hcju/isabelleps huggingface dataset:
priorities_set: fixes q :: "('d, 'c) pq" shows "set \<parallel>q\<parallel> = snd ` set (pq.alist_of q)"
filter_IFS_no_violations_accu_correct_induction: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and accu :: "('b \<times> 'b) list"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs (get_IFS M)"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = set accu\<rparr>)"   and "set accu \<union> set edgesList \<subseteq> E" shows "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = set (filter_IFS_no_violations_accu \<lparr>nodes = V, edges = E\<rparr> M accu edgesList)\<rparr>)"
less_asym: fixes x :: "'a list"   and y :: "'a list"   and P :: "bool" assumes "x <lex y"   and "\<not> P \<Longrightarrow> y <lex x" shows "P"
simps_2: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" shows "avl \<langle>l, (a, n), r\<rangle> = (\<bar>int (height l) - int (height r)\<bar> \<le> 1 \<and> n = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
mono_Sup: fixes f :: "'g \<Rightarrow> 'h"   and S :: "'g set" assumes "mono f" shows "Sup (f ` S) \<le> f (Sup S)"
antipar_1: shows "trg (tab\<^sub>0 r)\<^sup>* = src (tab\<^sub>0 r)"
snoc_4: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c'' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "P c t c'"   and "c' \<turnstile> ev \<mapsto> c''" shows "P c (t @ [ev]) c''"
less_imp_not_eq2: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "(y = x) = False"
prime_card_int: assumes "class.prime_card TYPE('b)" shows "prime (int CARD('b))"
simpleDefs_phiDefs_disjoint: fixes n :: "'a"   and g :: "('a, 'b, 'c) gen_cfg_wf" assumes "n \<in> set (gen_wf_\<alpha>n g)" shows "gen_wf_defs' g n \<inter> gen_wf.braun_ssa.phiDefs g n = {}"
that: fixes hb :: "real" assumes "0 < hb"   and "\<forall>\<^sub>F x in at_top. \<forall>h\<in>set hs'. \<bar>h x\<bar> \<le> hb * x / ln x powr (1 + e)" shows "thesis"
repl_fair_price_unique: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and \<pi> :: "real" assumes "replicating_portfolio pf der matur"   and "fair_price Mkt \<pi> der matur" shows "\<pi> = initial_value pf"
ereal_add_le_add_iff2: fixes a :: "ereal"   and c :: "ereal"   and b :: "ereal" shows "(a + c \<le> b + c) = (a \<le> b \<or> c = \<infinity> \<or> c = - \<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
evalC_def': fixes b :: "nat"   and n :: "C" shows "evalC b n = evalO b (C2O n)"
Cons_2: shows "\<forall>x y. x \<le> f x y \<and> y \<le> f x y"
mem: fixes X :: "'b set"   and x :: "'b" assumes "X \<in> P"   and "x \<in> X" shows "x \<in> f ` A"
getEnv_updEnv2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "(rho [x \<leftarrow> X]_xs) xs x = Some X"
widen_antisym: fixes m :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and T :: "ty"   and U :: "ty" assumes "wf_prog m P"   and "P \<turnstile> T \<le> U"   and "P \<turnstile> U \<le> T" shows "T = U"
changes_itv_smods_rec: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "poly (p * q) a \<noteq> 0"   and "poly (p * q) b \<noteq> 0" shows "changes_itv_smods a b p q = cross (p * q) a b + changes_itv_smods a b q (- (p mod q))"
hb: shows "monom 1 (p - j) \<circ>\<^sub>p [:1, - 1:] \<noteq> 0"
B_neq_T: shows "\<bottom> \<noteq> \<top>"
known_destinations_increase: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). kD (rt \<xi>) \<subseteq> kD (rt \<xi>'))"
fold_add_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "class.semigroup_add plus" shows "fold plus xs (plus x y) = plus (fold plus xs x) y"
map_to_set_upd: fixes m :: "'g \<Rightarrow> 'h option"   and k :: "'g"   and v :: "'h" shows "map_to_set (m(k \<mapsto> v)) = insert (k, v) (map_to_set m - {(k, v') |v'. True})"
v11_double_pair: fixes a :: "V"   and a' :: "V"   and b :: "V"   and b' :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "a' \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b"   and "r\<lparr>a'\<rparr> = b'" shows "(a = a') = (b = b')"
