hcju/isabelleps huggingface dataset:
bounded_iff: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "semilattice_order f less_eq less" shows "less_eq a (f b c) = (less_eq a b \<and> less_eq a c)"
add_opp: fixes a :: "'a"   and b :: "'a"   and p :: "'a point" assumes "on_curve a b p" shows "local.add a p (opp p) = Infinity"
l_set_mode_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_tag_name type_wf set_mode set_mode_locs get_tag_name get_tag_name_locs \<equiv> l_set_mode type_wf set_mode set_mode_locs \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_set_mode_get_tag_name_axioms set_mode_locs get_tag_name_locs"
nonzero_norm_divide: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "norm (a / b) = norm a / norm b"
scale_left_distrib: fixes x :: "'b"   and a :: "'a"   and b :: "'a" assumes "x \<in> U\<^sub>V\<^sub>S" shows "s (a + b) x = s a x + s b x"
sign_simps_1: fixes ad :: "'d"   and bd :: "'d"   and cd :: "'d" shows "ad + bd + cd = ad + (bd + cd)"
lim_H: shows "(^) (U s s) \<longlonglongrightarrow> H s s"
invT_into: fixes h :: "'h" assumes "h \<in> H" shows "invT h \<in> G"
basic_monos_6: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
type_wf\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap" assumes "DocumentClass.type_wf h" shows "ElementClass.type_wf h"
input2state_double_exists: fixes x :: "nat"   and a :: "value"   and y :: "nat" shows "\<exists>i. input2state i $ x = Some a \<and> input2state i $ y = Some a"
with_def: shows "max.with \<equiv> \<lambda>less_eq a b. if less_eq a b then b else a"
prv_\<phi>G_eqv: shows "prv (eqv \<phi>G (neg (PP \<langle>\<phi>G\<rangle>)))"
prc_compat: fixes f :: "'f"   and g :: "'f"   and s1 :: "bool"   and ns1 :: "bool"   and h :: "'f"   and s2 :: "bool"   and ns2 :: "bool"   and s :: "bool"   and ns :: "bool" assumes "prc f g = (s1, ns1)"   and "prc g h = (s2, ns2)"   and "prc f h = (s, ns)" shows "(ns1 \<and> ns2 \<longrightarrow> ns) \<and> (ns1 \<and> s2 \<longrightarrow> s) \<and> (s1 \<and> ns2 \<longrightarrow> s)"
bounded_iff: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(linordered_monoid.Max A \<preceq> x) = (\<forall>a\<in>A. a \<preceq> x)"
exec_to_exec_subseteq_guards: fixes c :: "('d, 'e, 'f) com"   and c' :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "c \<subseteq>\<^sub>g c'"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c',s\<rangle> \<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
subgraph_has_max_density: fixes G :: "nat set \<times> nat set set" assumes "finite_graph G"   and "nonempty_graph G"   and "uwellformed G" shows "\<exists>G'. Ugraph_Lemmas.density G' = max_density G \<and> subgraph G' G \<and> nonempty_graph G' \<and> finite_graph G' \<and> uwellformed G'"
suminf_ereal_eq_SUP: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "(\<Sum>x. f x) = (SUP n. sum f {..<n})"
cond: shows "\<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 < \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2"
distinct_1: fixes x1 :: "bool"   and x21 :: "'b aexp"   and x22 :: "'b aexp" shows "Bc x1 \<noteq> Eq x21 x22"
type_less_right_simps_9: fixes \<tau> :: "'a OCL_Types.type"   and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" shows "(\<tau> < Tuple \<xi>) = (\<exists>\<pi>. \<tau> = Tuple \<pi> \<and> strict_subtuple (\<le>) \<pi> \<xi>)"
PiE_mem: fixes f :: "'f \<Rightarrow> 'g"   and S :: "'f set"   and T :: "'f \<Rightarrow> 'g set"   and x :: "'f" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
div_mult_self1: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times c b)) b = plus c (divide a b)"
