lebesgue_on S. (\<lambda>n. g n x) \<longlonglongrightarrow> h x"
times_\<omega>_minus_of_nat: fixes m :: "hmultiset"   and n :: "nat" shows "m * \<omega> - of_nat n = m * \<omega>"
repeat_zmset_distrib2: fixes n :: "nat"   and A :: "'b zmultiset"   and B :: "'b zmultiset" shows "repeat_zmset n (A + B) = repeat_zmset n A + repeat_zmset n B"
lemma2_6_6_a: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset"   and Q :: "'a multiset" assumes "trans r"   and "(M, N) \<in> mul_eq r" shows "(Q + M, Q + N) \<in> mul_eq r"
is_sup_uniq: fixes x :: "'b"   and y :: "'b"   and sup :: "'b"   and sup' :: "'b" assumes "is_sup x y sup"   and "is_sup x y sup'" shows "sup = sup'"
simplex_1: fixes cs :: "constraint list"   and I :: "nat list" assumes "simplex cs = Inl I" shows "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s set cs"
char_inv: fixes a :: "'a \<Rightarrow> complex" assumes "a \<in> carrier C" shows "principal_char C (inv\<^bsub>C\<^esub> a) = cnj (principal_char C a)"
paint_in_traverse_pairs: fixes c :: "color"   and t :: "('a, 'b) rbt" shows "rbt_in_traverse_pairs (paint c t) = rbt_in_traverse_pairs t"
compE\<^sub>1_pres_wt: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and Vs :: "char list list"   and Ts :: "ty list"   and e :: "char list exp"   and U :: "ty"   and f :: "char list list \<times> char list exp \<Rightarrow> nat exp" assumes "P,[Vs [\<mapsto>] Ts] \<turnstile> e :: U"   and "length Ts = length Vs" shows "compP f P,Ts \<turnstile>\<^sub>1 compE\<^sub>1 Vs e :: U"
counit_simps_4: shows "src h.\<epsilon> = trg h"
commutative_monoid_axioms_def: fixes M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "commutative_monoid_axioms M composition \<equiv> \<forall>x y. x \<in> M \<longrightarrow> y \<in> M \<longrightarrow> composition x y = composition y x"
3: fixes i :: "int" assumes "odd i"   and "\<forall>n. even n \<or> real_of_int i \<noteq> - real n" shows "\<exists>n. odd n \<and> real_of_int i = real n"
is_arith_prog_on_def: fixes l :: "nat"   and start :: "int"   and step :: "nat"   and a :: "int"   and b :: "int" shows "is_arith_prog_on l start step a b = (a \<le> start \<and> arith_prog start step (l - 1) \<le> b)"
pos: shows "0 < (\<Sum>m<n. (- x) ^ m / fact m)"
emeasure_measure_pmf_finite: fixes S :: "'b set"   and M :: "'b pmf" assumes "finite S" shows "emeasure (measure_pmf M) S = ennreal (sum (pmf M) S)"
projection_def: fixes l :: "'b list"   and E :: "'b set" shows "l \<upharpoonleft> E \<equiv> filter (\<lambda>x. x \<in> E) l"
islimpt_Un_finite: fixes s :: "'a set"   and x :: "'a"   and t :: "'a set" assumes "finite s" shows "x islimpt s \<union> t = x islimpt t"
is_process_def: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "is_process P = (([], {}) \<in> FAILURES P \<and> (\<forall>s X. (s, X) \<in> FAILURES P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, Y) \<in> FAILURES P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, X) \<in> FAILURES P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> FAILURES P) \<longrightarrow> (s, X \<union> Y) \<in> FAILURES P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> FAILURES P \<longrightarrow> (s, X - {tick}) \<in> FAILURES P) \<and> (\<forall>s t. s \<in> DIVERGENCES P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> DIVERGENCES P) \<and> (\<forall>s X. s \<in> DIVERGENCES P \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s. s @ [tick] \<in> DIVERGENCES P \<longrightarrow> s \<in> DIVERGENCES P))"
falsifies\<^sub>l_def: fixes G :: "bool list"   and l :: "fterm literal" shows "falsifies\<^sub>l G l = (ground\<^sub>l l \<and> (let i = nat_of_fatom (get_atom l) in i < length G \<and> G ! i = (\<not> sign l)))"
