hcju/isabelleps huggingface dataset:
select_convs_2: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b" shows "cls \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = cls"
assms_2: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "fst (f a) = fst (f b)" shows "fst a = fst b"
ext: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "g \<bullet> f \<noteq> V.null" shows "V.seq g f"
normalize_alt_simps_5: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "metaop"   and b :: "'a expr" shows "(\<Gamma> \<turnstile> MetaOperationCall \<tau> op \<Rrightarrow> b) = (b = MetaOperationCall \<tau> op)"
boundedI: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> min b c"
continuous_on_Shleg: fixes A :: "real set"   and n :: "nat" shows "continuous_on A (Shleg n)"
approximating_fun_imp_semantics: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs :: "'b rule list"   and s :: "state"   and t :: "state" assumes "wf_ruleset \<gamma> p rs"   and "approximating_bigstep_fun \<gamma> p rs s = t" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t"
lm066: fixes R :: "('d \<times> 'c) set"   and X :: "'d set" shows "Range (R outside X) = R `` (Domain R - X)"
comp_ide_arr: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and f :: "'c" assumes "partial_magma C"   and "partial_magma.ide C b"   and "partial_magma.arr C (C b f)" shows "C b f = f"
obtain_point_with_rep: fixes r :: "int" assumes "r \<in> replication_numbers" shows "\<exists>x. x \<in> \<V> \<and> \<B> rep x = r"
Value_CValue_take_induct: fixes P :: "Value \<Rightarrow> (C \<rightarrow> CValue) \<Rightarrow> bool"   and x :: "Value"   and y :: "C \<rightarrow> CValue" assumes "adm (\<lambda>(x, y). P x y)"   and "\<And>n. P (\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x) (\<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>y)" shows "P x y"
subo: shows "Subobjs P C (Cs @ C' # rev Cs')"
linear: fixes x :: "'b"   and y :: "'b" shows "x \<le> y \<or> y \<le> x"
id_subst_A: fixes A :: "type_scheme list" shows "$ id_subst A = A"
strongly_connected_non_disj: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "wf_digraph H"   and "compatible G H"   and "strongly_connected G"   and "strongly_connected H"   and "verts G \<inter> verts H \<noteq> {}" shows "strongly_connected (Digraph_Component.union G H)"
type_wf_def: shows "DocumentClass.type_wf \<equiv> l_type_wf_def\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_type_wf"
baIA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and \<phi> :: "'KS \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'KS" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)" shows "sb.eval c t t' n [\<phi>]\<^sub>b"
jordan_nf_char_poly: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list" assumes "jordan_nf A n_as" shows "char_poly A = (\<Prod>(n, a)\<leftarrow>n_as. [:- a, 1::'a:] ^ n)"
open_openin: fixes S :: "'d set" shows "open S = openin euclidean S"
closest_point_le: fixes S :: "'a set"   and x :: "'a"   and a :: "'a" assumes "closed S"   and "x \<in> S" shows "dist a (closest_point S a) \<le> dist a x"
bind_assoc: fixes x :: "'m"   and f :: "'a \<times> 's \<Rightarrow> 'm"   and g :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
case_map_generat: fixes io :: "'j \<Rightarrow> 'k \<Rightarrow> 'h"   and a :: "'l \<Rightarrow> 'i"   and b :: "'m \<Rightarrow> 'j"   and d :: "'n \<Rightarrow> 'k"   and r :: "('l, 'm, 'n) generat" shows "(case map_generat a b d r of Pure x \<Rightarrow> pure x | IO x xa \<Rightarrow> io x xa) = (case r of Pure x \<Rightarrow> (pure \<circ> a) x | IO out x \<Rightarrow> (io (b out) \<circ> d) x)"
complement_symmetric: fixes x :: "'a"   and y :: "'a" assumes "complement x y" shows "complement y x"
