hcju/isabelleps huggingface dataset:
type_equiv_trans: fixes t :: "'BExp set"   and P :: "'BExp set"   and t' :: "'BExp set"   and P' :: "'BExp set"   and t'' :: "'BExp set" assumes "t =:\<^sub>P t'"   and "t' =:\<^sub>P' t''"   and "P \<turnstile> P'" shows "t =:\<^sub>P t''"
point_subset_in_group: fixes G :: "'a set"   and ps :: "'a set" assumes "G \<in> \<G>"   and "ps \<subseteq> G" shows "ps \<subseteq> \<V>"
Subgroup_imp_Subring: fixes H :: "'g set" assumes "ActingGroup.Subgroup H" shows "Ring1 (aezfun_setspan H) \<and> aezfun_setspan H \<subseteq> ActingGroup.group_ring"
mwb_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "mwb_lens x \<equiv> weak_lens x \<and> mwb_lens_axioms x"
rep_eq: fixes x :: "'a \<Longrightarrow> 'b"   and xa :: "('c, 'b) uexpr" shows "(x \<sharp> xa) = (\<forall>b v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
RMWReadOnly: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "\<not> cond (\<theta>(t \<mapsto> m a))"   and "a \<in> \<O> \<or> a \<in> dom \<S>"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> (\<R>s ! j) a \<noteq> Some False" shows "\<O>s,\<R>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
enum_ind_def: fixes x :: "'a" shows "enum_ind x = first_ind enum_class.enum x 0"
EF_fp: fixes p :: "'b set" shows "\<^bold>E\<^bold>F p = p \<union> \<^bold>E\<^bold>X \<^bold>E\<^bold>F p"
assms: shows "\<forall>l. wf\<^sub>l\<^sub>s\<^sub>t V (proj l A)"
aext_not: fixes P :: "'c upred"   and x :: "'c \<Longrightarrow> 'b" shows "(\<not> P) \<oplus>\<^sub>p x = (\<not> P \<oplus>\<^sub>p x)"
union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "\<Sum>\<^sub># (M + N) = \<Sum>\<^sub># M + \<Sum>\<^sub># N"
r_code_constn_prim: fixes n :: "nat" shows "prim_recfn 1 (r_code_constn n)"
comp_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and C :: "'s set"   and G :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)"   and "arr (mkArr B C G)" shows "mkArr B C G \<cdot> mkArr A B F = mkArr A C (G \<circ> F)"
count_ne_remove: fixes x :: "'c"   and t :: "'c"   and S :: "'c multiset" assumes "x \<noteq> t" shows "count S x = count (S - {#t#}) x"
dvd_imp_gcd_dvd_gcd: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b dvd c" shows "some_gcd a b dvd some_gcd a c"
upper_truncate_ivl: fixes p :: "nat"   and x :: "real interval" shows "upper (truncate_ivl p x) = truncate_up p (upper x)"
invar_insert_tries: fixes t :: "('c, 'd list) trie"   and as :: "'c list"   and v :: "'d" assumes "invar_trie t" shows "invar_trie (insert_tries as v t)"
prefixes_Suc_last: fixes j :: "nat"   and t :: "nat" shows "e_nth (local.prefixes j (Suc t)) (e_length (local.prefixes j (Suc t)) - 1) = 1"
set_minus_plus: fixes a :: "'a"   and b :: "'a"   and C :: "'a set" shows "(a - b \<in> C) = (a \<in> b +o C)"
NetsCollectedSepaux: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "first_bothNet x \<noteq> first_bothNet y"   and "\<not> member DenyAll y \<and> noDenyAll z"   and "(\<forall>a\<in>set z. first_bothNet x \<noteq> first_bothNet a) \<and> NetsCollected (y # z)"   and "NetsCollected (separate (y # z))"   and "first_bothNet x \<noteq> firstList (separate (y # z))"   and "a \<in> set (separate (y # z))" shows "first_bothNet x \<noteq> first_bothNet a"
is_partial_heap_smaller_front: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and m :: "int"   and r :: "int"   and m' :: "int" assumes "is_partial_heap cmp heap l m r"   and "m \<le> m'" shows "is_partial_heap cmp heap l m' r"
wf_len: shows "wf {(x, y). length x < length y}"
2: shows "(f \<cdot>\<^sub>C \<epsilon> (C.dom f)) \<cdot>\<^sub>C C.inv (\<epsilon> (C.dom f)) = f"
init_atoms_imply_bounds_index: fixes t :: "(nat \<times> linear_poly) list" shows "{} \<Turnstile>\<^sub>i \<B>\<I> (init t)"