to_subset: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "Worder D"   and "T \<subseteq> carrier D"   and "ord_isom D (Iod D T) f" shows "\<forall>a. a \<in> carrier D \<longrightarrow> a \<preceq>\<^bsub>D\<^esub> f a"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "vc (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (vc C\<^sub>1 Q \<and> vc C\<^sub>2 Q)"
zero_pfun_def: shows "{}\<^sub>p \<equiv> Abs_pfun Map.empty"
SUP_vifunion: fixes A :: "V \<Rightarrow> V"   and \<alpha> :: "V" shows "\<Squnion> (A ` elts \<alpha>) = \<Union>\<^sub>\<circ> (VLambda \<alpha> A `\<^sub>\<circ> \<alpha>)"
map_drop_def: fixes a :: "'c" shows "map_drop a = map_filter (\<lambda>a'. a' \<noteq> a)"
subset_not_subset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subset> B) = (A \<subseteq> B \<and> \<not> B \<subseteq> A)"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_prefix_subset_1: fixes A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and B :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ B)"
T1_def: shows "T1 \<equiv> \<lparr>verts = {}, arcs = {}\<rparr>"
ceiling_less_cancel: fixes x :: "'a"   and y :: "'a" assumes "\<lceil>x\<rceil> < \<lceil>y\<rceil>" shows "x < y"
radical_sqrt_line_circle_intersection: fixes A :: "point"   and B :: "point"   and C :: "point"   and D :: "point"   and E :: "point"   and X :: "point" assumes "abscissa A \<in> radical_sqrt"   and "ordinate A \<in> radical_sqrt"   and "abscissa B \<in> radical_sqrt"   and "ordinate B \<in> radical_sqrt"   and "abscissa C \<in> radical_sqrt"   and "ordinate C \<in> radical_sqrt"   and "abscissa D \<in> radical_sqrt"   and "ordinate D \<in> radical_sqrt"   and "abscissa E \<in> radical_sqrt"   and "ordinate E \<in> radical_sqrt"   and "A \<noteq> B"   and "collinear A X B"   and "distance C X = distance D E" shows "abscissa X \<in> radical_sqrt \<and> ordinate X \<in> radical_sqrt"
m1_inv2r_servD: fixes x :: "'a m1r_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and nlb :: "atom list" assumes "runs x Rb = Some (Resp, [A, B], aKey Kab # nlb)"   and "x \<in> m1_inv2r_serv"   and "B \<notin> bad" shows "\<exists>Rs Na. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na])"
linorder_rank_mono: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "linorder_on B R"   and "finite A"   and "A \<subseteq> B"   and "(x, y) \<in> R" shows "linorder_rank R A x \<le> linorder_rank R A y"
mset_set_set_mset': fixes M :: "'b multiset" assumes "\<forall>x. count M x \<le> 1" shows "mset_set (set_mset M) = M"
cxt_inv_if: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = Stmt.If e p q" shows "c = Stmt.If e p q \<and> E = []"
sqrt_roots: fixes y :: "real"   and x :: "real" assumes "y\<^sup>2 = x" shows "0 \<le> x \<and> (sqrt x = y \<or> sqrt x = - y)"
symmetric_mpoly_def: fixes A :: "nat set"   and p :: "'c mpoly" shows "symmetric_mpoly A p = (\<forall>\<pi>. \<pi> permutes A \<longrightarrow> mpoly_map_vars \<pi> p = p)"
ex_sum_in_interval_lemma: fixes la :: "'b"   and ua :: "'b"   and lb :: "'b"   and ub :: "'b"   and x :: "'b" assumes "la \<le> ua"   and "lb \<le> ub"   and "la + lb \<le> x"   and "x \<le> ua + ub"   and "ua - la \<le> ub - lb" shows "\<exists>xa\<in>{la..ua}. \<exists>xb\<in>{lb..ub}. x = xa + xb"
determ_maximalD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "determ t" shows "maximal t"
remove_code_2: fixes x :: "'b"   and xs :: "'b list" shows "Set.remove x (List.coset xs) = List.coset (List.insert x xs)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_self_inverse: fixes A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = A"
rimp_rel_def: fixes R :: "('a \<times> 'a) set"   and T :: "('a \<times> 'a) set" shows "rimp_rel R T = {(y, z) |y z. \<forall>x. (x, y) \<in> R \<longrightarrow> (x, z) \<in> T}"