suffixes_append: fixes xs :: "'a list"   and ys :: "'a list" shows "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>zs. zs @ ys) (suffixes xs)"
select_least_methods_subtypes: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and Ts' :: "ty list"   and T' :: "ty"   and pns' :: "char list list"   and body' :: "expr"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"   and "P \<turnstile> last Cs has least M = (Ts', T', pns', body') via Ds"   and "P \<turnstile> Path C to last Cs via Cs "   and "wf_prog wf_md P" shows "Ts' = Ts \<and> P \<turnstile> T \<le> T'"
normal_poly_def: fixes p :: "'b poly" shows "normal_poly p \<equiv> p \<noteq> 0 \<and> (\<forall>i. (0::'b) \<le> coeff p i) \<and> (\<forall>i. coeff p i * coeff p (i + 2) \<le> (coeff p (i + 1))\<^sup>2) \<and> (\<forall>i j k. i \<le> j \<longrightarrow> j \<le> k \<longrightarrow> (0::'b) < coeff p i \<longrightarrow> (0::'b) < coeff p k \<longrightarrow> (0::'b) < coeff p j)"
associatedE2: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b" obtains u :: "'a" where "is_unit u" and "b = u * a"
\<gamma>\<^sub>\<mu>: shows "\<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>)"
abs_vec_nth_blinfun_apply_lemma: fixes x :: "(real, 'b) vec \<Rightarrow>\<^sub>L (real, 'a) vec"   and ia :: "'b"   and i :: "'a" shows "\<bar>blinfun_apply x (axis ia 1) $ i\<bar> \<le> norm x"
sec_id_identified_2: fixes n :: "nat" shows "(Spy, PubKey {Tok_PriK n, Rev_PriK}) \<notin> s\<^sub>0"
Lyndon_mono_last_smallest: fixes ws :: "'a list list"   and s :: "'a list" assumes "Lyndon_mono ws"   and "s \<le>ns concat ws" shows "last ws \<le>lex s"
eq_refl: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
object_ptr_kinds_eq_h2: shows "object_ptr_kinds h3 = object_ptr_kinds h2"
fBall_alt_def: fixes S :: "'d fset"   and P :: "'d \<Rightarrow> bool" shows "fBall S P = (\<forall>x. x |\<in>| S \<longrightarrow> P x)"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "[xa \<leftrightarrow> x] = abs_prm (preprm_unit xa x)"
dres_Sup_chain_cases: fixes M :: "'e dres set" assumes "is_chain M" obtains "M \<subseteq> {dSUCCEED}" and "Sup M = dSUCCEED"   | r :: "'e" where "M \<subseteq> {dSUCCEED, dRETURN r}" and "dRETURN r \<in> M" and "Sup M = dRETURN r"   | "dFAIL \<in> M" and "Sup M = dFAIL"
rotate_3: assumes "round_robin_reschedule t0 (queue @ [t']) n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "round_robin_invar (queue @ [t'], n0) (dom (thr s))" shows "round_robin_invar \<sigma>' (dom (thr s))"
alphaAbs_preserves_qGoodAbs1: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and A' :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "A $= A'"   and "FixVars.qGoodAbs A" shows "FixVars.qGoodAbs A'"
ide_is_iso: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.iso a"
proots_count_leq_degree: fixes p :: "'b poly"   and s :: "'b set" assumes "p \<noteq> 0" shows "proots_count p s \<le> degree p"
X: shows "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
integral_spmf_of_set: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" shows "integral\<^sup>L (measure_spmf (spmf_of_set A)) f = sum f A / real (card A)"
Lp_Lq_duality_bound_1: fixes p :: "ennreal"   and q :: "ennreal"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "1 / p + 1 / q = 1"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "g \<in> space\<^sub>N (\<LL> q M)" shows "integrable M (\<lambda>x. f x * g x)"
valid_plans_def: fixes prob :: "'a set" shows "valid_plans prob \<equiv> {as. set as \<subseteq> prob}"
cod_eqI: fixes b :: "'a2"   and f :: "'a2" assumes "C2.ide b"   and "C2.seq b f" shows "C2.cod f = b"