boundU: assumes "w \<noteq> NEG_INF"   and "w \<noteq> POS_INF" shows "sint (Rep_bword bw) < sint POS_INF"
i3: shows "ideal_generated {a1', c1} = ideal_generated {1::'a}"
closed_closing_subst: fixes e :: "db"   and \<Gamma> :: "nat \<Rightarrow> db" assumes "\<forall>v. freedb e v \<longrightarrow> closed (\<Gamma> v)" shows "closed (closing_subst e \<Gamma> 0)"
distinct_15: fixes x51 :: "'b formula"   and x52 :: "'b formula" shows "\<bottom> \<noteq> x51 \<^bold>\<or> x52"
conj_imp: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and r :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(imp_PL (conj_PL p q) r \<in> Cn A) = (imp_PL p (imp_PL q r) \<in> Cn A)"
finite_measure_distr: fixes f :: "'a \<Rightarrow> 'c"   and M' :: "'c measure" assumes "random_variable M' f" shows "finite_measure (distr M M' f)"
l7_21: 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 P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "B \<noteq> D"   and "Cong A B C D"   and "Cong B C D A"   and "Tarski_neutral_dimensionless.Col Bet A P C"   and "Tarski_neutral_dimensionless.Col Bet B P D" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong P A C \<and> Tarski_neutral_dimensionless.Midpoint Bet Cong P B D"
sub_1: shows "\<Union> \<D> \<subseteq> cbox a b"
ccw_system_def: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" shows "ccw_system ccw S \<equiv> ccw_system1235 ccw S \<and> ccw_system4 ccw S"
noteq_def: fixes r :: "'a ref"   and s :: "'b ref" shows "r =!= s = (TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_ref r \<noteq> addr_of_ref s)"
assms_4: shows "Prj\<^sub>1\<^sub>0 \<cdot> h = Prj\<^sub>1\<^sub>0 \<cdot> h'"
set_of_regions: fixes R :: "('d \<Rightarrow> real) set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and v :: "'d \<Rightarrow> real"   and R' :: "('d \<Rightarrow> real) set" assumes "R \<in> {region X I r |I r. valid_region X k I r}"   and "v \<in> R"   and "R' \<in> Succ {region X I r |I r. valid_region X k I r} R"   and "finite X" shows "\<exists>t\<ge>0. [v \<oplus> t]\<^sub>{region X I r |I r. valid_region X k I r} = R'"
fun_app_def: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "f $ x \<equiv> f x"
inject_2: fixes x6 :: "ty"   and y6 :: "ty" shows "(x6\<lfloor>\<rceil> = y6\<lfloor>\<rceil>) = (x6 = y6)"
negate_valid: fixes x :: "trilean" shows "(\<not>? x \<noteq> invalid) = (x \<noteq> invalid)"
assms: shows "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
mapping_linorder_matrix_unique: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.mapping f = (\<forall>i. \<exists>!j. f (i, j) = top \<and> (\<forall>k. j \<noteq> k \<longrightarrow> f (i, k) = bot))"
idom_potent_nonzero: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Idomain R"   and "x \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x^\<^bsup>R n\<^esup> \<noteq> \<zero>\<^bsub>R\<^esub>"
rbt_lookup_rbt_delete: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt"   and k :: "'a" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_delete less k t) = ord.rbt_lookup less t |` (- {k})"
Cons_eq_append_conv: fixes x :: "'e"   and xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
distincts_Cons': fixes xs :: "'b list"   and xss :: "'b list list" shows "distincts (xs # xss) = (xs \<noteq> [] \<and> distinct xs \<and> distincts xss \<and> (\<forall>ys\<in>set xss. set xs \<inter> set ys = {}))"
n_subsets: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "card {B. B \<subseteq> A \<and> card B = k} = card A choose k"
assms_1: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"
cr_spmf_option_def: shows "??.Misc_CryptHOL.option_to_spmf.cr_option \<equiv> \<lambda>x y. x = return_pmf y"
order_def: fixes f :: "'a perm" shows "order f = card \<circ> orbit f"
rel_funD: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