longest_Lyndon_suf_nemp: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "LynSuf w \<noteq> \<epsilon>"
continuous_openin_preimage_gen: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "open T" shows "openin (top_of_set S) (S \<inter> f -` T)"
arr_char: fixes f :: "'c local.term set" shows "\<F>C.arr f = (f \<noteq> {} \<and> (\<forall>t. t \<in> f \<longrightarrow> f = \<F>C.mkarr t))"
row_Cons: fixes v :: "'b list"   and M :: "'b list list"   and i :: "nat" shows "row (v # M) i = v ! i # row M i"
rel_refl_strong: fixes x :: "'a tree"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set_tree x \<Longrightarrow> Ra z z" shows "rel_tree Ra x x"
chain_boundary_minus: fixes p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (- c) = - chain_boundary p c"
2: shows "degree p2 \<le> length zs_ws - 1 \<and> (\<forall>x y. (x, y) \<in> set zs_ws \<longrightarrow> poly p2 x = y)"
truncate_down_le: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x \<le> y" shows "truncate_down prec x \<le> y"
last_drop: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "last (drop n xs) = last xs"
scc_ofI_awalk: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "u \<in> G.scc_of v"
stream_smap_fromN: fixes s :: "'d stream"   and n :: "nat" shows "s = smap (\<lambda>j. Let (j - n) ((!!) s)) (fromN n)"
bigo_refl: fixes f :: "'a \<Rightarrow> 'b" shows "f \<in> O(f)"
r_def: shows "r = (\<lambda>x. integral {2..x} (\<lambda>t. \<theta> t / (t * (ln t)\<^sup>2)))"
Min_elem_dest: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "x = Min A" shows "x \<in> A"
inf_sls_def: shows "inf_sls \<equiv> Abs_semilattice_set inf"
subst\<^sub>0: fixes eq :: "'a"   and x :: "'b"   and xs :: "'b list"   and a :: "'a" assumes "solvable\<^sub>0 eq"   and "\<not> trivial eq"   and "I\<^sub>a eq (x # xs)"   and "depends\<^sub>0 a" shows "I\<^sub>a (subst\<^sub>0 eq a) xs = I\<^sub>a a (x # xs)"
lnull_cons_def: fixes s :: "'s" shows "local.lnull_cons s = lnull (lunstream g s)"
inf_unit_def: fixes uu :: "unit"   and uv :: "unit" shows "inf_class.inf uu uv = ()"
assoc: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m"   and c :: "'a\<^sub>m" shows "m a b \<bind> m c = m b c \<bind> m a"
disc_iff_2: fixes xs :: "'c llist"   and ys :: "'c llist" shows "(\<not> lnull (lappend xs ys)) = (\<not> lnull xs \<or> \<not> lnull ys)"
nemp_emp_power: fixes u :: "'b list"   and m :: "nat" assumes "u \<noteq> \<epsilon>" shows "(u \<^sup>@ m = \<epsilon>) = (m = 0)"
Ball_mem_monad_less_zero: fixes x :: "real star"   and u :: "real star" assumes "x < 0"   and "x \<notin> Infinitesimal"   and "u \<in> monad x" shows "u < 0"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and msg :: "msg" shows "P (case msg of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = (\<not> ((\<exists>x11 x12 x13 x14 x15 x16 x17 x18. msg = Rreq x11 x12 x13 x14 x15 x16 x17 x18 \<and> \<not> P (f1 x11 x12 x13 x14 x15 x16 x17 x18)) \<or> (\<exists>x21 x22 x23 x24 x25. msg = Rrep x21 x22 x23 x24 x25 \<and> \<not> P (f2 x21 x22 x23 x24 x25)) \<or> (\<exists>x31 x32. msg = Rerr x31 x32 \<and> \<not> P (f3 x31 x32)) \<or> (\<exists>x41 x42. msg = Newpkt x41 x42 \<and> \<not> P (f4 x41 x42)) \<or> (\<exists>x51 x52 x53. msg = Pkt x51 x52 x53 \<and> \<not> P (f5 x51 x52 x53))))"
function_factors_left: fixes g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "(\<forall>x y. g x = g y \<longrightarrow> f x = f y) = (\<exists>h. f = h \<circ> g)"
retractionI: fixes g :: "'a"   and f :: "'a" assumes "C.ide (g \<cdot> f)" shows "C.retraction g"
pot_comp_prod_def: fixes cmp :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> order" shows "pot_comp cmp = (\<lambda>u v. case comparator_of (snd u) (snd v) of Eq \<Rightarrow> cmp u v | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
