hcju/isabelleps huggingface dataset:
commutative_squares_4: shows "C.commutative_square \<p>\<^sub>1[\<mu>.cod.leg0, assoc\<nu>\<pi>\<rho>.cod.leg1] \<mu>.chine (chine_hcomp \<mu> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>) \<p>\<^sub>1[\<mu>.leg0, assoc\<nu>\<pi>\<rho>.dom.leg1]"
rel_map_1: fixes S1b :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S2b :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i1 :: "'e \<Rightarrow> 'a"   and i2 :: "'f \<Rightarrow> 'c"   and x :: "('e, 'f) tllist"   and y :: "('b, 'd) tllist" shows "tllist_all2 S1b S2b (tmap i1 i2 x) y = tllist_all2 (\<lambda>x. S1b (i1 x)) (\<lambda>x. S2b (i2 x)) x y"
pw_bind_le_iff: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres"   and S :: "'b nres" shows "(M \<bind> f \<le> S) = ((nofail S \<longrightarrow> nofail M) \<and> (\<forall>x. nofail M \<and> inres M x \<longrightarrow> f x \<le> S))"
bmod_fres_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'c"   and x :: "'e" shows "\<rhd> R f g x = \<Sqinter>{f y \<leftarrow> g z |y z. R y x z}"
inject_1: fixes x1 :: "xcpt"   and y1 :: "xcpt" shows "(Xcpt x1 = Xcpt y1) = (x1 = y1)"
finfun_to_list_const: fixes c :: "'b" shows "finfun_to_list (finfun_const c) = (if infinite UNIV \<or> c = undefined then [] else THE xs. set xs = UNIV \<and> sorted xs \<and> distinct xs)"
binomial_mono: fixes k :: "nat"   and k' :: "nat"   and n :: "nat" assumes "k \<le> k'"   and "2 * k' \<le> n" shows "n choose k \<le> n choose k'"
reachable_apath: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>* v) = (\<exists>p. apath u p v)"
N2: shows "nonneg_cexpr V \<Gamma> e2"
path_input_containment: fixes M :: "('e, 'f, 'g) FSM"   and p :: "(('e \<times> 'f) \<times> 'g) list"   and q :: "'g" assumes "well_formed M"   and "path M p q" shows "set (map fst (map fst p)) \<subseteq> inputs M"
assms_6: fixes x :: "'a option list" assumes "b"   and "wf_tuple n C x"   and "P x" shows "Q x = (Q1 (restrict A x) \<and> Q2 (restrict B x))"
nonempty_update_with_trie: fixes ks :: "'c list"   and f :: "'d option \<Rightarrow> 'd"   and t :: "('c, 'd) trie" shows "\<not> is_empty_trie (update_with_trie ks f t)"
distinct_1: fixes x11 :: "nat"   and x12 :: "nat set"   and x21 :: "net_tree"   and x22 :: "net_tree" shows "\<langle>x11; x12\<rangle> \<noteq> x21 \<parallel> x22"
agree_def: fixes s :: "'d \<Rightarrow> bool"   and env :: "('d, bool) mapping" shows "agree s env = (\<forall>x b. Mapping.lookup env x = Some b \<longrightarrow> s x = b)"
eval_ground\<^sub>t: fixes t :: "fterm"   and E :: "char list \<Rightarrow> hterm" assumes "ground\<^sub>t t" shows "eval\<^sub>t E HFun t = hterm_of_fterm t"
parallel_constructive_security_aux_obsf: fixes real1 :: "('y + 'z, 'aa + 'ab) resource"   and ideal1 :: "('ac + 'z, 'ad + 'ab) resource"   and sim1 :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real1 :: "('y, 'aa) \<I>"   and \<I>_inner1 :: "('ac, 'ad) \<I>"   and \<I>_common1 :: "('z, 'ab) \<I>"   and adv1 :: "real"   and real2 :: "('ae + 'af, 'ag + 'ah) resource"   and ideal2 :: "('ai + 'af, 'aj + 'ah) resource"   and sim2 :: "('ae, 'ag, 'ai, 'aj) converter"   and \<I>_real2 :: "('ae, 'ag) \<I>"   and \<I>_inner2 :: "('ai, 'aj) \<I>"   and \<I>_common2 :: "('af, 'ah) \<I>"   and adv2 :: "real" assumes "constructive_security_aux_obsf real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 adv1"   and "constructive_security_aux_obsf real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 adv2" shows "constructive_security_aux_obsf (parallel_wiring \<rhd> real1 \<parallel> real2) (parallel_wiring \<rhd> ideal1 \<parallel> ideal2) (sim1 |\<^sub>= sim2) (\<I>_real1 \<oplus>\<^sub>\<I> \<I>_real2) (\<I>_inner1 \<oplus>\<^sub>\<I> \<I>_inner2) (\<I>_common1 \<oplus>\<^sub>\<I> \<I>_common2) (adv1 + adv2)"