ball_empty: fixes P :: "'e \<Rightarrow> bool" shows "Ball {} P = True"
inc_above_dec_above_iff: fixes b :: "nat"   and i :: "nat"   and x :: "nat" shows "(inc_above b i (dec_above b i x) = x) = (x < b \<or> b + i \<le> x)"
inc_induct: fixes i :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i \<le> j"   and "P j"   and "\<And>n. \<lbrakk>i \<le> n; n < j; P (Suc n)\<rbrakk> \<Longrightarrow> P n" shows "P i"
fst_def: fixes prod :: "'d \<times> 'e" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
dom_mmupd: fixes m :: "'a \<Rightarrow> 'b option"   and K :: "'a set"   and v :: "'b" shows "dom (map_mmupd m K v) = dom m \<union> K"
successors_def: fixes g :: "'g"   and m :: "'node" shows "successors g m \<equiv> concat (map (\<lambda>n. if m \<in> set (predecessors g n) then [n] else []) (\<alpha>n g))"
IH: fixes y :: "nat"   and m :: "nat" assumes "y < n"   and "m \<le> y" shows "T'_mom_select m \<le> T'_mom_select y"
nonword_has_nonterminal: fixes a :: "symbol list" assumes "is_sentence a"   and "\<not> is_word a" shows "\<exists>k<length a. is_nonterminal (a ! k)"
applyBackjumpPreservedVariables: fixes state :: "State" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = applyBackjump state in getSATFlag state' = getSATFlag state"
reads_insert_writes_set_right: fixes getter :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and S :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x" assumes "reflp getter"   and "transp getter"   and "reads S f h h'" shows "reads (insert getter S) f h h'"
norm_Pair: fixes a :: "'a"   and b :: "'b" shows "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
is_finished_and_finished_item: fixes x :: "item" assumes "wellformed_item x" shows "is_finished x = (\<exists>\<alpha>. x = finished_item \<alpha>)"
wR: fixes x :: "'a" assumes "x \<cdot> x \<le> x" shows "x\<^sup>\<star> = (1::'a) + x"
1: shows "\<r>\<^sub>C[a] \<in> C.hom (a \<otimes>\<^sub>C \<I>\<^sub>C) a \<and> \<r>\<^sub>C[a] \<otimes>\<^sub>C \<I>\<^sub>C = (a \<otimes>\<^sub>C \<iota>\<^sub>C) \<cdot>\<^sub>C \<a>\<^sub>C[a, \<I>\<^sub>C, \<I>\<^sub>C]"
is_final_dstate_imp_not_weighted_RP: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "is_final_dstate St" shows "\<not> wstate_of_dstate St \<leadsto>\<^sub>w St'"
lzip_lappend: fixes xs :: "'b llist"   and us :: "'c llist"   and ys :: "'b llist"   and vs :: "'c llist" assumes "llength xs = llength us" shows "lzip (lappend xs ys) (lappend us vs) = lappend (lzip xs us) (lzip ys vs)"
path_less: shows "\<forall>z\<in>path_image \<gamma>. cmod (g z) < cmod (f z)"
r: shows "r = rat_of_int a' / rat_of_int b'"
bot_empty_eq: shows "bot_class.bot = (\<lambda>x. x \<in> {})"
that: fixes d' :: "'a" assumes "u(c := d') \<turnstile>\<^bsub>v,n\<^esub> M" shows "thesis"
ls_def: shows "ls = filter (\<lambda>y. y < x) xs"
lcm_D: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.left_complemented_monoid times inf imp_l one less_eq less" shows "imp_l a a = one"
Field_rel_restrict: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "Field (rel_restrict R A) \<subseteq> Field R - A"
3: shows "w = v\<^sub>2 @- v\<^sub>2'"
assms_2: shows "fst e1 \<noteq> fst e2"
is_mgu_is_unifiers: fixes \<sigma> :: "'s"   and AAA :: "'a set set" assumes "is_mgu \<sigma> AAA" shows "is_unifiers \<sigma> AAA"
matchChain: fixes P :: "pi"   and P' :: "pi"   and b :: "name" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P \<noteq> P'" shows "[b\<frown>b]P \<Longrightarrow>\<^sub>\<tau> P'"
Im_cinner_x_x: fixes x :: "'b" shows "Im \<langle>x, x\<rangle> = 0"
F1set2_JF1set_incl_JF1set: fixes a' :: "'d JF1"   and a :: "'d JF1" assumes "a' \<in> F1set2 (dtor1 a)" shows "JF1set a' \<subseteq> JF1set a"
blocksOf_nonInitBlks: fixes s :: "state"   and P :: "DiskBlock \<Rightarrow> bool"   and bk :: "DiskBlock"   and p :: "Proc" assumes "\<forall>p bk. bk \<in> blocksOf s p \<longrightarrow> P bk" shows "bk \<in> nonInitBlks s p \<longrightarrow> P bk"