length_int: fixes i :: "int" shows "length (int_to_bv i) = length_int i"
b: shows "drop (Suc x) Strat = drop (Suc x) (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat)"
assms_2: shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>lebesgue_on S < \<infinity>"
Ideal1_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool"   and out1 :: "bool" shows "and_secret_sharing.Ideal1 m1 m2 out1 = S1_and m1 out1 \<bind> (\<lambda>view1. out1 m1 m2 out1 \<bind> (\<lambda>out1. return_spmf (view1, out1)))"
finfun_apply_inverse: fixes x :: "'c \<Rightarrow>f 'd" shows "Abs_finfun (finfun_apply x) = x"
l2_inv5D: fixes A :: "agent"   and B :: "agent"   and M :: "msg"   and x :: "l2_state" assumes "Confid A B M \<in> chan x \<or> Secure A B M \<in> chan x"   and "x \<in> l2_inv5" shows "M \<in> dy_fake_msg (bad x) (ik x) (chan x)"
reduced_homology_group_pair: fixes X :: "'a topology"   and a :: "'a"   and b :: "'a"   and p :: "int" assumes "t1_space X"   and "a \<in> topspace X"   and "b \<in> topspace X"   and "a \<noteq> b" shows "reduced_homology_group p (subtopology X {a, b}) \<cong> homology_group p (subtopology X {a})"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and at :: "('c, 'd) atm" shows "substL \<pi> (Pos at) = Pos (substA \<pi> at)"
b_e_check_single_type_not_bot_sound: fixes ts :: "checker_type"   and t_in :: "t list"   and t_out :: "t list"   and ts' :: "checker_type"   and tm :: "t list"   and \<C> :: "t_context"   and e :: "b_e" assumes "type_update ts (to_ct_list t_in) (Type t_out) = ts'"   and "ts \<noteq> Bot"   and "ts' \<noteq> Bot"   and "c_types_agree ts' tm"   and "\<C> \<turnstile> [e] : t_in _> t_out" shows "\<exists>tn. c_types_agree ts tn \<and> \<C> \<turnstile> [e] : tn _> tm"
integrable_componentwise: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A" shows "f integrable_on A"
direct_lim_is_ring: fixes U :: "'c set" assumes "U \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" shows "ring stfx.carrier_direct_lim stfx.add_rel stfx.mult_rel (stfx.class_of U (zero_str\<^sub>Y U)) (stfx.class_of U (one_str\<^sub>Y U))"
trcl_rev_cases: fixes c :: "'c"   and w :: "'d list"   and c' :: "'c"   and S :: "('c \<times> 'd \<times> 'c) set" assumes "(c, w, c') \<in> trcl S" obtains "w = []" and "c = c'"   | ch :: "'c" and e :: "'d" and wh :: "'d list" where "w = wh @ [e]" and "(c, wh, ch) \<in> trcl S" and "(ch, e, c') \<in> S"
r_maxlt_prim: shows "prim_recfn 1 r_maxlt"
language_def: shows "S.language \<equiv> {smap int (p ## S.trace w p) |p w. init p \<and> S.run w p}"
h': shows "(\<Sum>i\<le>p. monom (f (monom (1::'a) i)) i) = (\<Sum>x\<in>{monom (1::'a) x |x. x \<le> p}. smult (f x) x)"
simps_4: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list"   and s :: "'state_t" assumes "\<not> interrupt (Suc n)"   and "\<not> thread_empty (execs (current s))" shows "run_total (Suc n) s execs = run_total n (step (next_state s execs) (next_action s execs)) (next_execs s execs)"
degree0isovarspar: fixes p :: "real mpoly"   and x :: "nat" assumes "MPoly_Type.degree p x = 0" shows "isolate_variable_sparse p x 0 = p"
assms_5: shows "ideal_points (poincare_line u v) = {i1, i2}"
distinct_disc_11: fixes event :: "('a, 'b, 'c) event" assumes "isSend event" shows "\<not> isSnapshot event"
field_sum_of_halves: fixes x :: "'d" shows "x / (2::'d) + x / (2::'d) = x"
pderiv_prod: fixes f :: "'c \<Rightarrow> 'b poly"   and as :: "'c set" shows "pderiv (prod f as) = (\<Sum>a\<in>as. prod f (as - {a}) * pderiv (f a))"