1: shows "\<Theta>,\<Gamma> \<turnstile> fold (\<lambda>single. subst_term [single]) (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) B"
inv_unique: fixes y :: "'a mat"   and x :: "'a mat"   and y' :: "'a mat" assumes "y \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"   and "x \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> y' = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"   and "x \<in> carrier (ring_mat TYPE('a) n b)"   and "y \<in> carrier (ring_mat TYPE('a) n b)"   and "y' \<in> carrier (ring_mat TYPE('a) n b)" shows "y = y'"
abs_eq: fixes xa :: "'b \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'b \<Rightarrow> 'd option" shows "map_values xa (Mapping x) = Mapping (\<lambda>xb. map_option (xa xb) (x xb))"
lprefix_down_linear: fixes xs :: "'b llist"   and zs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs zs"   and "lprefix ys zs" shows "lprefix xs ys \<or> lprefix ys xs"
span_closed_sum_list: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> pmdl B" shows "sum_list xs \<in> pmdl B"
fresh_list_rev: fixes a :: "'a"   and xs :: "'b list" shows "a \<sharp> rev xs = a \<sharp> xs"
prems_1: shows "n'' \<in> HRB_slice {n}"
call_slice1: fixes n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p\<rightarrow>\<^bsub>call\<^esub> n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
MkIde_Dom: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.arr a" shows "Maps.MkIde (Maps.Dom a) = Maps.dom a"
sheaf_spec_def: fixes U :: "nat set set" shows "cring0.sheaf_spec U \<equiv> {s \<in> \<Pi>\<^sub>E \<pp>\<in>U. {0} \<^bsub>\<pp> ring0.subtraction ring0.subtraction 0\<^esub>. cring0.is_regular s U}"
assms_4: shows "fst (snd f) = fst (snd g)"
simps_1: fixes x :: "('a, 'b) Combinators"   and y :: "('a, 'b) Combinators" shows "first_srcNet (x \<oplus> y) = first_srcNet x"
fls_shift_nonneg_subdegree: fixes m :: "int"   and f :: "'b fls" assumes "m \<le> fls_subdegree f" shows "0 \<le> fls_subdegree (fls_shift m f)"
IH: shows "Digamma (of_nat (Suc n)) = harm n - euler_mascheroni"
update_updates_conv_if: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c"   and y :: "'d"   and al :: "('c \<times> 'd) list" shows "map_of (updates xs ys (update x y al)) = map_of (if x \<in> set (take (length ys) xs) then updates xs ys al else update x y (updates xs ys al))"
ord_resolve_counterex_reducing: fixes N :: "'a literal multiset set"   and DA :: "'a literal multiset" assumes "{#} \<notin> N"   and "DA \<in> N"   and "\<not> INTERP N \<Turnstile> DA"   and "\<And>C. \<lbrakk>C \<in> N; \<not> INTERP N \<Turnstile> C\<rbrakk> \<Longrightarrow> DA \<le> C" obtains CAs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and E :: "'a literal multiset" where "set CAs \<subseteq> N" and "INTERP N \<Turnstile>m mset CAs" and "\<And>CA. CA \<in> set CAs \<Longrightarrow> production N CA \<noteq> {}" and "ord_resolve CAs DA AAs As E" and "\<not> INTERP N \<Turnstile> E" and "E < DA"
CollInt_iff: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "{s. P s} \<inter> {s. Q s} = {s. P s \<and> Q s}"
