hcju/isabelleps huggingface dataset:
start_of_ge: fixes as :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and e :: "nat"   and ly :: "nat list" assumes "abc_fetch as ap = Some (Dec n e)"   and "ly = layout_of ap"   and "as < e" shows "start_of ly as + 2 * n + 16 \<le> start_of ly e"
preservation_1: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and t' :: "trm" assumes "\<Gamma> \<turnstile> t : T"   and "t \<longmapsto> t'" shows "\<Gamma> \<turnstile> t' : T"
filter_empty_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(List.filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
t1_space: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
assms_2: shows "\<forall>a b c. ord a b c \<longrightarrow> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
incomplete_designI: assumes "\<k> < \<v>" shows "incomplete_design \<V> \<B> \<k>"
assms_1: fixes x :: "'b"   and s :: "'a" assumes "x \<in> outs_\<I> \<I>" shows "results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"
WT2: shows "\<I>', \<I>'' \<turnstile>\<^sub>C conv2 \<surd>"
sba4_inf_right_unit: fixes x :: "'a" shows "x \<sqinter> top = x"
lt_j: shows "lt (fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i))) = term_of_pair (punit.lt (fs ! j), i)"
rel_interior_sum_gen: fixes I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> convex (S i)" shows "rel_interior (sum S I) = (\<Sum>i\<in>I. rel_interior (S i))"
app_def: fixes f :: "hf"   and x :: "hf" shows "app f x = (THE y. \<langle>x, y\<rangle> \<^bold>\<in> f)"
succs_finite: fixes V :: "'v set"   and ws :: "'v \<Rightarrow> 'v list option"   and u :: "'v"   and p :: "'v list" assumes "finite (E\<^sup>* `` U0)"   and "inv (None, V, ws)"   and "ws u = Some p" shows "finite (E `` {u})"
simps_6: fixes x1 :: "'d" shows "\<infinity> \<noteq> Le x1"
elims: fixes x :: "'a list"   and xa :: "'a list"   and y :: "'a list" assumes "p_fusion x xa = y" obtains uu_ :: "'a list" where "x = []" and "xa = uu_" and "y = []"   | v :: "'a" and va :: "'a list" where "x = v__ # va__" and "xa = []" and "y = []"   | v :: "'a" and va :: "'a list" and q :: "'a" and qs :: "'a list" where "x = v # va" and "xa = q # qs" and "y = (v # va) @ qs"
scaleC_le_0_iff: fixes a :: "complex"   and b :: "'b" assumes "a \<in> \<real>" shows "(a *\<^sub>C b \<le> (0::'b)) = (0 < a \<and> b \<le> (0::'b) \<or> a < 0 \<and> (0::'b) \<le> b \<or> a = 0)"
lossless_S1_xor: fixes A :: "bool \<times> bool"   and out :: "bool" shows "lossless_spmf (S1_xor A out)"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (a \<otimes> \<I>)"
sup_compl_top_left1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) (sup x y) = top"
mult_one_associative: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> top \<sqinter> y = x \<sqinter> y"
PartialMeetContraction_def: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set" shows "PartialMeetContraction Cn selection \<equiv> Tarskian_logic Cn \<and> PartialMeetContraction_axioms Cn selection"
fSum_zeroTr: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<in>nset n (n + m). f j = \<zero>) \<longrightarrow> \<Sigma>\<^sub>f A f n (n + m) = \<zero>"
chine_assoc_props_2: shows "f_src_src.Prj\<^sub>1 \<cdot> f_src_src.chine_assoc = f_src_src.Prj\<^sub>1\<^sub>1"
intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "AXHp (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F)"
matches_ex_entry_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and pc :: "nat"   and xcp :: "nat \<times> nat \<times> char list \<times> nat \<times> nat" shows "matches_ex_entry P C pc xcp \<equiv> let (s, e, C', h, d) = xcp in s \<le> pc \<and> pc < e \<and> P \<turnstile> C \<preceq>\<^sup>* C'"
rel_fun_iff_leq_vimage2p: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "rel_fun R S f g = (R \<le> BNF_Def.vimage2p f g S)"
