hcju/isabelleps huggingface dataset:
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S" shows "\<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S"
disj_cancel_right: fixes x :: "'a" shows "x \<^bold>\<sqinter> \<sim> x = \<zero>"
simps_9: fixes v :: "('a, String.literal) phantom"   and \<sigma> :: "'a basic_type" shows "Enum v \<squnion> \<sigma> = (if Enum v \<le> \<sigma> then \<sigma> else if \<sigma> \<le> Enum v then Enum v else OclAny)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "prefixes (x # xs) = [] # map ((#) x) (prefixes xs)"
simps_1: shows "prefixes \<epsilon> = [\<epsilon>]"
mult_less_cancel_right2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times a c) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
numeral_code_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = (let m = numeral.numeral one plus n in plus m m)"
that: fixes A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "('t \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" assumes "finite A"   and "A \<subseteq> B"   and "\<And>b. q b \<in> punit.dgrad_p_set d m"   and "p = (\<Sum>b\<in>A. q b \<odot> b)" shows "thesis"
insert_def: fixes x :: "'b"   and t :: "('b \<times> color) tree" shows "RBT_Set.insert x t = paint Black (ins x t)"
wordinterval_setminus_set_eq: fixes r1 :: "'a wordinterval"   and r2 :: "'a wordinterval" shows "wordinterval_to_set (wordinterval_setminus r1 r2) = wordinterval_to_set r1 - wordinterval_to_set r2"
intros_2: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "'c set \<Rightarrow> 'c set \<Rightarrow> bool"   and u :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and a :: "'c set"   and b :: "'c set"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "timpl_closurep t TI u"   and "TI a b"   and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s" shows "timpl_closurep t TI s"
polyn_deg_add2: fixes p :: "'a"   and q :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>"   and "q \<in> carrier R"   and "q \<noteq> \<zero>"   and "p \<plusminus> q \<noteq> \<zero>"   and "deg_n R S X p = deg_n R S X q" shows "deg_n R S X (p \<plusminus> q) \<le> deg_n R S X q"
types_preserved_callcl_host_some: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and ves :: "e list"   and cl :: "cl"   and ts :: "t list"   and ts' :: "t list"   and t1s :: "t list"   and t2s :: "t list"   and f :: "host"   and vcs :: "v list"   and n :: "nat"   and m :: "nat"   and s :: "s"   and hs :: "host_state"   and s' :: "s"   and vcs' :: "v list" assumes "\<S>\<bullet>\<C> \<turnstile> ves @ [Callcl cl] : ts _> ts'"   and "cl = Func_host (t1s _> t2s) f"   and "ves = $$* vcs"   and "length vcs = n"   and "length t1s = n"   and "length t2s = m"   and "host_apply s (t1s _> t2s) f vcs hs = Some (s', vcs')"   and "store_typing s \<S>" shows "\<S>\<bullet>\<C> \<turnstile> $$* vcs' : ts _> ts'"
simps_1: fixes P :: "'b memref \<Rightarrow> bool" shows "outstanding_refs P [] = {}"
ordinal_mod_0: fixes x :: "ordinal" shows "x mod 0 = x"
intersect_def: fixes M :: "'b multiset"   and S :: "'b set" shows "M \<inter>s S = {#x \<in># M. x \<in> S#}"
range: fixes X :: "'a set" shows "linear_cf X \<subseteq> X \<inter> Field r"
abel_semigroup_axioms_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "abel_semigroup_axioms f \<equiv> \<forall>a b. f a b = f b a"
reflcl_of_refl_rel: fixes Rel :: "('a \<times> 'a) set" assumes "refl Rel" shows "Rel\<^sup>= = Rel"
hyps: assumes "\<And>x. x \<in> set xs - set ys \<Longrightarrow> \<exists>y. y \<in> set ys \<and> x \<le> y" shows "max_list xs \<le> max_list ys"
assms_2: shows "\<forall>x\<in>A. x < n"
degree_add_le_max: fixes p :: "'a poly"   and q :: "'a poly" shows "degree (p + q) \<le> max (degree p) (degree q)"
Node_2: assumes "wbt r"   and "r \<noteq> \<langle>\<rangle>" shows "2 \<le> (1 - \<alpha>) ^ (height r - 1) * real (size1 r)"