exe_coregular_tcsigs_imp_coregular_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "exe_coregular_tcsigs cs arss" shows "coregular_tcsigs (set cs) (translate_ars arss)"
simps_8: fixes x3 :: "num"   and x2 :: "num" shows "num.Bit1 x3 \<noteq> num.Bit0 x2"
bval_Some: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option" assumes "vars b \<subseteq> dom s" shows "\<exists>bv. Def_Init_Exp.bval b s = Some bv"
ah_leq_il: fixes h1 :: "'a \<Rightarrow> 'a set"   and h2 :: "'a \<Rightarrow> 'a set"   and h1' :: "'a \<Rightarrow> 'a set"   and h2' :: "'a \<Rightarrow> 'a set" assumes "h1 [*] h2"   and "h1' \<le> h1"   and "h2' \<le> h2" shows "h1' [*] h2'"
r_ifeq_prim: shows "prim_recfn 4 r_ifeq"
hcomp_closed: fixes f :: "'c"   and g :: "'c" assumes "Arr f"   and "Arr g"   and "trg\<^sub>C f = src\<^sub>C g" shows "Arr (g \<star>\<^sub>C f)"
B5: shows "\<^bold>\<forall>x xa xb. x \<cdot> (xa \<cdot> xb) \<cong> (x \<cdot> xa) \<cdot> xb"
finprod_Un_Int: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "g \<in> A \<rightarrow> carrier G"   and "g \<in> B \<rightarrow> carrier G" shows "finprod G g (A \<union> B) \<otimes> finprod G g (A \<inter> B) = finprod G g A \<otimes> finprod G g B"
biQuot_def: fixes x :: "'b"   and y :: "'b"   and A :: "'b list set" shows "biQuot x y A = {xs. x # xs @ [y] \<in> A}"
simps_5: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "safety"   and r :: "'b regex"   and s :: "'b regex" shows "Regex.safe_regex fv safe Past g (Times r s) = ((g = Lax \<or> fv_regex fv s \<subseteq> fv_regex fv r) \<and> Regex.safe_regex fv safe Past g r \<and> Regex.safe_regex fv safe Past Lax s)"
H_neq_zeroD: fixes x :: "'c" assumes "H x \<noteq> 0" shows "x \<in> ball (0::'c) 2"
idx_sequence_idx: fixes idx :: "nat \<Rightarrow> nat"   and k :: "nat" assumes "idx_sequence idx" shows "idx k \<in> {idx k..<idx (Suc k)}"
map_zip_assoc: fixes f :: "('c \<times> 'd) \<times> 'e \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list"   and zs :: "'e list" shows "map f (zip (zip xs ys) zs) = Map2.map2 (\<lambda>x (y, z). f ((x, y), z)) xs (zip ys zs)"
lessThan_borel: fixes a :: "'a" shows "{..<a} \<in> sets borel"
merge_sort_monoms_sort_monoms_spec: shows "(RETURN \<circ> merge_monoms_poly, sort_coeff) \<in> \<langle>Id\<rangle>list_rel \<rightarrow>\<^sub>f \<langle>\<langle>Id\<rangle>list_rel\<rangle>nres_rel"
that: fixes m :: "real" assumes "0 < m"   and "\<And>x. x \<in> span B \<Longrightarrow> \<bar>representation B x b\<bar> \<le> m * norm x" shows "thesis"
p_natpow_prod: fixes n :: "nat"   and m :: "nat" shows "[p] \<cdot> \<one> [^] n \<otimes> [p] \<cdot> \<one> [^] m = [p] \<cdot> \<one> [^] (n + m)"
compass_reflect_compass_10: shows "apply_cltn2 K2_centre equator_reflect = K2_centre"
insert_def: fixes a :: "'b"   and B :: "'b set" shows "a \<triangleright> B = {x. x = a} \<union> B"
invar_var_goal_def: fixes I :: "'c \<Rightarrow> bool"   and R :: "('d \<times> 'd) set"   and v :: "'c \<Rightarrow> 'd"   and s :: "'c"   and s' :: "'c" shows "invar_var_goal I R v s s' \<equiv> I s' \<and> (v s', v s) \<in> R"
pprogress_def: fixes \<phi> :: "Formula.formula"   and \<pi> :: "(String.literal \<times> event_data list) prefix" shows "pprogress \<phi> \<pi> = (THE n. \<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> Monitor.progress \<sigma> Map.empty \<phi> (plen \<pi>) = n)"
natural_extension_3: shows "mpt.mpt_factor J.natural_extension_measure J.natural_extension_map J.natural_extension_proj MJ TJ"
