hcju/isabelleps huggingface dataset:
a_\<beta>_ex: fixes p :: "fm"   and l :: "int"   and bbs :: "bool list"   and bs :: "int list" assumes "iszlfm p"   and "d_\<beta> p l"   and "0 < l" shows "(\<exists>x. l dvd x \<and> Ifm bbs (x # bs) (a_\<beta> p l)) = (\<exists>x. Ifm bbs (x # bs) p)"
kruskal_vc_1: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'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 uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and g :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star"   and "conv g = g" shows "m_kleene_algebra.kruskal_spanning_invariant inf less_eq uminus one times conv star bot g g"
Gromov_completion_cases: fixes z :: "'a Gromov_completion" obtains x :: "'a" where "z = to_Gromov_completion x"   | "z \<in> Gromov_boundary"
T2: fixes w :: "'b"   and w' :: "'b"   and \<theta> :: "'b"   and u :: "'b"   and \<theta>' :: "'b"   and \<beta> :: "'b" assumes "D.ide w"   and "D.ide w'"   and "\<guillemotleft>\<theta> : F f \<star>\<^sub>D w \<Rightarrow>\<^sub>D u\<guillemotright>"   and "\<guillemotleft>\<theta>' : F f \<star>\<^sub>D w' \<Rightarrow>\<^sub>D u\<guillemotright>"   and "\<guillemotleft>\<beta> : F g \<star>\<^sub>D w \<Rightarrow>\<^sub>D F g \<star>\<^sub>D w'\<guillemotright>"   and "(F r \<star>\<^sub>D \<theta>) \<cdot>\<^sub>D \<a>\<^sub>D[F r, F f, w] \<cdot>\<^sub>D (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> \<star>\<^sub>D w) = ((F r \<star>\<^sub>D \<theta>') \<cdot>\<^sub>D \<a>\<^sub>D[F r, F f, w'] \<cdot>\<^sub>D (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> \<star>\<^sub>D w')) \<cdot>\<^sub>D \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow>\<^sub>D w'\<guillemotright> \<and> \<beta> = F g \<star>\<^sub>D \<gamma> \<and> \<theta> = \<theta>' \<cdot>\<^sub>D (F f \<star>\<^sub>D \<gamma>)"
zero_index: fixes i :: "'d" shows "0 $ i = (0::'c)"
dim_eq_card_independent: fixes B :: "('a, 'b) vec set" assumes "vec.independent B" shows "vec.dim B = card B"
atLeastLessThan_inj_2: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "{a..<b} = {c..<d}"   and "a < b"   and "c < d" shows "b = d"
max_min_distrib2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linorder less_eq less" shows "ord.max less_eq a (ord.min less_eq b c) = ord.min less_eq (ord.max less_eq a b) (ord.max less_eq a c)"
graph_homomorphism_composes: fixes a :: "('a, 'b) labeled_graph"   and b :: "('a, 'c) labeled_graph"   and x :: "('b \<times> 'c) set"   and c :: "('a, 'd) labeled_graph"   and y :: "('c \<times> 'd) set" assumes "graph_homomorphism a b x"   and "graph_homomorphism b c y" shows "graph_homomorphism a c (x O y)"
su2_2: shows "flow0 x su2 \<in> {a<--<b}"
simps_3: fixes v1 :: "'c"   and l1 :: "'c Tree"   and r1 :: "'c Tree"   and v2 :: "'c"   and l2 :: "'c Tree"   and r2 :: "'c Tree" shows "merge (T v1 l1 r1) (T v2 l2 r2) = (if v2 \<le> v1 then T v1 (merge l1 (T v2 l2 r2)) r1 else T v2 (merge l2 (T v1 l1 r1)) r2)"
order_mult: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p * q \<noteq> 0" shows "order a (p * q) = order a p + order a q"
isCont_power: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "isCont f a" shows "isCont (\<lambda>x. f x ^ n) a"
assms_2: shows " t1 \<longrightarrow>\<^sub>\<beta> t2"
assms_4: shows "Collect J0.arr \<inter> Collect J1.arr = {}"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "Def_Init_Exp.bval b s = Some False" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