subsumeseq_term_size_emv: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "t \<le>\<cdot> s"   and "num_syms s = num_syms t"   and "num_funs s = num_funs t" shows "emv s t"
pNode_1: fixes bdt1 :: "bdt" assumes "Node lst stp rst \<le> lt"   and "bdt lt var = Some bdt1" shows "\<exists>bdt2. bdt (Node lst stp rst) var = Some bdt2"
liberal_witness_three: fixes x :: "'c"   and y :: "'c"   and v :: "'c"   and A :: "'c set"   and i :: "'d"   and j :: "'d"   and Is :: "'d set" assumes "hasw [x, y, v] A"   and "hasw [i, j] Is" obtains P :: "'d \<Rightarrow> ('c \<times> 'c) set" where "profile A Is P" and "x \<^bsub>(P i)\<^esub>\<prec> y" and "v \<^bsub>(P j)\<^esub>\<prec> x" and "\<forall>i\<in>Is. y \<^bsub>(P i)\<^esub>\<prec> v"
is_regular_one_sheaf_spec: fixes U :: "'a set set" assumes "is_zariski_open U" shows "is_regular (one_sheaf_spec U) U"
ennreal_inverse_eq_top_iff: fixes a :: "ennreal" shows "(inverse a = top) = (a = 0)"
simps_8: fixes Vs :: "char list list"   and e\<^sub>1 :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "compE\<^sub>1 Vs (e\<^sub>1\<bullet>F{D} := e\<^sub>2) = compE\<^sub>1 Vs e\<^sub>1\<bullet>F{D} := compE\<^sub>1 Vs e\<^sub>2"
has_integral_is_0_cbox: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = (0::'d)" shows "(f has_integral (0::'d)) (cbox a b)"
forget: fixes x :: "name"   and t :: "trm"   and t' :: "trm" assumes "x \<sharp> t" shows "t[x::=t'] = t"
big_prod: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'d \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> L F (g x)" shows "(\<lambda>y. \<Prod>x\<in>A. f x y) \<in> L F (\<lambda>y. \<Prod>x\<in>A. g x y)"
empty_sb: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "sb = []"
IH_1: fixes \<sigma>\<^sub>1 :: "nat \<Rightarrow> val"   and \<sigma>\<^sub>2 :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars b \<Longrightarrow> \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "expr_sem \<sigma>\<^sub>1 b = expr_sem \<sigma>\<^sub>2 b"
dvd_0_left_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times zero a = (a = zero)"
typeII_regions_integral: shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
inext_nth_cut_less_conv: fixes I :: "nat set"   and n :: "nat"   and t :: "nat" assumes "I \<rightarrow> n < t" shows "I \<down>< t \<rightarrow> n = I \<rightarrow> n"
compare_self_below_EQ: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b" assumes "class.Ord_linear eq compare" shows "compare\<cdot>x\<cdot>x \<sqsubseteq> EQ"
atoms_nInter: fixes r :: "nat rexp"   and s :: "nat rexp" shows "atoms (nInter r s) \<subseteq> atoms r \<union> atoms s"
a: shows "disj_of_clause a \<^bold>\<or> disj_of_clause b, {#disj_of_clause c \<^bold>\<or> disj_of_clause d. (c, d) \<in># C#} + \<Gamma> \<Rightarrow>\<^sub>n"
simps_43: fixes l :: "Formula.formula list" shows "safe_formula (formula.Ands l) = (let (pos, neg) = partition safe_formula l in pos \<noteq> [] \<and> list_all safe_formula (map remove_neg neg) \<and> \<Union> (set (map fv neg)) \<subseteq> \<Union> (set (map fv pos)))"
list_rel_dropD: fixes a :: "'a list"   and b :: "'b list"   and R :: "('a \<times> 'b) set"   and n :: "nat"   and n' :: "nat" assumes "(a, b) \<in> \<langle>R\<rangle>list_rel"   and "(n, n') \<in> nat_rel" shows "(drop n a, drop n' b) \<in> \<langle>R\<rangle>list_rel"
countable_support: fixes M :: "'b pmf" shows "countable {x. measure_pmf.prob M {x} \<noteq> 0}"
path_ret_ipd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> k \<noteq> local.return"   and "\<pi> n = local.return" obtains l :: "nat" where "k < l" and "l \<le> n" and "\<pi> l = ipd (\<pi> k)"
rep_eq: fixes x :: "'a" shows "blinfun_apply (blinfun_inner_left x) = (\<lambda>y. y \<bullet> x)"
mult_cancel_right1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and b :: "'d" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times b c) = (c = zero \<or> b = one)"