retrieve_polys_def: fixes qss :: "real poly list"   and index_list :: "nat list" shows "retrieve_polys qss index_list = map ((!) qss) index_list"
composites_are_identities_1: shows "ide (\<nu> \<star> \<pi> \<star> \<rho>)"
lemma19: fixes j :: "nat"   and \<psi> :: "'a ltl" assumes "j' \<le> j"   and "G \<psi> \<in> \<G>" shows "suffix j w \<Turnstile> eval\<^sub>G \<G> (\<F> \<psi> w \<G> j)"
mem_Sigma_iff: fixes a :: "'a"   and b :: "'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
ideD_1: fixes a :: "'a" assumes "hom'.ide a" shows "hom'.arr a"
map_of_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "('b \<times> 'c) list" shows "map_of (List.filter (\<lambda>x. P (fst x)) xs) = map_of xs |` Collect P"
1: shows "- (- - x \<sqinter> - - y) \<sqinter> - - - - x = - - x \<sqinter> - y"
cnf_of_encode_all_frame_axioms_structure: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_frame_axioms \<Pi> t) = \<Union> (\<Union>(k, v)\<in>{0..<t} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}}}) \<union> \<Union> (\<Union>(k, v)\<in>{0..<t} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}}})"
RETURN_rule: fixes \<Phi> :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
prv_LLq_Num_dsj: fixes t :: "'trm"   and n :: "nat" assumes "t \<in> atrm" shows "prv (dsj (LLq t (Num n)) (LLq (Num n) t))"
fresh_star_list_append: fixes A :: "name list"   and B :: "name list"   and C :: "name list" shows "A \<sharp>* (B @ C) = (A \<sharp>* B \<and> A \<sharp>* C)"
imp: fixes B :: "'a set" assumes "B \<in> sets M \<and> B \<subseteq> A \<and> 0 < emeasure M B" shows "\<exists>N>0. 0 < emeasure M ((induced_map A ^^ N) -` B \<inter> B)"
subspace_inter: fixes A :: "'b set"   and B :: "'b set" assumes "subspace A"   and "subspace B" shows "subspace (A \<inter> B)"
outstanding_is_Write\<^sub>s\<^sub>b_refs_conv: fixes sb :: "'b memref list" shows "outstanding_refs is_Write\<^sub>s\<^sub>b sb = outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<union> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb"
prems_4: shows "o\<^sub>0 \<in> out (Ps ! 0) t"
Icc_subset_Iic_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and h :: "'b"   and h' :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq l h \<subseteq> ord.atMost less_eq h') = (\<not> less_eq l h \<or> less_eq h h')"
d\<^sub>y': shows "profile_single d\<^sub>y"
flip_bit_def: 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 "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.flip_bit one push_bit xor n a = xor a (push_bit n one)"
distinct_7: shows "T_i64 \<noteq> T_f32"
flat_Stream: fixes xs :: "'d list"   and ws :: "'d list stream" assumes "xs \<noteq> []" shows "flat (xs ## ws) = xs @- flat ws"
not_call_simps_1: fixes l :: "'d"   and fg :: "'a \<Rightarrow> 'a set"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}\<langle>fg\<rangle> p)"
j2: shows "degree (q * u + Polynomial.smult m r) = degree (q * u)"
lec: shows "(c - a \<bullet> i) * sum ((\<lambda>K. content K / interv_diff K i) \<circ> (\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c})) \<D> \<le> content (cbox a b')"
edge_preserving_def: fixes h :: "('e \<times> 'f) set"   and e1 :: "('g \<times> 'e \<times> 'e) set"   and e2 :: "('g \<times> 'f \<times> 'f) set" shows "edge_preserving h e1 e2 \<equiv> \<forall>(k, v1, v2)\<in>e1. \<forall>v1' v2'. (v1, v1') \<in> h \<and> (v2, v2') \<in> h \<longrightarrow> (k, v1', v2') \<in> e2"
summable: shows "(\<lambda>(x, y). pmf N x) abs_summable_on Sigma {Suc 0..} (atLeastAtMost (Suc 0))"
cut_leE: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>\<le> t" obtains "x \<in> I" and "x \<le> t"
Nil_tl: fixes xs :: "'d list" shows "([] = tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