elims_2: fixes x :: "v" assumes "is_cupcake_value x" obtains uu_ :: "char list" and uv_ :: "char list" and vs :: "v list" where "x = Conv (Some (uu_, TypeId (Short uv_))) vs" and "list_all is_cupcake_value vs"   | env :: "v sem_env" and uw_ :: "char list" and e :: "exp0" where "x = Closure envb__ uw_ e" and "is_cupcake_exp e \<and> is_cupcake_all_env envb__"   | env :: "v sem_env" and es :: "(char list \<times> char list \<times> exp0) list" and ux_ :: "char list" where "x = Recclosure env es ux_" and "list_all (\<lambda>(_, uu_, y). is_cupcake_exp y) es \<and> is_cupcake_all_env env"
lm040: fixes P :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set"   and Q :: "('a \<times> 'b) set"   and Z :: "'a set" shows "Outside' (X \<union> Y) P \<inter> Q || X = {} \<and> Outside' X P \<inter> Q || (X \<inter> Z) = {}"
deg_sect_zero: fixes X :: "'b set" shows "deg_sect X 0 = {0}"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<And>x. x \<in> carrier L \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> x"   and "\<And>x y. \<lbrakk>x \<sqsubseteq>\<^bsub>L\<^esub> y; y \<sqsubseteq>\<^bsub>L\<^esub> x; x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> x .=\<^bsub>L\<^esub> y"   and "\<And>x y z. \<lbrakk>x \<sqsubseteq>\<^bsub>L\<^esub> y; y \<sqsubseteq>\<^bsub>L\<^esub> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> z"   and "\<And>x y z w. \<lbrakk>x .=\<^bsub>L\<^esub> y; z .=\<^bsub>L\<^esub> w; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L; w \<in> carrier L\<rbrakk> \<Longrightarrow> (x \<sqsubseteq>\<^bsub>L\<^esub> z) = (y \<sqsubseteq>\<^bsub>L\<^esub> w)" shows "weak_partial_order_axioms L"
mem_Complement: fixes x :: "'b"   and X :: "'b set" shows "(x \<in> Complement X) = (x \<notin> X)"
divide_pos_pos: 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 x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less zero y" shows "less zero (divide x y)"
Bisimulation_Invariant_axioms: shows "Bisimulation_Invariant A B (\<sim>) PA PB"
conjugate_id: fixes conjugate :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.conjugate conjugate" shows "conjugate (conjugate a) = a"
cop_F_increasing: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d set"   and ds :: "'e set" assumes "Contracts Xd Xh Pd Ch" shows "x \<subseteq> Contracts.cop_F Pd Ch ds x"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) Cop.H Cop.\<a> Cop.\<i> Cop.src Cop.trg g f \<eta> \<epsilon>"
col_cong_bet: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Col A B D"   and "Cong A B C D"   and "Bet A C B" shows "Bet C A D \<or> Bet C B D"
filterlim_at_right_to_top: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" shows "filterlim f F (at_right 0) = (LIM x at_top. f (inverse x) :> F)"
plus_iarray_def: fixes A :: "'b iarray"   and B :: "'b iarray" shows "A + B = IArray.of_fun (\<lambda>n. A !! n + B !! n) (IArray.length A)"
abs_eq: shows "{#} = Abs_multiset (\<lambda>a. 0)"
intros_2: fixes c1 :: "'a com"   and s :: "'a"   and s'' :: "'a"   and c2 :: "'a com"   and s' :: "'a" assumes "Sem c1 s s''"   and "Sem c2 s'' s'" shows "Sem (Seq c1 c2) s s'"
presheaf_of_rings_axioms: shows "presheaf_of_rings Spec is_zariski_open local.im_sheaf im_sheaf_morphisms \<O>b (\<lambda>V. add_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. mult_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. zero_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. one_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V))"
