hcju/isabelleps huggingface dataset:
redT_acquire: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x2"   and ln :: "'l \<Rightarrow>f nat"   and n :: "'l"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln"   and "0 < ln $ n"   and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)" shows "s -2-t\<triangleright>(K$ [], [], [], [], [], convert_RA ln)\<rightarrow> s'"
\<delta>_prod_2sng_2: fixes r1 :: "('a, 'b) ta_rule"   and r2 :: "('c, 'b) ta_rule" assumes "length (rhsq r1) \<noteq> length (rhsq r2)" shows "\<delta>_prod {r1} {r2} = {}"
MkArr_Map: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
boruvka_outer_invariant_when_e_not_bot: fixes j :: "'a"   and f :: "'a"   and h :: "'a"   and g :: "'a"   and d :: "'a" assumes "boruvka_inner_invariant j f h g d"   and "j \<noteq> bot"   and "selected_edge h j g \<le> - forest_components f"   and "selected_edge h j g \<noteq> bot" shows "boruvka_outer_invariant (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion> (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion> selected_edge h j g) g"
mult: fixes R :: "('c \<Rightarrow> real) set"   and S :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "function_ring_on R S"   and "f \<in> R"   and "g \<in> R" shows "(\<lambda>x. f x * g x) \<in> R"
litci_def: fixes v :: "nat"   and bdd :: "bddi" shows "litci v bdd \<equiv> tci bdd \<bind> (\<lambda>(t, bdd). fci bdd \<bind> (\<lambda>(f, bdd). ifci v t f bdd))"
intro: fixes Null :: "'a"   and Obj :: "'a set" assumes "Null \<notin> Obj" shows "discrete_category Obj Null"
galleryD_adj: fixes xs :: "'a set list" assumes "folding.gallery xs" shows "adjacentchain xs"
simple_labels_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp" shows "simple_labels \<Gamma> \<equiv> \<forall>pn. \<forall>p\<in>subterms (\<Gamma> pn). \<exists>!l. labels \<Gamma> p = {l}"
snd_inf: fixes x :: "'e \<times> 'd"   and y :: "'e \<times> 'd" shows "snd (inf x y) = inf (snd x) (snd y)"
pair_in_swap_image: fixes y :: "'d"   and x :: "'e"   and A :: "('e \<times> 'd) set" shows "((y, x) \<in> prod.swap ` A) = ((x, y) \<in> A)"
take_ind: shows "List.member (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))) x"
nat_eq_an_eq: fixes n :: "nat"   and m :: "nat" assumes "n = m" shows "an n = an m"
A: shows "{\<cdot> p } * post {\<cdot> q } \<le> S"
that: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "{u. u \<turnstile> cc} = [M]\<^bsub>v,n\<^esub>"   and "dbm_int M n" shows "thesis"
open_box: fixes a :: "'a"   and b :: "'a" shows "open (box a b)"
equivp_vimage2p: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd" assumes "equivp R" shows "equivp (BNF_Def.vimage2p f f R)"
set_times_intro2: fixes b :: "'a"   and C :: "'a set"   and a :: "'a" assumes "b \<in> C" shows "a * b \<in> a *o C"
inv_into_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "inv_into A f = (\<lambda>x. SOME y. y \<in> A \<and> f y = x)"
equivalence_in_bicategory_axioms: shows "internal_equivalence f g \<eta> \<epsilon>"
assms_1: shows "set (awalk_verts u p) \<subseteq> verts G"
comp1: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'a" assumes "bounded_bilinear prod"   and "bounded_linear g" shows "bounded_bilinear (\<lambda>x. prod (g x))"
graph_symmetric: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>G\<^esub> v" shows "v \<rightarrow>\<^bsub>G\<^esub> u"
is_singletonI': fixes A :: "'a set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