elims_3: fixes x :: "atom" assumes "\<not> is_Less x" obtains v :: "real" and va :: "real list" where "x = Eq v va"
filtered_directed_dual: shows "filtered \<circ> (`) \<partial> = directed"
all_star_eq: fixes P :: "'b star \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>X. P (star_n X))"
lconfluent_confluent: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "wfP R\<inverse>\<inverse>"   and "\<And>a b c. \<lbrakk>R a b; R a c\<rbrakk> \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d" shows "confluent R"
Seq_sound2: fixes x :: "('b com option \<times> 'b) list"   and P :: "'b com"   and Q :: "'b com"   and s :: "'b"   and i :: "nat" assumes "x \<in> cptn"   and "x ! 0 = (Some (Seq P Q), s)"   and "i < length x"   and "fst (x ! i) = Some Q"   and "\<And>j. j < i \<Longrightarrow> fst (x ! j) \<noteq> Some Q" shows "\<exists>xs ys. xs \<in> cp (Some P) s \<and> length xs = Suc i \<and> ys \<in> cp (Some Q) (snd (xs ! i)) \<and> x = map (lift Q) xs @ tl ys"
Abs_norm: fixes x :: "int" shows "word_of_int (take_bit LENGTH('b) x) = word_of_int x"
subcls_asym: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "(subcls1 P)\<^sup>+\<^sup>+ C D" shows "\<not> (subcls1 P)\<^sup>+\<^sup>+ D C"
sop_assocTr: fixes s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and i :: "'d \<Rightarrow> 'd"   and z :: "'d"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and n :: "nat" assumes "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z" shows "\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>add_set f (aug_pm_set z i (s_set R s A)) n. a \<^sub>s\<cdot> (b \<^sub>s\<cdot> x) = a \<cdot>\<^sub>r b \<^sub>s\<cdot> x"
emeasure_finite: fixes s :: "'a"   and A :: "'a stream set" shows "emeasure (DTMC.T s) A \<noteq> \<top>"
span_add: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> local.span S"   and "y \<in> local.span S" shows "x + y \<in> local.span S"
prime_sum_upto_altdef3: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "prime_sum_upto f x = sum_list (map f (primes_upto (nat \<lfloor>x\<rfloor>)))"
preds_split: fixes ets :: "('var, 'val, 'ret, 'pname) edge_kind list"   and ets' :: "('var, 'val, 'ret, 'pname) edge_kind list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "preds (ets @ ets') s = (preds ets s \<and> preds ets' (transfers ets s))"
single_vote_to_RPR_def: fixes A :: "'b set"   and a :: "'b" shows "single_vote_to_RPR A a \<equiv> {(a, x) |x. x \<in> A} \<union> (A - {a}) \<times> (A - {a})"
c_manifold_order_le: fixes l :: "enat" assumes "l \<le> k" shows "c_manifold charts1 l"
wf_append_suffix: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list" assumes "wf\<^sub>s\<^sub>t V S"   and "wfrestrictedvars\<^sub>s\<^sub>t S' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ S')"
simps_2: fixes s :: "nat \<Rightarrow> 'd list"   and k :: "nat" shows "inf_make_untimed1 s (Suc k) = (if Suc k < length (s 0) then s 0 ! Suc k else if s 0 = [] then inf_make_untimed1 (inf_tl (inf_drop (LEAST i. \<forall>j<i. s j = []) s)) k else inf_make_untimed1 (inf_tl s) k)"
ode_set_def: fixes X :: "'c set" shows "ode_set X = mk_safe X \<bind> (\<lambda>_. approx_slp_appr ode_e ode_slp (list_of_eucl ` X))"
subcls_is_class: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D" shows "\<exists>c. class G C = Some c"
y: shows "\<not> is_zero_row_upt_k y (Suc k) (bezout_iterate (interchange_rows A (0::'rows) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a))) (nrows A - Suc 0) (0::'rows) (mod_type_class.from_nat k) bezout)"
G_D: fixes f :: "'a \<Rightarrow> ennreal"   and A :: "'a set" assumes "f \<in> G"   and "A \<in> sets M" shows "\<integral>\<^sup>+ x. f x * indicator A x \<partial>M \<le> emeasure N A"
X_4: assumes "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map cyc_permutations xss)" shows "map hd xss = map hd yss"
ih: assumes "chain R xs" shows "R\<^sup>*\<^sup>* (lhd xs) (llast xs)"