assms: shows "\<forall>\<sigma>. small P \<sigma> = {} \<or> (\<exists>\<sigma>'. small P \<sigma> = {\<sigma>'})"
c_max_def: fixes S :: "('e, 'f) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "c_max S c = (if {j. j \<le> fst c \<and> snd c j \<noteq> \<zero>\<^bsub>S\<^esub>} = {} then 0 else n_max {j. j \<le> fst c \<and> snd c j \<noteq> \<zero>\<^bsub>S\<^esub>})"
finite_resources_suffice: fixes f :: "C \<rightarrow> 'a" assumes "f\<cdot>C\<^sup>\<infinity> \<noteq> \<bottom>" obtains n :: "nat" where "f\<cdot>C\<^bsup>n\<^esup> \<noteq> \<bottom>"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Left.in_hom f a b" obtains "Left.arr f" and "Left.dom f = a" and "Left.cod f = b"
lfinite_code_2: fixes x :: "'a"   and xs :: "'a llist" shows "lfinite (LCons x xs) = lfinite xs"
finprod_one_eqI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<one>" shows "finprod G f A = \<one>"
in_set_pmf_of_setD: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> set_pmf (pmf_of_set A)"   and "finite A"   and "A \<noteq> {}" shows "x \<in> A"
g2_im1: shows "gs2.gso (i - 1) = fs.gs.gso i + fs.gs.\<mu> i (i - 1) \<cdot>\<^sub>v fs.gs.gso (i - 1)"
pause_state_def: fixes pause :: "'k \<Rightarrow> ('l \<Rightarrow> 'j) \<Rightarrow> 'j"   and out :: "'k"   and c :: "'l \<Rightarrow> ('i, 'j) stateT" shows "pause_state pause out c = StateT (\<lambda>s. pause out (\<lambda>i. run_state (c i) s))"
hyps_2: assumes "d < length b \<Longrightarrow> p' ! d = b ! d" shows "p' \<in> grid b ds"
keys_add: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" assumes "keys f \<inter> keys g = {}" shows "keys f \<union> keys g = keys (f + g)"
map_ident: fixes t :: "('d, 'e, 'f) generat" shows "map_generat (\<lambda>x. x) (\<lambda>x. x) (\<lambda>x. x) t = t"
r_neg2: fixes x :: "'a"   and y :: "'a" shows "x + (- x + y) = y"
fst: assumes "Opair a b = Opair c d" shows "a = c"
unsigned_ucast_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'d word" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (ucast w) = take_bit LENGTH('c) (semiring_1.unsigned one plus zero w)"
sran_def: fixes m :: "'d \<Rightarrow> 'c set" shows "sran m = {b. \<exists>a. b \<in> m a}"
assms: shows "\<bar>x\<bar> \<noteq> \<infinity>"
P_def: shows "P = [:(cmod z)\<^sup>2, - 2 * Re z, 1:]"
conc_empty: fixes w :: "nat \<Rightarrow> 'b" shows "[] \<frown> w = w"
x_def: shows "x = (t1 * t2 - int (D * u1 * u2)) div k"
path_conc: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and la :: "'b list"   and v :: "'b"   and lb :: "'b list"   and w :: "'b" assumes "path E u la v"   and "path E v lb w" shows "path E u (la @ lb) w"
foldr_conv_foldl: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" shows "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
uniform_limit_sequentially_iff: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c" shows "uniform_limit S f l sequentially = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. dist (f n x) (l x) < e)"
generate_topology_on_coarsest: fixes T :: "'c set \<Rightarrow> bool"   and S :: "'c set set"   and s0 :: "'c set" assumes "istopology T"   and "\<And>s. s \<in> S \<Longrightarrow> T s"   and "generate_topology_on S s0" shows "T s0"