neutral_const: fixes A :: "'d set" shows "(\<Prod>_\<in>A. (1::'c)) = (1::'c)"
no_illegal_updates_code_aux: fixes u :: "(nat \<times> vname aexp) list"   and r :: "nat" shows "(\<forall>u\<in>set u. fst u \<noteq> r) = no_illegal_updates_code u r"
sandwich_apply_adj: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "sandwich A *\<^sub>V B* = (sandwich A *\<^sub>V B)*"
has_vderiv_on_subset: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set"   and T :: "real set" assumes " D f = f' on S"   and "T \<subseteq> S" shows " D f = f' on T"
refl: fixes x :: "'a" assumes "x \<in> \<Union> (set As)" shows "of_weak_ranking As x x"
elims_2: fixes x :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "closed x" obtains \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "x = (\<Gamma>, e, S)" and "fv (\<Gamma>, e, S) \<subseteq> domA \<Gamma> \<union> upds S"
ws_unique_fields: fixes G :: "prog"   and C :: "qtname" assumes "is_class G C"   and "ws_prog G"   and "\<And>C c. class G C = Some c \<Longrightarrow> unique (cfields c)" shows "unique (DeclConcepts.fields G C)"
connected_edgeRtc: fixes u :: "nat"   and v :: "nat" shows "cf.connected u v = ((u, v) \<in> cf.E\<^sup>*)"
partial_summation_integrable_strong: fixes X :: "real set"   and y :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "finite X"   and "0 \<le> y"   and "y < x"   and "\<And>z. z \<in> {y..x} - X \<Longrightarrow> (f has_vector_derivative f' z) (at z)"   and "continuous_on {y..x} f" shows "(\<lambda>t. sum_upto a t * f' t) integrable_on {y..x}"
lub_take: shows "(\<Squnion>n. ValD_take n) = ID"
union_assoc: fixes M :: "'k multiset"   and N :: "'k multiset"   and K :: "'k multiset" shows "M + N + K = M + (N + K)"
rep_eq: fixes x :: "'c ell2"   and xa :: "'c ell2" shows "Rep_ell2 (x - xa) = (\<lambda>xb. Rep_ell2 x xb - Rep_ell2 xa xb)"
subcocycle_avg_ereal_cmult: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and c :: "real" assumes "subcocycle u"   and "0 \<le> c" shows "subcocycle_avg_ereal (\<lambda>n x. c * u n x) = ereal c * subcocycle_avg_ereal u"
timpl_closure_mono: fixes TI :: "('a set \<times> 'a set) set"   and TI' :: "('a set \<times> 'a set) set"   and t :: "(('b, 'c, 'a) prot_fun, 'd) Term.term" assumes "TI \<subseteq> TI'" shows "timpl_closure t TI \<subseteq> timpl_closure t TI'"
inf_commute: fixes inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
expectation_less: fixes X :: "'a \<Rightarrow> real"   and b :: "real" assumes "integrable M X"   and "AE x in M. X x < b" shows "expectation X < b"
is_joinI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.cpo below"   and "below x z"   and "below y z"   and "\<And>a. \<lbrakk>below x a; below y a\<rbrakk> \<Longrightarrow> below z a" shows "cpo.join below x y = z"
leB: fixes t :: "real" assumes "t \<in> {0..1}" shows "cmod (vector_derivative \<gamma> (at t)) \<le> B"
drop_take_drop_unsplit: fixes i :: "nat"   and j :: "nat"   and l :: "'a list" assumes "i \<le> j" shows "drop i (take j l) @ drop j l = drop i l"
matrix: shows "(V, E) = access_matrix \<lparr>pc_iiface = p_iiface p, pc_oiface = p_oiface p, pc_proto = p_proto p, pc_sport = p_sport p, pc_dport = p_dport p\<rparr> (to_simple_firewall (preprocess rs))"
planning_dimacs_complete_code: fixes prob :: "(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 \<pi>s :: "char list list"   and h :: "nat" assumes "ast_problem.well_formed prob"   and "\<forall>\<pi>\<in>set (ast_problem.ast\<delta> prob). is_standard_operator' \<pi>"   and "ast_problem.valid_plan prob \<pi>s"   and "length \<pi>s \<le> h" shows "let cnf_formula = SASP_to_DIMACS h prob in \<exists>dimacs_M. dimacs_model dimacs_M cnf_formula"
listall_nil: fixes P :: "'a \<Rightarrow> bool" shows "listall P []"