Ide_Diagonalize_Can: fixes t :: "'c local.term" assumes "Can t" shows "Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
Cauchy_theorem_homotopic_loops: fixes S :: "complex set"   and g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "homotopic_loops S g h"   and "open S"   and "f holomorphic_on S"   and "valid_path g"   and "valid_path h" shows "contour_integral g f = contour_integral h f"
induct: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> (('c \<times> 'd) \<times> nat) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "'d"   and a2 :: "(('c \<times> 'd) \<times> nat) tree" assumes "\<And>x y. P x y \<langle>\<rangle>"   and "\<And>x y l a b h r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x y l; cmp x a = GT \<Longrightarrow> P x y r\<rbrakk> \<Longrightarrow> P x y \<langle>l, ((a, b), h), r\<rangle>" shows "P a0 a1 a2"
greaterThanAtMost_iff: fixes i :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "(i \<in> ord.greaterThanAtMost less_eq less l u) = (less l i \<and> less_eq i u)"
x_coord_inv_2: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real"   and x :: "real" assumes "circle i j d" shows "x_coord (x_coord_inv x) = x"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "N i \<in> null_sets M"
p: shows "p = (\<Prod>(a, i)\<in>as. a ^ i) * (\<Prod>(a, i)\<in>as. a)"
Says_C_PInitRes: fixes A :: "agent"   and C :: "agent"   and M :: "agent"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and Chall_M :: "nat"   and P :: "agent"   and EKj :: "nat"   and evs :: "event list" assumes "Says A C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P EKj onlyEnc (priSK RCA)\<rbrace>) \<in> set evs"   and "M \<notin> bad"   and "evs \<in> set_pur" shows "\<exists>j trans. Notes M \<lbrace>Number LID_M, Agent P, trans\<rbrace> \<in> set evs \<and> P = PG j \<and> EKj = pubEK (PG j)"
HInfinite_hypreal_sqrt_imp_HInfinite: fixes x :: "real star" assumes "0 \<le> x"   and "(*f* sqrt) x \<in> HInfinite" shows "x \<in> HInfinite"
subGA: shows "generate G (carrier G \<inter> A) \<subseteq> A"
parent_num_assms: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "dist v \<noteq> \<infinity>" shows "\<exists>e\<in>arcs G. parent_edge v = Some e \<and> head G e = v \<and> dist (tail G e) \<noteq> \<infinity> \<and> num v = num (tail G e) + 1"
h3b_a: shows "\<forall>x. List.member (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))) x \<longrightarrow> x < length subsets"
matrix_to_iarray_invertible: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = invertible_iarray (matrix_to_iarray A)"
Image_eq_UN: fixes r :: "('d \<times> 'c) set"   and B :: "'d set" shows "r `` B = \<mu> (\<P> (\<lambda>y. r `` \<eta> y) B)"
abs_mult_self_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (abs a) (abs a) = times a a"
pdg_index_in_Vset: shows "I \<in>\<^sub>\<circ> Vset \<alpha>"
fine_division_exists: fixes g :: "'a \<Rightarrow> 'a set"   and a :: "'a"   and b :: "'a" assumes "gauge g" obtains p :: "('a \<times> 'a set) set" where "p tagged_division_of cbox a b" and "g fine p"
sgn_if: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "sgn x = (if x = zero then zero else if less zero x then one else uminus one)"
ext_snoc_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_snoc_axioms ext \<equiv> \<forall>gt xs x. ext gt (xs @ [x]) xs"
lookup_mult: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and g :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "poly_mapping.lookup (f * g) k = (\<Sum>l. poly_mapping.lookup f l * (\<Sum>q. poly_mapping.lookup g q when k = l + q))"
hyps: shows "((\<lambda>t. inverse (t ^ k) / exp (inverse t)) \<longlongrightarrow> 0) (at_right 0)"
minimalVertex_def: fixes g :: "graph"   and f :: "face" shows "minimalVertex g f \<equiv> minimal (height g) (vertices f)"
dvd_imp_multiplicity_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "b \<noteq> zero" shows "factorial_semiring.multiplicity times one p a \<le> factorial_semiring.multiplicity times one p b"