constant_at_ide_is_constant: fixes x :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<chi> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "local.cone x \<chi>"   and "A.ide a" shows "at a (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) x) = constant_functor.map (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (A_B.Map x a)"
r\<^sub>\<A>_def: shows "r\<^sub>\<A> = run\<^sub>t (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) w"
add_node_def: fixes v :: "'b"   and G :: "'b graph" shows "add_node v G \<equiv> \<lparr>nodes = {v} \<union> nodes G, edges = edges G\<rparr>"
word_le_minus_one_leq: fixes x :: "'a word"   and y :: "'a word" assumes "x < y" shows "x \<le> y - 1"
R_def: shows "R = (\<lambda>x. integral {1..x} (\<lambda>t. M t / t))"
finite_measure_axioms: fixes M :: "'a pmf" shows "finite_measure (measure_pmf M)"
separable: fixes S :: "'a set" obtains T :: "'a set" where "countable T" and "T \<subseteq> S" and "S \<subseteq> closure T"
pos_def: fixes ts :: "traffic" shows "pos ts \<equiv> fst (Rep_traffic ts)"
Gamma_hurwitz_zeta_aux_integrable: fixes z :: "complex"   and a :: "real"   and n :: "nat" assumes "0 < Re z"   and "0 < a" shows "complex_set_integrable lebesgue {0<..} (\<lambda>s. complex_of_real s powr (z - 1) / complex_of_real (exp ((real n + a) * s)))"
stake_shift_stake_shift: fixes i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and j :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b" shows "stake i \<sigma> @- \<sigma>(i \<rightarrow> j) @- \<beta> = stake (i + j) \<sigma> @- \<beta>"
Pow_def: fixes A :: "'c set" shows "Pow A = {B. B \<subseteq> A}"
closed_def: fixes S :: "'a set" shows "closed S = open (- S)"
less_eq_bot_Bot: fixes x :: "'b bot" shows "bot.Bot \<le> x"
VDM_validn_valid: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "\<forall>n.  \<Turnstile>\<^sub>n c : A " shows " \<Turnstile> c : A "
verit_la_disequality: fixes a :: "'d"   and b :: "'d" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
seq_comp_secure: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process" assumes "secure_termination I D"   and "ref_union_closed P"   and "sequential P"   and "secure P I D"   and "secure Q I D" shows "secure (P ; Q) I D"
p_def: shows "p = (LEAST n. xs ! n \<noteq> (0::'a))"
approx_hrabs_zero_cancel: fixes x :: "real star" assumes "\<bar>x\<bar> \<approx> 0" shows "x \<approx> 0"
sym_iff_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "sympartp (\<sqsubset>) x y = (x = y \<and> y \<sqsubset> y)"
fds_shift_integral: fixes b :: "'b"   and a :: "'b"   and f :: "'b fds" shows "fds_shift b (fds_integral a f) = fds_integral a (fds_shift b f)"
spec\<^sub>4_def: fixes p :: "prog" shows "spec\<^sub>4 p \<equiv> para p = [''x'', ''y''] \<and> (\<exists>e\<^sub>1 e\<^sub>2. body p = Add e\<^sub>1 e\<^sub>2 \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>1 \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>2 \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>1 = Some x) \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>2 = Some (2 * x + 2 * y)) \<and> coste e\<^sub>1 + coste e\<^sub>2 \<le> 2)"
wadjust_loop_check_nonempty: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and b :: "cell list" assumes "wadjust_loop_check m rs (c, b)" shows "c \<noteq> []"