i_Exec_Comp_Stream_def: shows "i_Exec_Comp_Stream \<equiv> \<lambda>trans_fun input c n. f_Exec_Comp trans_fun (input \<Down> Suc n) c"
empty: shows "\<forall>\<R>\<in>set \<R>s. \<R> = Map.empty"
bidirected_digraph_axioms_def: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd" shows "bidirected_digraph_axioms G arev \<equiv> (\<forall>a. (a \<in> arcs G) = (arev a \<noteq> a)) \<and> (\<forall>a. a \<in> arcs G \<longrightarrow> arev (arev a) = a) \<and> (\<forall>a. a \<in> arcs G \<longrightarrow> tail G (arev a) = head G a)"
bot_covector_def: shows "bot \<equiv> Abs_covector bot"
simps_1: fixes t1 :: "trm"   and t2 :: "trm" shows "safe_formula (formula.Eq t1 t2) = (is_Const t1 \<and> (is_Const t2 \<or> is_Var t2) \<or> is_Var t1 \<and> is_Const t2)"
fls_zero_nth: fixes n :: "int" shows "fls_nth 0 n = (0::'b)"
ennreal_SUP_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> nat \<Rightarrow> ennreal" assumes "\<And>i. i \<in> I \<Longrightarrow> incseq (f i)" shows "(\<Squnion>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Squnion> range (f i))"
evtE: fixes t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" assumes "pb.eval the_bb t t' n (\<diamond>\<^sub>b\<gamma>)" shows "\<exists>n'\<ge>n. pb.eval the_bb t t' n' \<gamma>"
assms: shows "(x, y) \<in> S O (S O S\<^sup>* O NS\<^sup>* \<union> NS\<^sup>*)"
rem_cycles_distinct: fixes i :: "'b"   and j :: "'b"   and xs :: "'b list" shows "distinct (rem_cycles i j xs)"
toEnum_eq_to_fromEnum_eq: fixes n :: "nat"   and v :: "'b" assumes "n \<le> fromEnum maxBound" shows "(toEnum n = v) = (n = fromEnum v)"
adm_single_valued: fixes f :: "'a \<Rightarrow> ('b \<times> 'c) set" assumes "cont f" shows "adm (\<lambda>x. single_valued (f x))"
lm003: fixes R :: "('c \<times> 'd) set" assumes "runiq R" shows "R \<subseteq> graph (Domain R) (toFunction R)"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "xs \<down> m \<up> n = xs \<up> n \<down> (m - n)"
complex_mat_decomposition_to_hermitian: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "\<exists>B C. hermitian B \<and> hermitian C \<and> A = B + \<i> \<cdot>\<^sub>m C \<and> B \<in> carrier_mat n n \<and> C \<in> carrier_mat n n"
bisimilar_def: fixes Q :: "('e \<Rightarrow> 'f \<Rightarrow> bool) \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and s1 :: "'e \<Rightarrow> 'g"   and s2 :: "'f \<Rightarrow> 'h"   and x :: "'e"   and y :: "'f" shows "bisimilar Q s1 s2 x y \<equiv> \<exists>R. R x y \<and> (\<forall>x y. R x y \<longrightarrow> Q R (s1 x) (s2 y))"
char_eq_0_iff: fixes a :: "'a" shows "(\<chi> a = 0) = (a \<notin> carrier G)"
lsu_def: fixes x :: "'b" shows "\<nu>\<^sup>\<natural> x = \<top> \<rightarrow> x"
return_bind: fixes x :: "'a \<times> 'w list"   and f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (return x) f = f x"
lemma_Infinitesimal: fixes x :: "real" shows "(\<forall>r>0. x < r) = (\<forall>n. x < inverse (real (Suc n)))"
scaleR_right_distrib: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Cod (t \<^bold>\<star> u) = Cod t \<^bold>\<star> Cod u"
new_element_ptr_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "new_element_ptr |\<in>| element_ptr_kinds h'"
sumId: fixes P :: "ccs" shows "P \<oplus> \<zero> \<sim> P"
set_drop_subset_set_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list" assumes "n \<le> m" shows "set (drop m xs) \<subseteq> set (drop n xs)"