subst_lsst_append: fixes A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list"   and B :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list"   and \<delta> :: "'g \<Rightarrow> ('f, 'g) Term.term" shows "A @ B \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) @ (B \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
Q': shows "Q' \<in> carrier_mat (dim_col (Mod_Type_Connect.from_hma\<^sub>m A)) (dim_col (Mod_Type_Connect.from_hma\<^sub>m A))"
is_adjoint_sym: fixes F :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'a" assumes "is_cadjoint F G" shows "is_cadjoint G F"
iSPEC_rule: fixes \<Phi> :: "'a \<Rightarrow> bool"   and \<Psi> :: "'a \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "SPEC \<Phi> \<le> SPEC \<Psi>"
Variable_inj_var: shows "inj Variable \<and> range Variable \<subseteq> var"
symmetry_preserves_per: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and A' :: "'p"   and P' :: "'p" assumes "Per B P A"   and "B Midpoint A A'"   and "B Midpoint P P'" shows "Per B P' A'"
face_cycle_succ_def: shows "H.face_cycle_succ \<equiv> edge_succ HM \<circ> rev_H"
add_left_mono: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "c + a \<subseteq># c + b"
remdups_adj_replicate: fixes n :: "nat"   and x :: "'g" shows "remdups_adj (replicate n x) = (if n = 0 then [] else [x])"
SES_valid_def: fixes SES :: "('a, 'b) SES_rec" shows "SES_valid SES \<equiv> s0_is_state SES \<and> ses_inputs_are_events SES \<and> ses_outputs_are_events SES \<and> ses_inputs_outputs_disjoint SES \<and> correct_transition_relation SES"
unique_connecting_path_properties_1: fixes v :: "nat"   and w :: "nat" assumes "v \<in> V"   and "w \<in> V" shows "path (v \<leadsto> w)"
ereal_open_closed: fixes S :: "ereal set" shows "(open S \<and> closed S) = (S = {} \<or> S = UNIV)"
test_comp_test_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> top \<sqinter> y \<sqinter> (z \<sqinter> top) = x \<sqinter> top \<sqinter> y \<sqinter> (y \<sqinter> (z \<sqinter> top))"
comp_wb_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'b \<Longrightarrow> 'c" assumes "wb_lens x"   and "wb_lens y" shows "wb_lens (x ;\<^sub>L y)"
conversep_le_swap: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(r \<le> s\<inverse>\<inverse>) = (r\<inverse>\<inverse> \<le> s)"
reachable_snapshot_inv_dequeue_Mark: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and r :: "'g"   and fl :: "bool"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers p s = mw_Mark r fl # ws"   and "mut_m.reachable_snapshot_inv m s"   and "valid_W_inv s" shows "mut_m.reachable_snapshot_inv m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (obj_mark_update (\<lambda>_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))"
one_cblinfun_def: shows "1 \<equiv> CBlinfun one_dim_iso"
concat_in_hull: fixes ws :: "'b list list"   and G :: "'b list set" assumes "set ws \<subseteq> G" shows "concat ws \<in> \<langle>G\<rangle>"
fun_mult_closed: fixes R :: "('a, 'b) ring_scheme"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "f \<otimes>\<^bsub>function_ring S R\<^esub> g \<in> carrier (function_ring S R)"
ground_subst_\<eta>: shows "is_ground_subst \<eta>"
the_coext_prop_2: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' (C.cod f__) f" shows "f = \<epsilon>o (C.cod f__) \<cdot>\<^sub>C F (x'\<epsilon>.the_coext x' f)"
intro: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "UP_cring R"
poly_mapping_eqI: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and g :: "'c \<Rightarrow>\<^sub>0 'd" assumes "\<And>k. poly_mapping.lookup f k = poly_mapping.lookup g k" shows "f = g"
vwelldefined_alt_def_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vwelldefined (Vabs cs \<Gamma>) = (fmpred (\<lambda>_. vwelldefined) \<Gamma> \<and> list_all (\<lambda>(uu_, t). consts t |\<subseteq>| all_consts) cs)"