stream_space_eq_sstart: fixes S :: "'c set"   and M :: "'c stream measure"   and N :: "'c stream measure" assumes "countable S"   and "prob_space M"   and "prob_space N"   and "AE x in M. x \<in> streams S"   and "AE x in N. x \<in> streams S"   and "sets M = sets (stream_space (count_space UNIV))"   and "sets N = sets (stream_space (count_space UNIV))"   and "\<And>xs. \<lbrakk>xs \<noteq> []; xs \<in> lists S\<rbrakk> \<Longrightarrow> emeasure M (sstart S xs) = emeasure N (sstart S xs)" shows "M = N"
asym: fixes a :: "'a"   and b :: "'a" assumes "a \<sqsubset> b"   and "b \<sqsubset> a" shows "False"
simps_3: fixes e :: "exp"   and x :: "Vars.var" shows "isVal (App e x) = False"
step_push: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and d :: "'dyn"   and H :: "'henv"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IPush d" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (d # \<Sigma>) # st))"
fps_mult_fps_X_commute: fixes a :: "'b fps" shows "fps_X * a = a * fps_X"
image_payer_dc_crypto: shows "payer ` dc_crypto = Some ` {0..<n}"
Max_ge_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Max less_eq A) = (\<exists>a\<in>A. less_eq x a)"
AE_completion_iff: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" shows "almost_everywhere (completion M) P = almost_everywhere M P"
cases: fixes a1 :: "('a, 'b, 'c) var0"   and a2 :: "('a, 'b, 'c) var0" assumes "a1 \<sim> a2" obtains x :: "('a, 'b, 'c) var0" where "a1 = x__" and "a2 = x__"   | a :: "'a" and x :: "'b" where "a1 = PMF (return_pmf (a__, xa__))" and "a2 = BPS (bsingleton (a__, xa__))"   | a :: "'a" and x :: "'b" where "a1 = BPS (bsingleton (a, x))" and "a2 = PMF (return_pmf (a, x))"
rd_ind_wait: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (\<not> P\<^sup>f\<^sub>f \<turnstile> P\<^sup>t\<^sub>f) = R (\<not> (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> \<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>))"
step_4: shows "\<I> \<turnstile>res res \<surd>"
walk_to_path: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "walk xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "\<exists>ys. v \<leadsto>ys\<leadsto> w \<and> set ys \<subseteq> set xs"
cEnum_option_def: shows "CENUM('a option) = (case ID CENUM('a) of None \<Rightarrow> None | Some (enum_a, enum_all_a, enum_ex_a) \<Rightarrow> Some (None # map Some enum_a, \<lambda>P. P None \<and> enum_all_a (\<lambda>x. P (Some x)), \<lambda>P. P None \<or> enum_ex_a (\<lambda>x. P (Some x))))"
winding_number_prop_def: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and e :: "real"   and p :: "real \<Rightarrow> complex"   and n :: "complex" shows "winding_number_prop \<gamma> z e p n \<equiv> valid_path p \<and> z \<notin> path_image p \<and> pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * n"
trans_fm_graph_aux_def: shows "trans_fm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>trans_fm_aux e t u. x1 = (e, t IN u) \<and> x2 = DBMem (trans_tm e t) (trans_tm e u)) \<or> (\<exists>trans_fm_aux e t u. x1 = (e, t EQ u) \<and> x2 = DBEq (trans_tm e t) (trans_tm e u)) \<or> (\<exists>trans_fm_aux e A B. x1 = (e, A OR B) \<and> x2 = DBDisj (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, A)) (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, B)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, A)) \<and> p (e, B) (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, B))) \<or> (\<exists>trans_fm_aux e A. x1 = (e, Neg A) \<and> x2 = DBNeg (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, A)) \<and> p (e, A) (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (e, A))) \<or> (\<exists>trans_fm_aux k e A. x1 = (e, SyntaxN.Ex k A) \<and> x2 = DBEx (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (k # e, A)) \<and> atom k \<sharp> e \<and> p (k # e, A) (((\<lambda>uu_. uu_) \<circ> trans_fm_aux) (k # e, A))))"