assms_2: shows "II2 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and more :: "'a" shows "signalsInit \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, \<dots> = more\<rparr> = signalsInit"
wp_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp (c\<^sub>1;; c\<^sub>2) Q = wp c\<^sub>1 (wp c\<^sub>2 Q)"
pred_def: fixes n :: "'a" shows "pred n = n - (1::'a)"
equal_ceq: assumes "ID CEQ('c) \<noteq> None" shows "equal ceq'"
filtersD3: fixes F :: "'b set set"   and T :: "'b set set"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> Filters\<^bsub>T\<^esub>"   and "A \<in> F"   and "B \<in> F" shows "A \<inter> B \<in> F"
drinks_step_2_none: fixes r :: "nat \<Rightarrow>f value option"   and e :: "String.literal \<times> value list" shows "ltl_step drinks (Some 2) r e = (None, [], r)"
Marker_in_channel_implies_not_done: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "Marker \<in> set (msgs (distributed_system.s channel trans send recv init t i) cid)"   and "channel cid = Some (p, q)"   and "i \<le> length t" shows "snd (cs (distributed_system.s channel trans send recv init t i) cid) \<noteq> Done"
mset_ran_cong: fixes a :: "int \<Rightarrow> 'a"   and b :: "int \<Rightarrow> 'a"   and r :: "int set" assumes "a = b on r" shows "mset_ran a r = mset_ran b r"
generate_empty: shows "generate K {} = {\<one>\<^bsub>K\<^esub>}"
op_defs_12: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "ev \<phi> \<equiv> until tr \<phi>"
is_finished_def: fixes x :: "item" shows "is_finished x = (item_nonterminal x = \<SS> \<and> item_origin x = 0 \<and> item_end x = length Doc \<and> is_complete x)"
P_closed: shows "P \<subseteq> Pow (space (Pi\<^sub>F {I} M))"
iso_unit: fixes a :: "'a" assumes "S.obj a" shows "S.iso \<i>[a]"
less_two: fixes j :: "nat" shows "(j < Suc (Suc 0)) = (j = 0 \<or> j = 1)"
lfinite_not_LNil_nth_llast: assumes "lfinite Sts"   and "Sts \<noteq> LNil" shows "\<exists>i. enat i < llength Sts \<and> lnth Sts i = llast Sts \<and> (\<forall>j<llength Sts. j \<le> enat i)"
bn_comp_fst_nonzero: fixes n :: "nat"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "bn_valid n p q"   and "0 < n" shows "0 < fst (bn_comp n p q r)"
induct: fixes x1 :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and x2 :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and x3 :: "('e, 'd, 'f) ann"   and x4 :: "('e, 'd, 'f) com"   and P :: "('d \<Rightarrow> ('e, 'd, 'f) com option) \<Rightarrow> ('d \<Rightarrow> ('e, 'd, 'f) ann list option) \<Rightarrow> ('e, 'd, 'f) ann \<Rightarrow> ('e, 'd, 'f) com \<Rightarrow> bool" assumes "ann_matches x1 x2 x3 x4"   and "\<And>\<Gamma> \<Theta> a. P \<Gamma> \<Theta> (AnnExpr a) Skip"   and "\<And>\<Gamma> \<Theta> a f. P \<Gamma> \<Theta> (AnnExpr a) (Basic f)"   and "\<And>\<Gamma> \<Theta> a r. P \<Gamma> \<Theta> (AnnExpr a) (Spec r)"   and "\<And>\<Gamma> \<Theta> a. P \<Gamma> \<Theta> (AnnExpr a) Throw"   and "\<And>\<Gamma> \<Theta> a e r b. \<lbrakk>ann_matches \<Gamma> \<Theta> a e; P \<Gamma> \<Theta> a e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec r a) (Await b e)"   and "\<And>\<Gamma> \<Theta> a1 p1 a2 p2. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 p1; P \<Gamma> \<Theta> a1 p1; ann_matches \<Gamma> \<Theta> a2 p2; P \<Gamma> \<Theta> a2 p2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnComp a1 a2) (Seq p1 p2)"   and "\<And>\<Gamma> \<Theta> a1 c1 a2 c2 a b. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 c1; P \<Gamma> \<Theta> a1 c1; ann_matches \<Gamma> \<Theta> a2 c2; P \<Gamma> \<Theta> a2 c2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnBin a a1 a2) (Cond b c1 c2)"   and "\<And>\<Gamma> \<Theta> a1 c1 a2 c2. \<lbrakk>ann_matches \<Gamma> \<Theta> a1 c1; P \<Gamma> \<Theta> a1 c1; ann_matches \<Gamma> \<Theta> a2 c2; P \<Gamma> \<Theta> a2 c2\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnComp a1 a2) (Catch c1 c2)"   and "\<And>\<Gamma> \<Theta> a' e a i b. \<lbrakk>ann_matches \<Gamma> \<Theta> a' e; P \<Gamma> \<Theta> a' e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnWhile a i a') (While b e)"   and "\<And>\<Gamma> \<Theta> a' e a f b. \<lbrakk>ann_matches \<Gamma> \<Theta> a' e; P \<Gamma> \<Theta> a' e\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec a a') (Guard f b e)"   and "\<And>\<Theta> f as \<Gamma> b n a. \<lbrakk>\<Theta> f = Some as; \<Gamma> f = Some b; n < length as; ann_matches \<Gamma> \<Theta> (as ! n) b; P \<Gamma> \<Theta> (as ! n) b\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnCall a n) (Call f)"   and "\<And>r \<Gamma> \<Theta> a c. \<forall>s\<in>r. ann_matches \<Gamma> \<Theta> a (c s) \<and> P \<Gamma> \<Theta> a (c s) \<Longrightarrow> P \<Gamma> \<Theta> (AnnRec r a) (DynCom c)"   and "\<And>as cs \<Gamma> \<Theta>. \<lbrakk>length as = length cs; \<forall>i<length cs. ann_matches \<Gamma> \<Theta> (fst (as ! i)) (cs ! i) \<and> P \<Gamma> \<Theta> (fst (as ! i)) (cs ! i)\<rbrakk> \<Longrightarrow> P \<Gamma> \<Theta> (AnnPar as) (Parallel cs)" shows "P x1 x2 x3 x4"
advantage2_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "advantage2 \<A> \<equiv> spmf (dis_log2 \<A>) True"
state_to_strips_state_map_le_iff: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option"   and t :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v" shows "(s \<subseteq>\<^sub>m t) = (\<phi>\<^sub>S \<Psi> s \<subseteq>\<^sub>m \<phi>\<^sub>S \<Psi> t)"
shEx_lift_seq_1: fixes P :: "'e \<Rightarrow> ('b, 'd) urel"   and Q :: "('d, 'c) urel" shows "ushEx P ;; Q = (\<^bold>\<exists> x \<bullet> P x ;; Q)"
CardinalityPrognosisEdom_axioms: shows "CardinalityPrognosisEdom prognosis"
fminus_disjoint: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<inter>| (B |-| A) = {||}"
norm_less_lminus: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x - n < y" shows "x \<le> y + (n - 1)"
f1_def: shows "f1 = (\<lambda>x. f x - real_cond_exp M Invariants f x)"
S_inductive_sup: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X" shows "minimum_elem D (upper_bounds D X) (Algebra2.Sup D X)"
convex_lincomb_def: fixes n :: "nat"   and c :: "'b vec \<Rightarrow> 'b"   and Vs :: "'b vec set"   and b :: "'b vec" shows "gram_schmidt.convex_lincomb n c Vs b = (gram_schmidt.nonneg_lincomb n c Vs b \<and> sum c Vs = (1::'b))"
proper_signed_list_map_uniform_snd: fixes b :: "bool"   and as :: "'a list" shows "proper_signed_list (map (\<lambda>s. (s, b)) as)"
wt_subst: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and \<pi> :: "Preliminaries.var \<Rightarrow> 'a trm"   and T :: "'a trm" assumes "Signature wtFsym wtPsym"   and "Signature.wtSB wtFsym arOf \<pi>"   and "Signature.wt wtFsym arOf T" shows "Signature.wt wtFsym arOf (subst \<pi> T)"
nip_eq: shows "PP x y = (P x y \<and> x \<noteq> y)"