assms_2: shows "elem \<lbrakk>\<Psi> (l', u') i'\<rbrakk> (l, u)"
mod_eq_0_imp_udvd: fixes w :: "'b word"   and v :: "'b word" assumes "w mod v = 0" shows "v udvd w"
card_of_Un_Plus_ordLeq: fixes A :: "'c set"   and B :: "'c set" shows "|A \<union> B| \<le>o |A <+> B|"
all_imp_iall: fixes P :: "nat \<Rightarrow> bool"   and I :: "nat set" assumes "\<forall>x. P x" shows "\<box> t I. P t"
core_iff: fixes a :: "'b set" shows "(a open\<^bsub>S\<^esub>) = (core\<^bsub>S\<^esub> a = a)"
module_hom_compose: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and s3 :: "'d \<Rightarrow> 'g \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "module_hom s1 s2 f"   and "module_hom s2 s3 g" shows "module_hom s1 s3 (g \<circ> f)"
inv_rule_basic: fixes T :: "('a, 'b) TS_scheme"   and P :: "'a set" assumes "init T \<subseteq> P"   and "{P} TS.trans T {> P}" shows "reach T \<subseteq> P"
z_before_x: shows "z < x in s_TS init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i + 1)"
bind_returns_result_E2: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and y :: "'z" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'aa" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
prefix: shows "g \<turnstile> Entry g-butlast sns\<rightarrow>last (butlast sns) \<and> last (butlast sns) \<in> set (predecessors g n)"
substNext_code_alt_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst_code.substNext_code phis g next = id(snd next := uninst_code.substitution_code phis g next)"
case_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case None of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
approximation_preproc_floatarith_11: fixes x :: "real" shows "sin x = cos (pi / 2 - x)"
homeomorphic_def: fixes s :: "'a set"   and t :: "'b set" shows "s homeomorphic t \<equiv> \<exists>f g. homeomorphism s t f g"
b: shows "Nml b \<and> \<not> is_Prim\<^sub>0 b"
filtration: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" assumes "filtrated_prob_space M F" shows "Filtration.filtration M F"
0: shows "v =\<^sub>F [a] @ remove1 a v"
eq_e: fixes l :: "'c"   and x :: "'a"   and y :: "'a"   and j :: "nat"   and x' :: "'b"   and i :: "nat"   and y' :: "'b" assumes "(l, x, y) \<in> edges (S j)"   and "(x, x') \<in> g i"   and "(y, y') \<in> g i" shows "(l, x', y') \<in> edges C"
le_sup_cong_c: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a \<squnion> c \<le> b \<squnion> c"
append_self_conv2: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<cdot> ys = ys) = (xs = [])"
Sem_eval_cases_3: fixes s :: "Var \<Rightarrow> Val"   and c1 :: "IMP"   and c2 :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , Comp c1 c2 \<rightarrow>\<^sub>n  t " obtains na :: "nat" and r :: "Var \<Rightarrow> Val" and m :: "nat" where "n = Suc (max na m)" and " s , c1 \<rightarrow>\<^sub>na  r " and " r , c2 \<rightarrow>\<^sub>m  t "
Vinv2_v_round: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{Vinv2 \<inter> Vinv1} v_round r v_f d_f {> Vinv2}"
insert_node: shows "h \<turnstile> a_insert_node ptr new_child reference_child_opt \<rightarrow>\<^sub>h h2"
permutes_bij: fixes p :: "'c \<Rightarrow> 'c"   and s :: "'c set" assumes "p permutes s" shows "bij p"
iso_tuple_update_accessor_cong_assist_id: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a" assumes "Record.iso_tuple_update_accessor_cong_assist upd ac" shows "upd id = id"
assms: shows "timpls_transformable_to TI (Fun f T) (Fun g S)"
seq_mono_left: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice)" shows "(c\<^sub>0::'a::refinement_lattice) ; (d::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice) ; d"
FL_bisimilar_is_L_bisimulation: shows "is_L_bisimulation FL_bisimilar"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and generat :: "('b, 'c, 'd) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
id1: shows "subresultant j (F (i - 2)) (F (i - 1)) = 0"