the_apartment_iso_middle_im: shows "the_apartment_iso A B ` D = E"
assms_2: shows "1 < degree (poly_real_alg_1 rai)"
simps_35: fixes P :: "'c prog"   and uu :: "char list"   and v :: "'d exp"   and va :: "'d exp" shows "icheck P uu (while (v) va) = False"
bezout_iterate_invertible: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and n :: "nat"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" assumes "is_bezout_ext bezout"   and "n < nrows A"   and "mod_type_class.to_nat i \<le> n"   and "A $ i $ j \<noteq> (0::'d)" shows "\<exists>P. invertible P \<and> P ** A = bezout_iterate A n i j bezout"
simps_1: shows "\<lbrakk>\<lbrakk> [] \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. True}"
SubstTermP_Zero: fixes t :: "tm"   and v :: "name" shows "{TermP t} \<turnstile> SubstTermP \<guillemotleft>Var v\<guillemotright> t Zero Zero"
sup_compl_top: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (uminus x) = top"
intro: fixes equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>x y. \<lbrakk>equiv x y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> equiv y x" shows "symmetric A equiv"
stoch_proc_filt_space: fixes M :: "'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure"   and n :: "nat" shows "space (stoch_proc_filt M X N n) = space M"
finfun_apply_inverse: fixes x :: "('a, 'b) finfun" shows "Abs_finfun (finfun_apply x) = x"
bijective: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "bijective \<alpha> S T" shows "bij_betw \<alpha> S T"
Nml_Nmlize_1: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Nml V src trg (bicategorical_language.Nmlize V t)"
current_of_bipartite_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "('a + 'a) \<times> ('a + 'a) \<Rightarrow> ennreal" assumes "web \<Gamma>" shows "web.current_of_bipartite \<Gamma> f = (\<lambda>(x, y). f (Inl x, Inr y) * indicator \<^bold>E\<^bsub>\<Gamma>\<^esub> (x, y))"
inj_on_g': fixes H :: "'a set"   and a :: "'a" assumes "H \<subseteq> carrier G"   and "a \<in> carrier G" shows "inj_on ((\<otimes>) a) H"
var1: fixes env :: "v sem_env"   and n :: "(char list, char list) id0"   and v1 :: "v"   and ck :: "bool"   and s :: "'b state" assumes "nsLookup (sem_env.v env) n = Some v1" shows "evaluate ck env s (Var n) (s, Rval v1)"
\<dd>_Times: fixes r :: "'a language"   and s :: "'a language" shows "\<dd> (Times r s) = (\<lambda>a. if \<oo> r then Plus (Times (\<dd> r a) s) (\<dd> s a) else Times (\<dd> r a) s)"
invariance_of_domain_gen: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open S"   and "continuous_on S f"   and "inj_on f S"   and "DIM('c) \<le> DIM('b)" shows "open (f ` S)"
steps_Single_True: fixes E :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "Graph_Defs.steps E [x] = True"
less_eq_dvd_minus: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(m dvd n) = (m dvd n - m)"
ExI: fixes w :: "vbl"   and A :: "formula"   and Gamma :: "formula list"   and R :: "(formula list \<times> formula list set) set" assumes "instanceF w A # Gamma \<in> deductions R"   and "Exs \<subseteq> R" shows "FAll Neg A # Gamma \<in> deductions R"
vsv_appE: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A"
absorb_iff2: fixes b :: "'e multiset"   and a :: "'e multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
inf_compl_bot_left2: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (- x \<sqinter> y) = \<bottom>"
snd_fst_flip: fixes xy :: "'c \<times> 'b" shows "snd xy = (fst \<circ> (\<lambda>(x, y). (y, x))) xy"
PiE_empty_domain: fixes T :: "'e \<Rightarrow> 'f set" shows "Pi\<^sub>E {} T = {\<lambda>x. undefined}"
exec_iff_execn: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t = (\<exists>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t)"