assms_2: fixes y :: "variable \<times> typ" assumes "y \<in> \<Union> (fv ` snd ` set (xs @ [x]))" shows "y \<notin> fst ` set (xs @ [x])"
set_integrable_subset: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "set_integrable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_integrable M B f"
rep_eq: fixes x :: "'a list" shows "fset (fset_from_list x) = set x"
infsetsum_Sigma': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) (B x)) = (\<Sum>\<^sub>a(x, y)\<in>Sigma A B. f x y)"
set_nebset_to_set_nebset: fixes A :: "'d set" assumes "A \<noteq> {}"   and "|A| <o natLeq +c |UNIV|" shows "set_nebset (the_inv set_nebset A) = A"
simplify_involutive: fixes S :: "'at Literal set set" shows "simplify (simplify S) = simplify S"
measure_Union_le: fixes F :: "'b set set"   and M :: "'b measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> F) \<le> sum (Sigma_Algebra.measure M) F"
real_cond_exp_int_2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f" shows "integral\<^sup>L M (real_cond_exp M F f) = integral\<^sup>L M f"
alg_by_fold: fixes ts :: "((nat \<times> nat) \<times> int) list"   and x :: "nat \<Rightarrow> int" shows "alg ts x = fold (\<lambda>((r, c), v) y. y(c := y c + x r * v)) ts (\<lambda>_. 0)"
stateful_policy_wf: shows "wf_stateful_policy \<T>"
\<phi>_p: shows "\<tau> \<cdot> \<phi> = \<eta> \<cdot> \<sigma>"
a0i_commit_def: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "a0i_commit A B M \<equiv> {(s, s'). a0n_state.signals s (Commit A B M) < a0n_state.signals s (Running A B M) \<and> s' = s\<lparr>a0n_state.signals := addSignal (a0n_state.signals s) (Commit A B M)\<rparr>}"
Incr: assumes "lvl (insert x t) = lvl t + 1"   and "sngl (insert x t)" shows "thesis"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "local.map (g \<cdot> f) = local.map g \<cdot> local.map f"
sym_preproc_addnew_pmdl: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" shows "pmdl (set gs \<union> set (snd (sym_preproc_addnew gs vs fs v))) = pmdl (set gs \<union> set fs)"
iT_cut_le_2: fixes n :: "nat"   and t :: "nat" shows "[n\<dots>] \<down>\<le> t = (if t < n then {} else [n\<dots>,t - n])"
PFF: fixes r :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! \<^bold>\<bottom>\<^sub>! \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! \<^bold>\<bottom>\<^sub>!)"
pick: fixes x :: "'c"   and X :: "'c set" assumes "x \<in> X" shows "pick X \<in> X"
slg_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>slg_rel = (R \<rightarrow> \<langle>R\<rangle>list_set_rel) O br (\<lambda>succs. {(u, v). v \<in> succs u}) (\<lambda>_. True)"
matchIdRight: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and a :: "name" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] [a\<frown>a]P"
l12: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - (x \<squnion> y)"
set_disconnected_nodes_get_tag_name_is_l_set_disconnected_nodes_get_tag_name: shows "l_set_disconnected_nodes_get_tag_name ShadowRootClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
that: fixes k :: "nat \<Rightarrow> nat"   and v :: "'b \<Rightarrow> nat"   and n :: "nat" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub>* \<langle>l', D'\<rangle> \<and> valid_dbm D n \<and> [D']\<^bsub>v,n\<^esub> \<noteq> {} \<longrightarrow> (\<exists>Z. A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>* \<langle>l', Z\<rangle> \<and> Z \<noteq> {})" shows "thesis"
step_def_exec: shows "step \<equiv> exec P mxs T\<^sub>r xt is"
cond_spmf_spmf_of_set: fixes A :: "'c set"   and B :: "'c set" assumes "finite A" shows "cond_spmf (spmf_of_set A) B = spmf_of_set (A \<inter> B)"
A: shows "A = prod_emb I E J (Pi\<^sub>E J G)"
inv_uptoD: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "inv_upto n p A m"   and "i < n"   and "j < n"   and "j < m" shows "p A i j"