assms_5: shows "kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
decode_instr_result_2: fixes v1 :: "word32" assumes "\<exists>e. decode_instruction v1 = Inl e" shows "\<nexists>v2. decode_instruction v1 = Inr v2"
can_lock_preserved: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and wf_state :: "(('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set) set"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t' :: "'b"   and ta' :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'a" assumes "preserve_deadlocked final r convert_RA wf_state"   and "s \<in> wf_state"   and "multithreaded_base.redT final r convert_RA s (t', ta') s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "multithreaded_base.must_sync final r t x (shr s)" shows "multithreaded_base.must_sync final r t x (shr s')"
subset_cball: fixes d :: "real"   and e :: "real"   and x :: "'b" assumes "d \<le> e" shows "cball x d \<subseteq> cball x e"
simps: fixes x :: "nat"   and y :: "nat" shows "(LnChain x \<le> LnChain y) = (x \<le> y)"
cases: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'g"   and a2 :: "'g" assumes "\<tau>trsys.silent_move trsys \<tau>move a1 a2" obtains s :: "'g" and s' :: "'g" and tl :: "'h" where "a1 = s" and "a2 = s'" and "trsys s tl s'" and "\<tau>move s tl s'"
Seq_ins: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "Seq s k"   and "k \<^bold>\<notin> hdomain s" shows "Seq (s \<triangleleft> \<langle>k, y\<rangle>) (succ k)"
all_total_iff_totalize: fixes F :: "'c program" shows "all_total F = (totalize F = F)"
fun_evaluate_single_relation_def: shows "fun_evaluate_single_relation = inv_image (less_than <*lex*> less_than) (\<lambda>x. case x of (x, s, e) \<Rightarrow> (clock s, size_exp' e))"
Max_wrt_altdef: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Max_wrt R = {x. R x x \<and> (\<forall>y. R x y \<longrightarrow> R y x)}"
SN_iff_wf: fixes A :: "('c \<times> 'c) set" shows "SN A = wf (A\<inverse>)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(OpDyn x1 = OpDyn y1) = (x1 = y1)"
ww: shows "w = sorted_list_of_set ({q. \<exists>t. (q, t) \<in> set (step (hd b) (a ! hd b)) \<and> t \<squnion>\<^bsub>f\<^esub> a ! q \<noteq> a ! q} \<union> set (tl b))"
\<C>_vars_def: fixes x :: "Eg1.var" shows "\<C>_vars x \<equiv> if x = buffer then {control_var} else {}"
sol_deriv: fixes s :: "real" assumes "s \<in> {0..t}" shows "(sol has_derivative (\<lambda>xa. xa *\<^sub>R ODE_sem I ODE (sol s))) (at s within {0..t})"
ostep_invariantD: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "('e \<times> 'f) \<times> 'g \<times> 'e \<times> 'f \<Rightarrow> bool"   and \<sigma> :: "'e"   and s :: "'f"   and a :: "'g"   and \<sigma>' :: "'e"   and s' :: "'f" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, s), a, \<sigma>', s')"
raise_trap_result: fixes t :: "Trap"   and s :: "'b sparc_state" shows "snd (raise_trap t s) = False"
disjoint_paths_new_path: assumes "\<And>S. Separation G v0 v1 S \<Longrightarrow> Suc (card paths) \<le> card S" shows "\<exists>P_new. v0 \<leadsto>P_new\<leadsto>\<^bsub>G\<^esub> v1 \<and> set P_new \<inter> G.second_vertices = {}"
isomorphicI: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "Maps.isomorphic (Maps.dom f) (Maps.cod f)"
mult_mv_mv: fixes B :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and v :: "complex \<times> complex" shows "B *\<^sub>m\<^sub>v (A *\<^sub>m\<^sub>v v) = B *\<^sub>m\<^sub>m A *\<^sub>m\<^sub>v v"