R_def: fixes A :: "'b set"   and u :: "nat"   and l :: "nat" shows "R A u l = map_pmf (\<lambda>f. steps A f 0 u l) (SL A)"
node_ptr_kinds_eq_h2: shows "node_ptr_kinds h2 = node_ptr_kinds h'"
FOREACHci_def: shows "FOREACHci \<equiv> FOREACHoci (\<lambda>_ _. True)"
complement_design: assumes "\<And>bl. bl \<in># \<B> \<Longrightarrow> incomplete_block bl" shows "design \<V> \<B>\<^sup>C"
leg1_simps_4: shows "trg\<^sub>C g = trg\<^sub>C r"
create_character_data_is_weakly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> create_character_data document_ptr text|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
threshold: shows "\<bar>x\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
red_Src: fixes t :: "'a local.term" assumes "Ide t" shows "Src t\<^bold>\<down> = Src t"
CCfix_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "CCfix \<Gamma>\<cdot>(ae, G) = (\<mu> x. ccBindsExtra \<Gamma>\<cdot>(ae, x) \<squnion> G)"
insert: fixes A :: "'g set"   and x :: "'g" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
diff_diff_commute_ereal: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" shows "x - y - z = x - z - y"
angle_actrans_sem: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "|~ \<langle>F\<rangle>_v = (F \<and> v$ \<noteq> $v)"
norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. norm (f x)) \<in> O[F](\<lambda>x. norm (g x))) = (f \<in> O[F](g))"
yy: shows "y \<notin> Fvars (subst \<phi> t1 x)"
eval_mmuaux_eq: shows "eval_mmuaux args nt aux = (rev done, tp, tss, len, maskL, maskR, a1_map, a2_map, [], 0)"
mod_ring_normalize64: shows "(mod_ring_rel64 ===> mod_ring_rel64) (\<lambda>x. if x = 0 then 0 else 1) normalize"
almost_full_on_Sigma: fixes P1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A1 :: "'c set"   and P2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A2 :: "'d set" assumes "almost_full_on P1 A1"   and "almost_full_on P2 A2" shows "almost_full_on (prod_le P1 P2) (A1 \<times> A2)"
compat_add_opp: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "local.add a p\<^sub>1 p\<^sub>2 = local.add a p\<^sub>1 (opp p\<^sub>2)"   and "p\<^sub>1 \<noteq> opp p\<^sub>1" shows "p\<^sub>2 = opp p\<^sub>2"
max_density_def: fixes G :: "nat set \<times> nat set set" shows "max_density G = Max (Ugraph_Lemmas.density ` nonempty_subgraphs G)"
inorder_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (RBT_Set2.del x t) = del_list x (Tree2.inorder t)"
correctCompositionIn_L1: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "ch \<notin> loc PQ"   and "ch \<in> ins P" shows "ch \<in> ins PQ"
borel_integrable_atLeastAtMost': fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "continuous_on {a..b} f" shows "set_integrable lborel {a..b} f"
swapVal_involutive: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b" shows "val ^[x \<and> y]_zs ^[x \<and> y]_zs = val"
finite_psubset_def: shows "finite_psubset = {(A, B). A \<subset> B \<and> finite B}"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xb :: "'a + 'b"   and xc :: "'a + 'b" assumes "sum_le x xa xb xc" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xd :: "'a" and y :: "'a" where "x = P__" and "xa = Q__" and "xb = Inl xd__" and "xc = Inl y__" and "P__ xd__ y__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xd :: "'b" and y :: "'b" where "x = P" and "xa = Q" and "xb = Inr xd" and "xc = Inr y" and "Q xd y"
new_bound: fixes k :: "real" assumes "0 \<le> k"   and "k \<le> 1"   and "\<lambda>s. k \<tturnstile> wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) (\<lambda>s. 1)" shows "\<lambda>s. p * (1 - k) + k \<tturnstile> wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) (\<lambda>s. 1)"
NotBoxD: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<not> [\<^bold>\<box>\<phi> in v]" shows "\<exists>v. \<not> [\<phi> in v]"