sum_of_2_squares_nat_power: fixes m :: "nat"   and n :: "nat" assumes "sum_of_2_squares_nat m" shows "sum_of_2_squares_nat (m ^ n)"
span_Int_subset: fixes A :: "'b set"   and B :: "'b set" shows "local.span (A \<inter> B) \<subseteq> local.span A \<inter> local.span B"
conjugate_dist_mul: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (times a b) = times (conjugate a) (conjugate b)"
j_decisive_xy: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b"   and u :: "'a"   and v :: "'a" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y"   and "hasw [u, v] {x, y, z}" shows "decisive swf A Is {j} u v"
Sup_bot_conv_2: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup A) = (\<forall>x\<in>A. x = bot)"
eucl_of_list_inner: fixes i :: "'e"   and xs :: "real list" assumes "i \<in> Basis"   and "length xs = DIM('e)" shows "eucl_of_list xs \<bullet> i = xs ! index Basis_list i"
orderE: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" obtains "a = min a b"
real_sqrt_le_0_iff: fixes x :: "real" shows "(sqrt x \<le> 0) = (x \<le> 0)"
fcard_seteq: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B"   and "fcard B \<le> fcard A" shows "A = B"
emeasure_T_observation_Cons: fixes s :: "'s"   and o\<^sub>0 :: "'t"   and o\<^sub>1 :: "'t"   and os :: "'t list" shows "emeasure (T (s, o\<^sub>0)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = o\<^sub>1 # os} = \<integral>\<^sup>+ t. ennreal (pmf (\<O> t) o\<^sub>1) * emeasure (T (t, o\<^sub>1)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>xs \<omega>'. \<omega> = xs @- \<omega>' \<and> map snd xs = os} \<partial>measure_pmf (\<K> s)"
has_vderiv_on_sq_mtx_affine: fixes t\<^sub>0 :: "real"   and A :: "'b sq_mtx"   and s :: "(real, 'b) vec"   and B :: "(real, 'b) vec"   and t :: "real" shows " D (\<lambda>t. exp ((1 * (t - t\<^sub>0)) *\<^sub>R A) *\<^sub>V s + exp ((1 * (t - t\<^sub>0)) *\<^sub>R A) *\<^sub>V (\<integral>\<^sub>t\<^sub>0\<^sup>t exp ((- 1 * (\<tau> - t\<^sub>0)) *\<^sub>R A) *\<^sub>V B\<partial>\<tau>)) = (\<lambda>t. A *\<^sub>V (exp ((1 * (t - t\<^sub>0)) *\<^sub>R A) *\<^sub>V s + exp ((1 * (t - t\<^sub>0)) *\<^sub>R A) *\<^sub>V (\<integral>\<^sub>t\<^sub>0\<^sup>t exp ((- 1 * (\<tau> - t\<^sub>0)) *\<^sub>R A) *\<^sub>V B\<partial>\<tau>)) + B) on {t\<^sub>0--t}"
Rep_ttree_inverse: fixes x :: "'b ttree" shows "Abs_ttree (Rep_ttree x) = x"
double_coreflexive_complement: fixes 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 one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_allegory uminus inf less_eq less sup bot top one times conv" shows "inf (uminus (inf (uminus x) one)) one = inf (uminus (uminus x)) one"
semiring_norm_26: shows "Num.BitM num.One = num.One"
if_split_eq1: fixes Q :: "bool"   and x :: "'d"   and y :: "'d"   and b :: "'d" shows "((if Q then x else y) = b) = ((Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b))"
lemma_3_reachability_s: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "card (reachable_s PROB s) - 1 < length as" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> length as' < length as \<and> subseq as' as"
agree_iff_subset: fixes G :: "('d, 'e) labeled_graph"   and X :: "('d, 'f) labeled_graph"   and f :: "('e \<times> 'f) set"   and g :: "('e \<times> 'f) set" assumes "graph_homomorphism G X f"   and "univalent g" shows "agree_on G f g = (f \<subseteq> g)"
bit_take_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (take_bit m a) n = (n < m \<and> bit a n)"
the_trivial_next_args: shows "the_trivial (snd next) next_args = Some v"
cond_nsg: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H "   and "\<forall>a\<in>carrier G. \<forall>h\<in>H. a \<cdot>\<^bsub>G\<^esub> h \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a \<in> H" shows "G \<triangleright> H"