resumption_ord_resultD: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "Resumption.resumption.is_Done r'" shows "option_ord (Resumption.resumption.result r) (Resumption.resumption.result r')"
op_project_set_def: fixes X :: "'c set"   and b :: "'c"   and y :: "real" shows "op_project_set X b y = inter_sctn_spec X (Sctn b y)"
P3_3: shows "\<not> fst a \<subseteq>\<^sub>f s1"
eval_cases_10: fixes 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"   and 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 e' :: "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)" assumes "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" where "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1__\<rangle>" and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1__\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" where "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | e'a :: "char list exp" where "e' = throw e'a" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
SCF_stable: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('f, 'd) Term.term" assumes "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)" shows "vars_term_ms (SCF (s \<cdot> \<sigma>)) \<subseteq># vars_term_ms (SCF (t \<cdot> \<sigma>))"
h2: fixes v :: "'c" assumes "v \<in> S" shows "\<not> P v"
lsum_list_inf: fixes xs :: "'a llist" assumes "\<not> lfinite xs" shows "lsum_list xs = (0::'a)"
epi_implies_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.epi C f" shows "partial_magma.arr C f"
uminus_uminus: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "mbt_algebra.neg_assert one times inf dual bot (mbt_algebra.neg_assert one times inf dual bot x) = x"
tab_simps_2: shows "src\<^sub>C \<rho> = src\<^sub>C f"
"1_1": shows "H \<subseteq> map_to_set g"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "??.Sublist.prefix a b" shows "strict_prefix a b"
idem2_star: fixes s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" assumes "contains s r"   and "\<And>r'. r' \<in> set rs \<Longrightarrow> fst r' \<noteq> fst r" shows "\<gamma> s r = \<gamma> (s \<star> rs) r"
rat_zero_code: shows "quotient_of 0 = (0, 1)"
bot_not_top: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c" assumes "class.non_trivial_bounded_order less_eq less bot top" shows "bot \<noteq> top"
PAC_checker_specification_def: fixes spec :: "int mpoly"   and A :: "int mpoly multiset" shows "PAC_checker_specification spec A = SPEC (\<lambda>(b, \<V>, B). (\<not> is_failed b \<longrightarrow> restricted_ideal_to\<^sub>I (\<Union> (vars ` set_mset A) \<union> vars spec) B \<subseteq> restricted_ideal_to\<^sub>I (\<Union> (vars ` set_mset A) \<union> vars spec) A) \<and> (is_found b \<longrightarrow> spec \<in> pac_ideal (set_mset A)))"
flatten_monomial: fixes t :: "'c"   and c :: "'c \<Rightarrow>\<^sub>0 'd" shows "flatten (monomial c t) = punit.monom_mult (1::'d) t c"
no_deadlock: fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "B.reachable s" shows "\<exists>s'. B.step s s'"
field_differentiable_add: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable F"   and "g field_differentiable F" shows "(\<lambda>z. f z + g z) field_differentiable F"
select_convs_1: fixes entries :: "'a list"   and getentry :: "'a \<Rightarrow> nat option"   and more :: "'b" shows "entries \<lparr>entries = entries, getentry = getentry, \<dots> = more\<rparr> = entries"