simps_3: fixes n :: "nat" shows "stirling (Suc n) 0 = 0"
DFIG_def: shows "DFIG.DFIG = Var (DFIG_alphabet.v, DFIG_alphabet.v_update) (Schema DFIG.Init `;` lfp ((`;`) (req \<rightarrow> Schema DFIG.Out `;` out`!`list_select DFIG_alphabet.v \<rightarrow> Skip \<box> read ret (DFIG_alphabet.x, DFIG_alphabet.x_update) (Schema DFIG.Remove))))"
bisim1BlockSome4: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and V :: "nat"   and T :: "ty"   and v :: "'addr val" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,{V:T=\<lfloor>v\<rfloor>; e},h \<turnstile> ({V:T=None; e'}, xs) \<leftrightarrow> (stk, loc, Suc (Suc pc), xcp)"
eq: shows "f ` S = k ` (h \<circ> f) ` S"
one_stalk_def: fixes V :: "'a set" shows "stx.one_stalk V \<equiv> stx.class_of V (one_str\<^sub>X V)"
take_f_join_eq1: fixes n :: "nat"   and I :: "nat set"   and xs :: "'b list" assumes "n < card (I \<down>< length xs)" shows "xs \<Join>\<^sub> I \<down> n = xs \<Join>\<^sub> (I \<down>< (I \<rightarrow> n))"
is_ft_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
assms: shows "\<guillemotleft>f : a \<cong> b\<guillemotright>"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "pre_well_ordered_set A less_eq" shows "well_related_set A less_eq"
nth_map: fixes n :: "nat"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
a_notin_Ssegment: fixes a :: "'c"   and D :: "'c Order" shows "a \<notin> Ssegment D a"
IsProperInXY_def: shows "IsProperInXY \<equiv> map_fun (map_fun id (map_fun id eval\<o>)) id (\<lambda>\<phi>. \<forall>x y v. (\<exists>a b. \<nu>\<upsilon> a = \<nu>\<upsilon> x \<and> \<nu>\<upsilon> b = \<nu>\<upsilon> y \<and> \<phi> (a\<^sup>P) (b\<^sup>P) dj v) = \<phi> (x\<^sup>P) (y\<^sup>P) dj v)"
LEK_EQN: shows "\<forall>k'. i \<le> k' \<and> k' < k \<longrightarrow> snd (r k') = n"
induct: fixes P :: "(nat \<times> (nat \<times> 'a) list) list \<Rightarrow> (nat \<times> (nat \<times> 'a) list) list \<Rightarrow> bool"   and a0 :: "(nat \<times> (nat \<times> 'a) list) list"   and a1 :: "(nat \<times> (nat \<times> 'a) list) list" assumes "P [] []"   and "\<And>i a as. P as [] \<Longrightarrow> P ((i, a) # as) []"   and "\<And>j b bs. P [] bs \<Longrightarrow> P [] ((j, b) # bs)"   and "\<And>i a as j b bs. \<lbrakk>i < j \<Longrightarrow> P as ((j, b) # bs); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P ((i, a) # as) bs; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P as bs\<rbrakk> \<Longrightarrow> P ((i, a) # as) ((j, b) # bs)" shows "P a0 a1"
infinite_bf: fixes M :: "nat set"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and u :: "nat \<times> nat" shows "infinite (snd (bf M q u)) = infinite M"
reachObjLoc: fixes C :: "CTypeId"   and cf :: "CAttId"   and s :: "Store"   and a :: "nat" assumes "C = cls cf" shows "s\<turnstile> objLoc cf a reachable_from objV C a"
swap_in_iff_inv: fixes p :: "'b \<times> 'a"   and S :: "('a \<times> 'b) set" shows "(prod.swap p \<in> S) = (p \<in> S\<inverse>)"
size_prod: fixes l0 :: "'a"   and r0 :: "'b"   and l1 :: "'a"   and r1 :: "'b" shows "prod_index_def.size index_class.size index_class.size (Bound (l0, r0) (l1, r1)) = index_class.size (Bound l0 l1) * index_class.size (Bound r0 r1)"
Quotient_mapping_rbt: shows "Quotient (eq_onp (\<lambda>t. ord.is_rbt cless t \<or> ID ccompare = None)) Mapping_RBT impl_of cr_mapping_rbt"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "in_hom f a b" obtains "arr f" and "local.dom f = a" and "cod f = b"