drop_bit_of_Suc_0: fixes n :: "nat" shows "drop_bit n (Suc 0) = of_bool (n = 0)"
get_bounds_2: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "real_of_2 x \<in>\<^sub>i get_itvl_2 x"
UP_l_neg_ex: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<exists>q\<in>carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
S: fixes s :: "nat"   and t :: "nat" assumes "s \<in> S"   and "t \<in> S" shows "s + t \<in> S"
identity\<^sub>\<kappa>_def: shows "(\<^bold>=) \<equiv> \<lambda>x y. x \<^bold>=\<^sub>E y \<^bold>\<or> \<lparr>A!,x\<rparr> \<^bold>& \<lparr>A!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,F\<rbrace>)"
frac_le: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   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 y :: "'c"   and x :: "'c"   and w :: "'c"   and z :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero y"   and "less_eq x y"   and "less zero w"   and "less_eq w z" shows "less_eq (divide x z) (divide y w)"
osp_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B C OSP P Q" shows "A \<noteq> B \<and> A \<noteq> C \<and> B \<noteq> C \<and> A \<noteq> P \<and> B \<noteq> P \<and> C \<noteq> P \<and> A \<noteq> Q \<and> B \<noteq> Q \<and> C \<noteq> Q"
Init: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and s0 :: "st"   and s3 :: "state"   and n :: "nat"   and s1 :: "state"   and s2 :: "state" assumes "the (class G C) = c"   and "if inited C (globs s0) then s3 = Norm s0 else G\<turnstile>Norm (init_class_obj G C s0) \<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n\<rightarrow> s1 \<and> G\<turnstile>set_lvars Map.empty s1 \<midarrow>init c\<midarrow>n\<rightarrow> s2 \<and> s3 = set_lvars (locals (snd s1)) s2" shows "G\<turnstile>Norm s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
pair_wf_digraph_axioms: shows "pair_wf_digraph (list_digraph G_list)"
size_hps_pass2: fixes hs :: "'b hp list"   and h :: "'b hp" assumes "pass\<^sub>2 hs = Some h" shows "size_hps hs = size_hps (hps h) + 1"
multiplicity_decompose': fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" obtains y :: "'a" where "x = p ^ multiplicity p x * y" and "\<not> p dvd y"
associatedD2: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b" shows "b dvd a"
elims: fixes x :: "'a \<Rightarrow> bool option"   and xa :: "'a strips_operator list list"   and y :: "('a \<Rightarrow> bool option) list" assumes "trace_parallel_plan_strips x xa = y" obtains s :: "'a \<Rightarrow> bool option" where "x = s__" and "xa = []" and "y = [s__]"   | s :: "'a \<Rightarrow> bool option" and ops :: "'a strips_operator list" and opss :: "'a strips_operator list list" where "x = s" and "xa = ops # opss" and "y = s # (if are_all_operators_applicable s ops \<and> are_all_operator_effects_consistent ops then trace_parallel_plan_strips (execute_parallel_operator s ops) opss else [])"
apx_empty: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and k :: "'d \<Rightarrow> nat" assumes "Beta_Regions' X v n not_in_X" shows "Beta_Regions'.Approx\<^sub>\<beta> X k v n not_in_X {} = {}"
lookup_operator'_correct: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list" assumes "wf_ast_problem problem" shows "lookup_operator' problem name = ast_problem.lookup_operator problem name"
le_take: fixes n1 :: "nat"   and n2 :: "nat"   and al :: "'a list" assumes "n1 \<le> n2" shows "take n1 al \<le> take n2 al"
Abs_inverse: fixes y :: "int" assumes "y \<in> sints LENGTH('a)" shows "sint (word_of_int y) = y"
is_minimal_basis_replace: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B"   and "(red (B - {p}))\<^sup>*\<^sup>* p r" shows "is_minimal_basis (insert r (B - {p}))"