in_gamma_update: fixes s :: "char list \<Rightarrow> int"   and S :: "'av st"   and i :: "int"   and a :: "'av"   and x :: "char list" assumes "s \<in> \<gamma>\<^sub>s S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>\<^sub>s (update S x a)"
sel_19: fixes x71 :: "'c list"   and x72 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and x73 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle> = x71"
Ioi_le_Ico: fixes a :: "'b multiset" shows "subset_mset.greaterThan a \<subseteq> subset_mset.atLeast a"
Gcd_real_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
intro: fixes 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" assumes "valid_write_sops ts"   and "valid_store_sops ts" shows "valid_sops ts"
left_neutral: fixes a :: "'c \<Rightarrow> 'c" shows "id \<circ> a = a"
lossless_protocol: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool" shows "lossless_spmf (protocol M \<sigma>)"
right_inverse: fixes y :: "nat"   and A :: "nat set" assumes "y \<in> A" shows "nth_least A (card {i \<in> A. i < y}) = y"
independent_vnmI1: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" assumes "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r" shows "independent_vnm C P"
If: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. P s + \<up> (bval b s)} c\<^sub>1 {Q}"   and "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. P s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}" shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. eSuc (P s)} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"
T_itrev: fixes xs :: "'b list"   and ys :: "'b list" shows "T_itrev xs ys = length xs + 1"
connected_edge: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "Graph.connected G v w"
bisimSubstParComm: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<sim>\<^sub>s Q \<parallel> P"
SCallInit\<^sub>1: fixes P :: "nat exp prog"   and ps :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and ls\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list"   and v' :: "val"   and h\<^sub>2 :: "heap"   and ls\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and ls\<^sub>2' :: "val list"   and e' :: "nat exp"   and h\<^sub>3 :: "heap"   and ls\<^sub>3 :: "val list"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = body in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>Val v',(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"   and "length vs = length Ts"   and "ls\<^sub>2' = vs @ replicate (max_vars body) undefined"   and "P \<turnstile>\<^sub>1 \<langle>body,(h\<^sub>2, ls\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>2, sh\<^sub>3)\<rangle>"
has_domain_iff_has_codomain: fixes f :: "'a1 \<times> 'a2" shows "(A1xA2.domains f \<noteq> {}) = (A1xA2.codomains f \<noteq> {})"
vimage_Inl_Plus: fixes A :: "'e set"   and B :: "'f set" shows "Inl -` (A <+> B) = A"
wp1_alternative_not_mt: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "p \<noteq> []"
wait_def: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" shows "wait \<phi> xs \<equiv> LEAST n. (nxt ^^ n) \<phi> xs"
simps_2: fixes lt :: "interval_tree"   and x :: "nat idx_interval"   and m :: "nat"   and rt :: "interval_tree" shows "delete_elt_tree (Node lt x m rt) = (if lt = Tip then rt else if rt = Tip then lt else let x' = fst (del_min rt); rt' = snd (del_min rt); m' = max3 x' (tmax lt) (tmax rt') in Node lt (fst (del_min rt)) m' rt')"
P_Suc: fixes n :: "nat" shows "F (Suc n) = (SOME N'. N' \<subseteq> F n \<and> infinite N' \<and> decides \<F> (f (Suc n)) N')"
eq_upred_def: fixes x :: "('f, 'e) uexpr"   and y :: "('f, 'e) uexpr" shows "(x =\<^sub>u y) = bop (=) x y"
vimage_eq: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" shows "(a \<in> f -` B) = (f a \<in> B)"
seq_lexord_transitive: fixes R :: "('b \<times> 'b) set" assumes "trans R" shows "trans (seq_lexord R)"