endpoints_in_S_perp_foot_cross_ratio_correct: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "a \<noteq> b"   and "proj2_incident a l"   and "proj2_incident b l" shows "cross_ratio_correct (endpoint_in_S a b) (endpoint_in_S b a) a (perp_foot c l)"
grounding_of_clss_def: fixes CC :: "('a, 'b) Term.term literal multiset set" shows "grounding_of_clss CC = \<Union> (grounding_of_cls ` CC)"
refl_on_converse: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "refl_on A (r\<inverse>) = refl_on A r"
diagonal_step_dvd2: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat" assumes "is_bezout_ext bezout" shows "diagonal_step A i j (case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> d) (case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> v) $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j"
to_nat_mono': fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "mod_type_class.to_nat a \<le> mod_type_class.to_nat b"
poly_eq: fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1"   and "t \<noteq> 1 / 2" shows "(Re ((poly p \<circ> circlepath z0 r) t) = 0) = (poly qq (tan (pi * t)) = 0)"
adds_term_alt: fixes u :: "'t"   and v :: "'t" shows "(u adds\<^sub>t v) = (\<exists>t. v = t \<oplus> u)"
converse_rtrancl3p_step: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and bs :: "'b list"   and a'' :: "'a" assumes "rtrancl3p r a (b # bs) a''" shows "\<exists>a'. r a b a' \<and> rtrancl3p r a' bs a''"
z: shows "z \<in> set (zippers_rose_tree e h (p, t))"
assms: shows "t \<in> trees (A \<times> UNIV)"
sum_diff: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "B \<subseteq> A" shows "sum f (A - B) = sum f A - sum f B"
simps_1: fixes x :: "'c" shows "remove1 x [] = []"
cinner_simps_1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "\<langle>x + y, z\<rangle> = \<langle>x, z\<rangle> + \<langle>y, z\<rangle>"
subtopology_nsphere_equator: fixes n :: "nat" shows "subtopology (nsphere (Suc n)) {x. x (Suc n) = 0} = nsphere n"
store_sops_append: fixes xs :: "instr list"   and ys :: "instr list" shows "store_sops (xs @ ys) = store_sops xs \<union> store_sops ys"
cases: fixes x :: "'a rose_tree" obtains data :: "'a" and subtrees :: "'a rose_tree list" where "x = Tree (data, subtrees)"
spr_absI: fixes t' :: "'s Trace"   and a :: "'a"   and t :: "'s Trace"   and v :: "'s" assumes "t' \<in> SPR.jkbpC"   and "spr_jview a t' = spr_jview a t"   and "v = tLast t'" shows "v \<in> spr_abs t"
axioms_1: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" assumes "module_hom s1 s2 f" shows "module s1"
deg_aboveD: fixes p :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "deg R p < m"   and "p \<in> carrier P" shows "coeff P p m = \<zero>"
subgroup_def: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" shows "H \<le> G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
inverse_real_alg_def: shows "inverse \<equiv> (rep_real_alg ---> Real_Alg_Quotient) inverse_3"
decompose_Max: fixes A :: "'b set" assumes "A \<subseteq> alts" shows "restrict_relation A (R i) = (\<lambda>x y. x \<in> A \<and> y \<in> Max_wrt_among (R i) A \<or> y \<notin> Max_wrt_among (R i) A \<and> restrict_relation (A - Max_wrt_among (R i) A) (R i) x y)"
CondReds2T: fixes e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(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"   and e\<^sub>1 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>"
matrix_matrix_mult_def: fixes m :: "(('d, 'g) vec, 'f) vec"   and m' :: "(('d, 'e) vec, 'g) vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $ i $ k * m' $ k $ j"
scale_measure_def: fixes r :: "ennreal"   and M :: "'b measure" shows "scale_measure r M = measure_of (space M) (sets M) (\<lambda>A. r * emeasure M A)"
var_prm_inaction: fixes a :: "'b"   and x :: "'b"   and b :: "'b" assumes "a \<noteq> x"   and "b \<noteq> x" shows "[a \<leftrightarrow> b] \<cdot> Var x = Var x"
right_space: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "right (space ts v c) = pos ts c + sensors (own v) ts c"