hoare_meaning: fixes P :: "'a upred"   and S :: "'a hrel"   and Q :: "'a upred" shows "\<lbrace>P\<rbrace> S \<lbrace>Q\<rbrace>\<^sub>u = (\<forall>s s'. \<lbrakk>P\<rbrakk>\<^sub>e s \<and> \<lbrakk>S\<rbrakk>\<^sub>e (s, s') \<longrightarrow> \<lbrakk>Q\<rbrakk>\<^sub>e s')"
inv_product_group: fixes f :: "'f \<Rightarrow> 'g"   and I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" assumes "f \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "inv\<^bsub>product_group I G\<^esub> f = (\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> f i)"
bounded_linear_scaleR_left: fixes x :: "'c" shows "bounded_linear (\<lambda>r. r *\<^sub>R x)"
hyps: shows "f ` set (remdups_wrt f xs) = f ` set xs"
Sup_less_def: fixes x :: "'c \<Rightarrow> 'a"   and w :: "'c" shows "Sup_less x w = Sup {y. \<exists>v<w. y = x v}"
assms_5: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"
assms: assumes "a \<noteq> 0" shows "b \<noteq> 0"
open_dist: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.metric_space dist uniformity open" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
Diff_eq_empty_iff_mset: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A - B = \<Empt>) = (A \<subseteq># B)"
partial_state2_axioms: shows "partial_state2 {Suc m} {0..<Suc m}"
bres_canc1: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (x \<rightarrow> y) \<le> y"
exhaust: fixes y :: "'a process" obtains x11 :: "'a" and x12 :: "'a process" where "y = Action x11 x12"   | x21 :: "'a process" and x22 :: "'a process" where "y = Choice x21 x22"
induct: fixes P :: "bexp \<times> bool \<times> int \<Rightarrow> bool"   and a0 :: "bexp \<times> bool \<times> int" assumes "\<And>v f i. P (Bc v, f, i)"   and "\<And>b f i. P (b, \<not> f, i) \<Longrightarrow> P (bexp.Not b, f, i)"   and "\<And>b\<^sub>1 b\<^sub>2 f i. \<lbrakk>P (b\<^sub>2, f, i); \<And>x. x = bcomp (b\<^sub>2, f, i) \<Longrightarrow> P (b\<^sub>1, False, size x + (if f then 0 else i))\<rbrakk> \<Longrightarrow> P (And b\<^sub>1 b\<^sub>2, f, i)"   and "\<And>a\<^sub>1 a\<^sub>2 f i. P (Less a\<^sub>1 a\<^sub>2, f, i)" shows "P a0"
R_def: fixes h :: "'a \<times> 'a"   and w :: "nat"   and e :: "nat" shows "chaum_ped_sigma.R h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
ENR_openin: fixes T :: "'b set"   and S :: "'b set" assumes "ENR T"   and "openin (top_of_set T) S" shows "ENR S"
soconf_def: fixes P :: "'a prog"   and h :: "heap"   and C :: "char list"   and sfs :: "sfields" shows "P,h,C \<turnstile>\<^sub>s sfs \<surd> \<equiv> \<forall>F T. P \<turnstile> C has F,Static:T in C \<longrightarrow> (\<exists>v. sfs F = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
dependent_on_def: fixes s :: "'a set" shows "subspace.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *\<^sub>R v) = (0::'a) \<and> (\<exists>v\<in>t. u v \<noteq> 0))"
slim_nodes_sharp: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "nodes (slim_node k a h) * k = bound k h"
inject_1: fixes x11 :: "char list"   and x12 :: "fterm list"   and y11 :: "char list"   and y12 :: "fterm list" shows "(Fun x11 x12 = Fun y11 y12) = (x11 = y11 \<and> x12 = y12)"
interior_of_def: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S \<equiv> {x. \<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> S}"
diff_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = x \<sqinter> - y"
nonempty_prefix_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<le>np v \<equiv> u \<noteq> \<epsilon> \<and> u \<le>p v"
lsimp: shows "length \<GG> - 1 - 1 + 1 = length \<GG> - 1"
map_1: fixes f :: "'c \<Rightarrow> 'b" shows "map_rexp f Zero = Zero"