OFSC_def: 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 C :: "'a"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.OFSC Bet Cong A B C D A' B' C' D' \<equiv> Bet A B C \<and> Bet A' B' C' \<and> Cong A B A' B' \<and> Cong B C B' C' \<and> Cong A D A' D' \<and> Cong B D B' D'"
iNext_iEx_iff_singleton: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P = (\<diamond> t {inext t0 I}. P t)"
less_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "(a < b) = flt a b"
fake_StatCh: fixes M :: "atom list"   and T :: "atom set"   and H :: "chmsg set"   and c :: "secprop set"   and A :: "agent"   and B :: "agent"   and U :: "fid_t set" assumes "set M \<subseteq> extr T H"   and "auth \<notin> c \<or> A \<in> bad \<or> B \<in> bad" shows "StatCh c A B (Msg M) \<in> fake T U H"
root_pref_cancel: fixes t :: "'b list"   and m :: "nat"   and y :: "'b list"   and k :: "nat" assumes "t \<^sup>@ m \<cdot> y = t \<^sup>@ k" shows "y = t \<^sup>@ (k - m)"
nprovRules_def: shows "nprovRules \<equiv> {(xa, x). nprovRulesp xa x}"
ext_inject: fixes xpos :: "nat"   and ypos :: "nat"   and more :: "'a"   and xpos' :: "nat"   and ypos' :: "nat"   and more' :: "'a" shows "(\<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = \<lparr>xpos = xpos', ypos = ypos', \<dots> = more'\<rparr>) = (xpos = xpos' \<and> ypos = ypos' \<and> more = more')"
Cnext_solve_tagrel: fixes K\<^sub>1 :: "clock"   and n :: "nat"   and K\<^sub>2 :: "clock"   and R :: "'c tag_const \<times> 'c tag_const \<Rightarrow> bool"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "{\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi>"
tllist_of_llist_LNil: fixes b :: "'b" shows "tllist_of_llist b LNil = TNil b"
product_def: fixes A :: "'g set"   and B :: "'h set" shows "Product_Type.product A B = A \<times> B"
abs_conv_abscissa_leI_weak: fixes d :: "ereal"   and f :: "'b fds" assumes "\<And>x. d < ereal x \<Longrightarrow> fds_abs_converges f (of_real x)" shows "abs_conv_abscissa f \<le> d"
le_zero_eq_hmset: fixes M :: "hmultiset" shows "(M \<le> 0) = (M = 0)"
preserves_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.dom (B.\<alpha>' f) = B.HoVH (B.VVV.dom f)"
projection_sequence: fixes xs :: "'a list"   and X :: "'a set"   and Y :: "'a set" shows "xs \<upharpoonleft> X \<upharpoonleft> Y = xs \<upharpoonleft> (X \<inter> Y)"
sum_rel_simp_2: fixes a :: "'b"   and a' :: "'d"   and Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "((Inr a, Inr a') \<in> \<langle>Rl, Rr\<rangle>sum_rel) = ((a, a') \<in> Rr)"
primitive_irreducible_imp_degree: fixes p :: "'a poly" assumes "primitive p"   and "irreducible p" shows "0 < degree p"
normalize_Lcm: fixes A :: "'a set" shows "normalize (Lcm A) = Lcm A"
lincomb_coordinates: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "module.lincomb (module_vec TYPE('b) n) (\<lambda>u. v $ (THE i. u = unit_vec n i)) (set (unit_vecs n)) = v"
AE_uniform_measureI: fixes A :: "'b set"   and M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "A \<in> sets M"   and "set_almost_everywhere A M P" shows "almost_everywhere (uniform_measure M A) P"
pos_mult_1: fixes c :: "'a"   and b :: "'a" assumes "(0::'a) < c" shows "((0::'a) < c * b) = ((0::'a) < b)"
permutes_conv_has_dom: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" shows "(f permutes S) = (bij f \<and> has_dom f S)"
valid_history_def: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_history program_step ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> program.history_consistent program_step \<theta> (hd_prog p sb) sb"