le_imp_less_Suc: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "m < Suc n"
z_property_def: fixes bullet :: "'a \<Rightarrow> 'a"   and R :: "('a \<times> 'a) set" shows "z_property bullet R \<equiv> \<forall>a b. (a, b) \<in> R \<longrightarrow> (b, bullet a) \<in> R\<^sup>* \<and> (bullet a, bullet b) \<in> R\<^sup>*"
surjective: fixes r :: "'a args_scheme" shows "r = \<lparr>args_ivl = args_ivl r, args_n = args_n r, args_L = args_L r, args_R = args_R r, args_pos = args_pos r, \<dots> = args.more r\<rparr>"
xsj: shows "xs ! index xs y = as ! (index xs y - index xs x - 1)"
strand_sem_c_imp_ineq_model: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<lbrakk>M; S\<rbrakk>\<^sub>c \<I>"   and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "ineq_model \<I> X F"
assms_4: shows "ideal_generated {a, b} = ideal_generated {d'}"
\<K>_power: fixes k :: "nat" shows "\<K> ^ k = (\<Sum>w | length w = k \<and> set w \<subseteq> L. 1 / b ^ cw_len_concat w)"
Pi_closed: fixes ct :: "'s \<Rightarrow> 's pmf"   and s :: "'s"   and t :: "'s" assumes "ct \<in> Pi S K"   and "s \<in> S"   and "t \<in> set_pmf (ct s)" shows "t \<in> S"
IH1: fixes p :: "net_tree" assumes "s1 \<in> init (pnet np p)"   and "i \<in> net_ips s1"   and "wf_net_tree p" shows "the (fst (netgmap sr s1) i) \<in> (fst \<circ> sr) ` init (np i)"
reads_writes_separate_forwards: fixes SR :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and getter :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x"   and SW :: "('x, 'aa, 'ab) prog set"   and setter :: "('x, 'aa, 'ac) prog"   and x :: "'z" assumes "reads SR getter h h'"   and "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "h \<turnstile> getter \<rightarrow>\<^sub>r x"   and "\<And>h h'. \<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')" shows "h' \<turnstile> getter \<rightarrow>\<^sub>r x"
abs_summable_Re: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set" assumes "f abs_summable_on A" shows "(\<lambda>x. Re (f x)) abs_summable_on A"
groups_finite: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set" assumes "group_design point_set block_collection groups" shows "finite groups"
fps_compose_uminus': fixes f :: "'a fps" shows "f oo - fps_X = Abs_fps (\<lambda>n. (- (1::'a)) ^ n * f $ n)"
cdj: shows "k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
simps_1: fixes l :: "'c extended"   and uu :: "'c"   and u :: "'c extended"   and uv :: "'d" shows "deps (Insert (InsertMessage l uu u uv)) = extended_to_set l \<union> extended_to_set u"
circline_set_unit_circle_I: fixes z :: "complex" assumes "cmod z = 1" shows "of_complex z \<in> circline_set unit_circle"
relative_homology_group_def: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "relative_homology_group p X S \<equiv> if p < 0 then singleton_group undefined else relcycle_group (nat p) X S Mod singular_relboundary_set (nat p) X S"
Sup_bot_conv_1: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup A = bot) = (\<forall>x\<in>A. x = bot)"
except_Int: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set"   and V :: "'a set" assumes "keys p \<subseteq> U \<union> V" shows "except p (U \<inter> V) = except p U + except p V"
rev_is_Nil_conv: fixes xs :: "'a list" shows "(rev xs = []) = (xs = [])"
unobtainable_if: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r' :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and r :: "nat \<Rightarrow>f value option" assumes "\<not> visits s e s' r' t" shows "\<not> obtains s r e s' r' t"
compl_inf_bot: fixes x :: "'a" shows "- x \<cdot> x = 0"
absolutely_integrable_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "(f absolutely_integrable_on S) = (f integrable_on S \<and> (\<lambda>x. norm (f x)) integrable_on S)"