matrix_def: fixes M :: "'b list list"   and rs :: "nat"   and cs :: "nat" shows "matrix M rs cs = ((\<forall>row\<in>set M. length row = cs) \<and> length M = rs)"
L_conc: fixes M1 :: "nat rexp set"   and M2 :: "nat rexp set" shows "L (concS M1 M2) = L M1 @@ L M2"
n_distrib: fixes n :: "nat" shows "A ^ n * B ^ n = (A * B) ^ n"
subst_apply_pairs_nil: fixes \<delta> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "[] \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta> = []"
share_domain_changes: fixes a :: "nat"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "a \<in> all_shared sb \<union> all_unshared sb" shows "share sb \<S>' a = share sb \<S> a"
dmu_array: fixes mm :: "nat" assumes "mm \<le> length fs" shows "dmu_array (IArray fs) (length fs) (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) mm) mm = IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) (length fs)"
assms_4: shows "Q_I_neg = filterQueryNeg I Qn"
rel_gpv''_neg_distr: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and A :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and A' :: "'n \<Rightarrow> 'l \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'o \<Rightarrow> bool"   and C' :: "'o \<Rightarrow> 'm \<Rightarrow> bool" assumes "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_gpv'' (A OO A') (C OO C') (R OO R') \<le> rel_gpv'' A C R OO rel_gpv'' A' C' R'"
erf_remainder_integral_nonneg: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "0 \<le> erf_remainder_integral n x"
lookup_pair_map_val_pair: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and k :: "'a" assumes "tc.oalist_inv_raw xs"   and "f k (0::'b) = (0::'c)" shows "lookup_pair_tc (map_val_pair f xs) k = f k (lookup_pair_tc xs k)"
spn_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "src (bicategory_of_spans.spn V H \<a> \<i> src trg \<mu>) = src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.dom V \<mu>))"
IH: assumes "NewThread t x m'' \<in> set (concat (map (thr_a \<circ> snd) TTAS))"   and "wset_thread_ok (wset S) (thr S)" shows "thr S' t \<noteq> None"
comp_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.comp C f g = (if partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.comp C f g) then free_monoidal_category.mkarr C (free_monoidal_category.rep C f \<^bold>\<cdot> free_monoidal_category.rep C g) else partial_magma.null (free_monoidal_category.comp C))"
list_update_append1: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "i < length xs" shows "(xs @ ys)[i := x] = xs[i := x] @ ys"
boundedI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> b \<sqinter> c"
tag_name_eq2_h: fixes ptr' :: "(_) element_ptr" shows "|h \<turnstile> get_tag_name ptr'|\<^sub>r = |h2 \<turnstile> get_tag_name ptr'|\<^sub>r"
steps_hd_drop_suffix_finite: fixes f :: "nat \<Rightarrow> ('s, 'p, 'f) com list \<times> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list \<times> ('s, 'f) xstate"   and c :: "('s, 'p, 'f) com"   and cs :: "('s, 'p, 'f) com list"   and css :: "(('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and s :: "('s, 'f) xstate"   and \<Gamma> :: "'p \<Rightarrow> ('s, 'p, 'f) com option"   and k :: "nat"   and pcss :: "nat \<Rightarrow> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and pcs :: "nat \<Rightarrow> ('s, 'p, 'f) com list" assumes "f 0 = (c # cs, css, s)"   and "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "\<forall>i<k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"   and "\<forall>i\<le>k. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css" shows "\<Gamma>\<turnstile> ([c], [], s) \<rightarrow>\<^sup>* (pcs k, pcss k, S (f k))"
psi_fair_m1: shows "\<turnstile> psi \<longrightarrow> SF(m1)_(x, y)"