tm_comp_wf: fixes A :: "(action \<times> nat) list"   and B :: "(action \<times> nat) list" assumes "tm_wf0 A"   and "tm_wf0 B" shows "tm_wf0 (A |+| B)"
phi_equiv_mapping_def: fixes g :: "'g"   and nou\<^sub>1 :: "('val, 'a set) mapping"   and nou\<^sub>2 :: "('val, 'a set) mapping" shows "g \<turnstile> nou\<^sub>1 \<approx>\<^sub>\<phi> nou\<^sub>2 \<equiv> \<forall>v\<in>Mapping.keys (uninst_code.ssa.phidefNodes phis g). (case Mapping.lookup nou\<^sub>1 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x) = (case Mapping.lookup nou\<^sub>2 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x)"
is_bcv_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and A :: "'b set"   and bcv :: "'b list \<Rightarrow> 'b list" shows "is_bcv r T step n A bcv = (\<forall>\<tau>s\<^sub>0\<in>list n A. (\<forall>p<n. bcv \<tau>s\<^sub>0 ! p \<noteq> T) = (\<exists>\<tau>s\<in>list n A. \<tau>s\<^sub>0 [\<sqsubseteq>\<^bsub>r\<^esub>] \<tau>s \<and> wt_step r T step \<tau>s))"
dim_substandard_cart: fixes d :: "'c set" shows "vec.dim {x. \<forall>i. i \<notin> d \<longrightarrow> x $ i = (0::'b)} = card d"
division_filter_def: fixes s :: "'b set" shows "division_filter s = (INF g\<in>{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})"
wt_elim_cases_13: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and v :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (va:=v)\<Colon>T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Inl T'" and "E,dt\<Turnstile>va\<Colon>=T" and "va \<noteq> LVar This" and "E,dt\<Turnstile>v\<Colon>-T'" and "prg E\<turnstile>T'\<preceq>T"
absorb2: fixes b :: "'k multiset"   and a :: "'k multiset" assumes "b \<subseteq># a" shows "a \<inter># b = b"
event_ex_aux: fixes a :: "nat \<Rightarrow> int"   and xa :: "nat \<Rightarrow> bool" assumes "a 0 = 0"   and "\<forall>n. xa (Suc n) = (\<not> xa n)"   and "\<forall>n. xa n \<and> a (Suc n) = a n + 1 \<or> \<not> xa n \<and> a (Suc n) = a n - 1 \<or> a (Suc n) = 0" shows "\<forall>n. (a n = - 1 \<longrightarrow> xa n) \<and> (a n = 1 \<longrightarrow> \<not> xa n) \<and> - 1 \<le> a n \<and> a n \<le> 1"
Infinite_cong: fixes X :: "'c set"   and Y :: "'c set" assumes "X = Y" shows "infinite X = infinite Y"
disjoint_effects_no_effects: fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and vs :: "'c set"   and s :: "('c, 'd) fmap" assumes "\<forall>a. ListMem a as \<longrightarrow> fmdom' (fmrestrict_set vs (snd a)) = {}" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs s"
Gamma_complex_altdef: fixes z :: "complex" shows "Gamma z = (if z \<in> \<int>\<^sub>\<le>\<^sub>0 then 0 else exp (ln_Gamma z))"
linorder_sorted_wrt_exists: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and B :: "'a set" assumes "linorder_on A R"   and "finite B"   and "B \<subseteq> A" shows "\<exists>xs. set xs = B \<and> distinct xs \<and> Linorder_Relations.sorted_wrt R xs"
semiring_hom_axioms: shows "semiring_hom (\<lambda>p. coeff p 0)"
assms_1: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"
inv_begin_step: fixes n :: "nat"   and cf :: "nat \<times> cell list \<times> cell list" assumes "inv_begin n cf"   and "0 < n" shows "inv_begin n (step0 cf tcopy_begin)"
eq_rat_3: fixes a :: "int"   and c :: "int" shows "Fract 0 a = Fract 0 c"
fls_zero_subdegree: shows "fls_subdegree 0 = 0"