vec_first_zero: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "vec_first (0\<^sub>v n) m = 0\<^sub>v m"
remdups_mset_singleton_sum: fixes a :: "'b"   and A :: "'b multiset" shows "remdups_mset (add_mset a A) = (if a \<in># A then remdups_mset A else add_mset a (remdups_mset A))"
pmod_image_characterization: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p \<noteq> []" shows "(\<lambda>q. UP.pmod q p) ` carrier (K [X]\<^bsub>K [X]\<^esub>) = {q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>). length q \<le> degree p}"
kcomp_distl: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'b set"   and h :: "'c \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K (g \<squnion> h) = f \<circ>\<^sub>K g \<squnion> f \<circ>\<^sub>K h"
eint_minus_distl: fixes b :: "eint"   and d :: "eint"   and c :: "eint" assumes "b \<noteq> d" shows "b * c - d * c = (b - d) * c"
bet2_le2__le2356: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A' B' C'"   and "Tarski_neutral_dimensionless.Le Bet Cong A B A' B'"   and "Tarski_neutral_dimensionless.Le Bet Cong A' C' A C" shows "Tarski_neutral_dimensionless.Le Bet Cong B' C' B C"
simps_1: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" shows "tree.root (tree v p pth) = (hyps_along ((adjacentTo v p, v, p) # pth) \<turnstile> labelAtIn v p, case adjacentTo v p of (x, xa) \<Rightarrow> toNatRule x xa)"
cod_dom: fixes f :: "'a" assumes "hom.arr f" shows "hom.cod (hom.dom f) = hom.dom f"
2: shows "(B * D) $$ (i mod dim_row B, j mod dim_col D) = row B (i mod dim_row B) \<bullet> col D (j mod dim_col D)"
oth_class_taut_3_a: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>& \<phi> \<^bold>\<equiv> \<phi> in v]"
pre_set: fixes c :: "('d, 'e, 'f) ann" shows "pre c = \<Inter> (pre_set c)"
ccFromList_Cons: fixes x :: "Vars.var"   and xs :: "Vars.var list" shows "ccFromList (x # xs) = {x} G\<times> set xs \<squnion> ccFromList xs"
elims_3: fixes x :: "'b list"   and xa :: "'b list set" assumes "\<not>  x \<in>B xa " obtains b :: "'b list" and G :: "'b list set" where "x = b" and "xa = G" and "\<not> (b \<in> G \<and> (\<forall>us. us \<in> lists G\<^sub>+ \<and> concat us = b \<longrightarrow> \<^bold>|us\<^bold>| = 1))"
INV_from_Refinement_basic: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec"   and I :: "'f set" assumes "refines R pi Sa Sc"   and "Range R \<subseteq> I" shows "reach Sc \<subseteq> I"
pcr_integer_def: shows "pcr_integer \<equiv> (=) OO cr_integer"
powser_0_nonzero: fixes r :: "real"   and \<xi> :: "'a"   and a :: "nat \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and m :: "nat" assumes "0 < r"   and "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums f x"   and "f \<xi> = (0::'a)"   and "a m \<noteq> (0::'a)"   and "0 < m" obtains s :: "real" where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> (0::'a)"
SKIP_minimal: fixes F :: "'c program" assumes "F \<le> \<bottom>" shows "F = \<bottom>"
rel_distinct_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y2 :: "'b" shows "\<not> rel_option R None (Some y2)"
refl: fixes x :: "'alt" assumes "x \<in> alts" shows "Pareto R x x"
sel_2: fixes x1 :: "'c"   and x2 :: "'a" shows "snd (x1, x2) = x2"
all_conv_all_member: fixes P :: "'b \<Rightarrow> bool"   and t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.all P t = (\<forall>x. RBT_Set2.member t x \<longrightarrow> P x)"