not_reset_mono: fixes xs :: "('c \<Rightarrow> 'a) stream"   and c2 :: "'c"   and c1 :: "'c" assumes "stream_trans xs"   and "shd xs c2 \<le> shd xs c1"   and "stream_all (\<lambda>u. u \<in> V) xs"   and "c2 \<in> \<X>" shows "(holds (\<lambda>u. u c2 \<le> u c1) until holds (\<lambda>u. u c1 = (0::'a))) xs"
halfspace_eq_empty_le: fixes a :: "'a"   and b :: "real" shows "({x. a \<bullet> x \<le> b} = {}) = (a = (0::'a) \<and> b < 0)"
eval_prim_inv: fixes \<rho> :: "(nat \<times> bval) list"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp"   and v :: "bval" assumes "\<rho> \<turnstile> EPrim f e1 e2 \<Down> v" obtains n1 :: "nat" and n2 :: "nat" where "v = BNat (f n1 n2)" and "\<rho> \<turnstile> e1 \<Down> BNat n1" and "\<rho> \<turnstile> e2 \<Down> BNat n2"
hcomp_simps_1: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.hseq \<nu> \<mu>" shows "src\<^sub>D (\<nu> \<star>\<^sub>D \<mu>) = src\<^sub>D \<mu>"
T_d: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "d (p \<leadsto> x \<leadsto> q) = p \<leadsto> x \<leadsto> q"
is_zero_row_upt_k_suc: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" assumes "is_zero_row_upt_k i k A"   and "A $ i $ mod_type_class.from_nat k = (0::'b)" shows "is_zero_row_upt_k i (Suc k) A"
nun: shows "real_of_int n * u - s < real_of_int n - s"
hyps: shows "TypeExn (mk_id mn_ cn_) \<in> defined_types s_"
T_opt_def: fixes s :: "'c list"   and rs :: "'c list" shows "T_opt s rs = \<Sqinter> {T s rs as |as. length as = length rs}"
indecomposable_imp_Limit: fixes \<alpha> :: "V" assumes "indecomposable \<alpha>"   and "1 < \<alpha>" shows "Limit \<alpha>"
\<i>_simps_1: fixes a :: "'a" assumes "obj a" shows "S.seq (S.UP \<i>[a]) (S.cmp\<^sub>U\<^sub>P (a, a))"
assms_2: shows "\<not> (l = 0 \<and> \<bar>c\<bar> = \<infinity>)"
paths_hits_winning_strategy_is_winning: fixes v :: "'a" assumes "winning_strategy p** \<sigma>' v"   and "v \<in> lset P" shows "winning_path p** P"
det_four_block_mat_lower_left_zero_col: fixes A1 :: "'a mat"   and A2 :: "'a mat"   and n :: "nat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A1 \<in> carrier_mat 1 1"   and "A2 \<in> carrier_mat 1 n"   and "A3 = 0\<^sub>m n 1"   and "A4 \<in> carrier_mat n n" shows "Determinant.det (four_block_mat A1 A2 A3 A4) = Determinant.det A1 * Determinant.det A4"
cases: fixes a1 :: "'a OCL_Types.type"   and a2 :: "metaop"   and a3 :: "'a OCL_Types.type" assumes "mataop_type a1 a2 a3" obtains \<tau> :: "'a OCL_Types.type" where "a1 = \<tau>" and "a2 = AllInstancesOp" and "a3 = Set \<tau>"
entt_star_mono: fixes A :: "assn"   and B :: "assn"   and C :: "assn"   and D :: "assn" assumes "A \<Longrightarrow>\<^sub>t B"   and "C \<Longrightarrow>\<^sub>t D" shows "A * C \<Longrightarrow>\<^sub>t B * D"
normalize_gives_polynomial: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and K :: "'c set" assumes "ring R"   and "set p \<subseteq> K" shows "polynomial\<^bsub>R\<^esub> K (ring.normalize R p)"
digraph_map_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" shows "digraph_map_axioms G M \<equiv> edge_succ M permutes arcs G \<and> (\<forall>v. v \<in> verts G \<longrightarrow> out_arcs G v \<noteq> {} \<longrightarrow> cyclic_on (edge_succ M) (out_arcs G v))"
transfer_return1: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(R ===> crel_vs1 R) Wrap return"
abs_eq: shows "fls_X_inv = Abs_fls (\<lambda>n. if n = - 1 then 1::'a else (0::'a))"