one_subst: fixes M :: "lam"   and M' :: "lam"   and N :: "lam"   and N' :: "lam"   and x :: "name" assumes " M \<longrightarrow>\<^sub>1 M'"   and " N \<longrightarrow>\<^sub>1 N'" shows " M[x::=N] \<longrightarrow>\<^sub>1 M'[x::=N']"
a_exp: fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> y) = d x + ad y"
1: shows "rawpsubstT (substT r (Var u) x) (zip (map Var vs) xs) = substT (rawpsubstT r (zip (map Var vs) xs)) (Var u) x"
min_height_le_height: fixes t :: "'a tree" shows "min_height t \<le> height t"
pickE_asEnv: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv qrho" shows "pickE (asEnv qrho) &= qrho"
B_blocks: shows "B = A @\<^sub>c mat_of_col b"
Rep_wf_prog: fixes x :: "wf_prog" shows "Rep_wf_prog x \<in> wf_prog"
IH: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "as \<in> valid_plans PROB"   and "s \<in> valid_states PROB"   and "\<forall>a. ListMem a as \<longrightarrow> agree (snd a) s" shows "exec_plan s as = s"
hyps_1: fixes s :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile>sequence Seq (flatten c1) \<down> s" shows "\<Gamma>\<turnstile>c1 \<down> s"
comp_cod_arr: fixes f :: "'c"   and b :: "'c" assumes "Cop.arr f"   and "Cop.cod f = b" shows "b \<cdot>\<^sup>o\<^sup>p f = f"
triangularizable_m0: fixes A :: "'b mat"   and m :: "nat" assumes "A \<in> carrier_mat m 0" shows "\<exists>U. U \<in> carrier_mat 0 0 \<and> invertible_mat U \<and> lower_triangular (A * U)"
\<beta>: shows "\<guillemotleft>\<beta> : p\<^sub>1 \<star> w \<Rightarrow> p\<^sub>1 \<star> w'\<guillemotright>"
cases: fixes S :: "'d \<Rightarrow> bool"   and a :: "'d tree" assumes "treesp S a" obtains "a = \<langle>\<rangle>"   | l :: "'d tree" and r :: "'d tree" and v :: "'d" where "a = \<langle>l, v, r\<rangle>" and "treesp S l" and "treesp S r" and "S v"
fs_at_inst: assumes "at TYPE('a)" shows "fs TYPE('a) TYPE('a)"
p_circ_def: shows "p_circ \<equiv> circlepath p (h p)"
l1_refines_a0i_i: shows "refines (R01iai \<inter> reach a0i \<times> (l1_inv1 \<inter> l1_inv2)) med01iai a0i l1"
list_of_oalist_sorted: fixes xs :: "'x" shows "sorted_wrt (lt (snd (list_of_oalist xs))) (map fst (fst (list_of_oalist xs)))"
Liminf_le: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "F \<noteq> \<bottom>"   and "\<forall>\<^sub>F x in F. f x \<le> l" shows "Liminf F f \<le> l"
CondThrow: fixes e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>if (throw e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
emeasure_measure_of_conv: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "emeasure (measure_of \<Omega> A \<mu>) = (\<lambda>B. if B \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> B else 0)"
in_sccs_verts_conv_reachable: fixes S :: "'a set" shows "(S \<in> sccs_verts) = (S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>* v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>* v \<or> \<not> v \<rightarrow>\<^sup>* u))"
sample_primes_def: shows "sample_primes = spmf_of_set prime_set"
h7a: assumes "Av < 0"   and "0 < Bv"   and "Av + Bv * sqrt Cv \<le> 0" shows "Bv\<^sup>2 * Cv \<le> Av\<^sup>2"
rt_graph_not_dip: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> rt_graph \<sigma> dip" shows "ip \<noteq> dip"
induced_arrowI_1: fixes \<chi>' :: "J.Arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "\<guillemotleft>induced_arrow a' \<chi>' : a' \<rightarrow> C.dom p0\<guillemotright>"
delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_not_in_heap: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "shadow_root_ptr |\<notin>| shadow_root_ptr_kinds h'"
N: shows "N \<in> M \<rightarrow>\<^sub>M subprob_algebra K"
\<iota>: fixes t :: "'s"   and t' :: "'s" assumes "(s, t) \<in> acc_on (- H)"   and "t' \<in> set_pmf (K t)" shows "\<iota> t t' < \<infinity>"