\<F>_semantics: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<mu> \<phi>. w \<Turnstile>\<^sub>n F\<^sub>n \<psi>}"
rbl_mul_def: fixes xs :: "bool list"   and ys :: "bool list" shows "rbl_mul xs ys = foldr (\<lambda>x sm. rbl_plus False (map ((\<and>) x) ys) (False # sm)) xs []"
suminf_emeasure: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "range A \<subseteq> sets M"   and "disjoint_family A" shows "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union> (range A))"
simps_7: fixes m :: "nat"   and p :: "('a, 'b) form" shows "closed m (Neg p) = closed m p"
Lim_component_ge: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and net :: "'b filter"   and b :: "real"   and i :: "'c" assumes "(f \<longlongrightarrow> l) net"   and "net \<noteq> bot"   and "\<forall>\<^sub>F x in net. b \<le> f x \<bullet> i" shows "b \<le> l \<bullet> i"
seq_SUP_distr: fixes P :: "'e \<Rightarrow> ('b, 'd) urel"   and A :: "'e set"   and Q :: "('d, 'c) urel" shows "\<Or>(P ` A) ;; Q = \<Or>((\<lambda>i. P i ;; Q) ` A)"
tllist_all2_TNil2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and ts :: "('a, 'c) tllist"   and b' :: "'d" shows "tllist_all2 P Q ts (TNil b') = (\<exists>b. ts = TNil b \<and> Q b b')"
bnd_c: shows "real_of_float c \<in>\<^sub>r ivlx"
mzero_im_sub_ker: fixes L :: "('i, 'a, 'j) Module_scheme"   and M :: "('k, 'a, 'l) Module_scheme"   and N :: "('m, 'a, 'n) Module_scheme"   and f :: "'i \<Rightarrow> 'k"   and g :: "'k \<Rightarrow> 'm" assumes "R module L"   and "R module M"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "compos L g f = mzeromap L N" shows "f ` carrier L \<subseteq> ker\<^bsub>M,N\<^esub> g"
subst_cls_def: fixes AA :: "('a, 'b) Term.term literal multiset"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_cls AA \<sigma> = {#subst_lit A \<sigma>. A \<in># AA#}"
bot_dres_def: shows "bot \<equiv> dSUCCEEDi"
transE: fixes r :: "('d \<times> 'd) set"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" obtains "(x, z) \<in> r"
cloU: shows "closedin (top_of_set (S \<inter> f -` T')) U"
less_b_least_impl_zero2: fixes y :: "nat"   and x :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "y < x"   and "b_least f x = x" shows "f x y = 0"
wt_subst_TAtom_fv: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and X :: "'var set" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "\<forall>x. wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"   and "\<forall>x\<in>fv t - X. \<exists>a. \<Gamma> (Var x) = Var a" shows "\<forall>x\<in>fv (t \<cdot> \<theta>) - fv\<^sub>s\<^sub>e\<^sub>t (\<theta> ` X). \<exists>a. \<Gamma> (Var x) = Var a"
size_4: fixes x21 :: "'a"   and x22 :: "'a list" shows "length (x21 # x22) = length x22 + Suc 0"
elims_1: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list"   and y :: "bool" assumes "successive x xa = y" obtains uu_ :: "'b \<Rightarrow> bool" where "x = uu_" and "xa = []" and "y"   | P :: "'b \<Rightarrow> bool" and uv_ :: "'b" where "x = Pa__" and "xa = [uv_]" and "y"   | P :: "'b \<Rightarrow> bool" and xb :: "'b" and ya :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # ya # xs" and "y = (\<not> P ya \<and> successive P xs \<or> \<not> P xb \<and> successive P (ya # xs))"
elims_2: fixes x :: "bool"   and xa :: "v sem_env"   and xb :: "'b state"   and xc :: "top0 list"   and xd :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_whole_prog x xa xb xc xd" obtains ck :: "bool" and env :: "v sem_env" and s1 :: "'b state" and tops :: "top0 list" and s2 :: "'b state" and res :: "(v sem_env, v) result" where "x = ck" and "xa = env" and "xb = s1" and "xc = tops" and "xd = (s2, res)" and "if no_dup_mods tops (defined_mods s1) \<and> no_dup_top_types tops (defined_types s1) then evaluate_prog ck env s1 tops (s2, res) else s1 = s2 \<and> res = Rerr (Rabort Rtype_error)"
t_set_multiset: fixes xt :: "'b bintree" shows "t_set xt = set_mset (t_multiset xt)"