Complex_Im_express_cnj: fixes z :: "complex" shows "Complex 0 (Im z) = (z - cnj z) / 2"
adm_subst2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. f (fst x) = g (snd x))"
gf_U_def: fixes K :: "'b \<Rightarrow> 'b pmf"   and x :: "'b"   and y :: "'b"   and z :: "'a" shows "MC_syntax.gf_U K x y z = (\<Sum>n. MC_syntax.u K x y n *\<^sub>R z ^ Suc n)"
h_chop_middle2: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> left (ext w) \<le> right (ext v)"
simps_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list"   and ys :: "'b list"   and yss :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_mat_Tensor f xs (ys # yss) = mult.vec_vec_Tensor f xs ys # mult.vec_mat_Tensor f xs yss"
assm: shows "u' \<le> u \<and> v = c > u"
l8_21_aux: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<exists>P T. A B Perp P A \<and> Col A B T \<and> Bet C T P"
weaken_left_Un1: fixes G :: "'b pl set"   and p :: "'b pl"   and B :: "'b pl set" assumes "G |- p" shows "G \<union> B |- p"
agree_union: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec"   and A :: "('d + 'd) set"   and B :: "('d + 'd) set" assumes "Vagree \<nu> \<omega> A"   and "Vagree \<nu> \<omega> B" shows "Vagree \<nu> \<omega> (A \<union> B)"
lookup_useNodes_of: fixes g :: "'g"   and v :: "'val" shows "Mapping.lookup (uninst_code.ssa.useNodes_of uses g) v = (if \<exists>n\<in>set (\<alpha>n g). v \<in> (usesOf \<circ> uses) g n then Some {n \<in> set (\<alpha>n g). v \<in> (usesOf \<circ> uses) g n} else None)"
yk_2: shows "y = nat (snd res) + k2 * q"
from_nat_to_nat_id: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (mod_type.to_nat Rep x) = x"
LT4: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> F \<longrightarrow> (F \<leadsto> G) \<longrightarrow> \<diamond>G"
valid_own: shows "valid_ownership \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
assms: shows "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vconst_on A c"
fun_pair_subst: fixes d :: "('fun, 'c) Term.term \<times> ('fun, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('fun, 'b) Term.term" shows "pair d \<cdot> \<delta> = pair (d \<cdot>\<^sub>p \<delta>)"
G_from_list_def: shows "G_from_list \<equiv> adt_wgraph.from_list (wgraph_by_map.empty RBT_Set.empty) (wgraph_by_map.add_edge RBT_Map.update lookup)"
simps_7: fixes b :: "nat"   and I :: "\<I>"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Next I \<phi>) = MFOTL.fvi b \<phi>"
perm_cont: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "perm" assumes "class.cont_pt permute below" shows "cont (permute p)"
inf_cases: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" assumes "AlternativeSmallStep.inf \<Gamma> (c # cs) css s" shows "AlternativeSmallStep.inf \<Gamma> [c] [] s \<or> (\<exists>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<and> AlternativeSmallStep.inf \<Gamma> cs css t)"
paths_ccTTree: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "paths (ccTTree S G) = valid_lists S G"
long_dividesI: fixes b :: "'a list list"   and r :: "'a list list"   and p :: "'a list list"   and q :: "'a list list" assumes "b \<in> carrier (poly_ring (poly_ring R))"   and "r \<in> carrier (poly_ring (poly_ring R))"   and "p = q \<otimes>\<^bsub>poly_ring (poly_ring R)\<^esub> b \<oplus>\<^bsub>poly_ring (poly_ring R)\<^esub> r"   and "r = [] \<or> degree r < degree q" shows "UP.long_divides p q (b, r)"
3: shows "e\<^sup>\<star> = (1::'a) \<squnion> e"
list_emb_Cons_iff2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "\<not> P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P (x # xs) ys"
LIMSEQ_I: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r" shows "X \<longlonglongrightarrow> L"