a_zero: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.antidomain_semiringl antidomain_op plus times one zero less_eq less" shows "antidomain_op zero = one"
eq_cbox: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(cbox a b = cbox c d) = (cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d)"
assms_2: shows "(x1, K1) \<in> \<D>"
simps: fixes Q_pos :: "(nat set \<times> 'b option list set) set"   and Q_neg :: "(nat set \<times> 'b option list set) set"   and V :: "nat set" shows "max_getIJ Q_pos Q_neg V = (let l = sorted_list_of_set V in if Set.is_empty Q_neg then let x = arg_max_list (score Q_pos) l in ({x}, V - {x}) else let x = arg_max_list (score Q_neg) l in (V - {x}, {x}))"
not_col_permutation_4: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col B A C"
height_mkt_bal_r: fixes r :: "'b tree"   and l :: "'b tree"   and n :: "'b" assumes "height r = height l + 2"   and "avl l"   and "avl r" shows "height (mkt_bal_r n l r) = height l + 2 \<or> height (mkt_bal_r n l r) = height l + 3"
empty_iff: fixes c :: "'b" shows "((c::'b) \<in> {}) = False"
IH: fixes L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "length xs' = var" shows "(\<forall>f\<in>set L. eval (substNegInfinity var f) (xs' @ x # xs)) = (\<forall>f\<in>set L'. evalUni (substNegInfinityUni f) x)"
Rep_pos_inject: fixes x :: "'a\<^sup>+"   and y :: "'a\<^sup>+" shows "(Rep_pos x = Rep_pos y) = (x = y)"
word_and_le2: fixes a :: "'b word"   and y :: "'b word" shows "a && y \<le> a"
eSubstAbs_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and us :: "'varSort"   and s :: "'sort"   and A :: "'d"   and y :: "'var" assumes "igWls MOD (asSort ys) Y"   and "igWlsAbs MOD (us, s) A" shows "eSubstAbs MOD ys (OK Y) y (OK A) = OK (igSubstAbs MOD ys Y y A)"
int_def_4: shows "\<C>_int = \<C>\<lparr>label := [tls] @ label \<C>\<rparr>"
sub1_preserves_diagnal: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 q k l (r, A) = (r', A')"   and "l < dim_col A"   and "k + l < dim_row A" shows "A' $$ (l, l) = A $$ (l, l)"
LCOSETS_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "lcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {a <#\<^bsub>G\<^esub> H})"
set_bit_integer_conv_masks: fixes x :: "integer"   and i :: "nat"   and b :: "bool" shows "set_bit_class.set_bit x i b = (if b then x OR (1 << i) else x AND NOT (1 << i))"
capacity_A_weight: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT cap \<langle>x\<rangle> \<le> 2 * weight \<Gamma> x"
factgroup_subgroup_union_factor: fixes A :: "'a set set" assumes "subgroup A (G Mod H)" shows "A = rcosets\<^bsub>G\<lparr>carrier := \<Union> A\<rparr>\<^esub> H"
wt_App_ge_fun: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "wt (App s t) \<ge>\<^sub>p wt s"
at_within_open: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and S :: "'a set" assumes "class.topological_space open"   and "a \<in> S"   and "open S" shows "topological_space.at_within open a S = topological_space.at_within open a UNIV"
cc: fixes S :: "'a set" assumes "S \<in> \<F>" shows "compact S \<and> connected S"
o_bij: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "g \<circ> f = id"   and "f \<circ> g = id" shows "bij f"
simps_1: fixes x :: "nat"   and y :: "nat" shows "pnat_plus (pnat x) (pnat y) = pnat (x + y)"
AllS: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" shows "[\<^bold>\<forall>x. \<phi> x in v] = (\<forall>x. [\<phi> x in v])"
le_less: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
runit_naturality: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "\<mu> \<cdot>\<^sub>D \<r>\<^sub>D[D.dom \<mu>] = \<r>\<^sub>D[D.cod \<mu>] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D src\<^sub>D \<mu>)"