hcju/isabelleps huggingface dataset:
ring_hom_add: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
poly_y_minus_x_pCons: fixes a :: "'b"   and p :: "'b poly" shows "poly_y_minus_x (pCons a p) = [:[:a:]:] + poly_y_minus_x p * y_x"
Abs_y: shows "[\<lparr>A!,y\<^sup>P\<rparr> in v]"
action_proj_disj_diff: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs1 :: "'d set"   and vs2 :: "'d set"   and vs3 :: "'d set" assumes "action_dom (fst a) (snd a) \<subseteq> vs1"   and "vs2 \<inter> vs3 = {}" shows "action_proj (action_proj a (vs1 - vs2)) vs3 = action_proj a vs3"
containsI: fixes iA :: "nat \<times> formula"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and n :: "nat" assumes "iA \<in> set (nforms (f n))" shows "contains f iA n"
fold_rec: fixes A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "finite A"   and "x \<in> A" shows "Finite_Set.fold f z A = f x (Finite_Set.fold f z (A - {x}))"
lm058: fixes N :: "'c set"   and G :: "'d set" shows "allocationsUniverse \<inter> {a. Domain a \<subseteq> N \<and> \<Union> (Range a) = G} \<subseteq> allAllocations N G"
is_abelian_group_hom: shows "abelian_group_hom G H h"
inject: fixes x2 :: "'c"   and y2 :: "'c" shows "(Ref x2 = Ref y2) = (x2 = y2)"
assms_3: shows "ys \<noteq> zs \<or> y \<notin> {z, z1}"
P_def: fixes x :: "real"   and n :: "nat" shows "P x n = (\<Prod>k = 1..n. 1 - x\<^sup>2 / (real k)\<^sup>2)"
safe_step_preserves_unowned_shared: fixes ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>d (ts', m', \<S>')"   and "safe_delayed (ts, m, \<S>)"   and "simple_ownership_distinct ts"   and "unowned_shared \<S> ts" shows "unowned_shared \<S>' ts'"
simps_1: fixes f :: "'fsym"   and al :: "univ list" shows "GE_intF (Oldf f) al = intF f al"
beukers_integral2_def: shows "local.beukers_integral2 = LBINT (x, y):D. - ln (x * y) / (1 - x * y) * P x * P y"
CallParams: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s,b\<rangle> \<rightarrow> \<langle>Val v\<bullet>M(es'),s',b'\<rangle>"
exist: shows "{t1..t2} \<subseteq> existence_ivl0 x"
l_get_host_wf_axioms_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" shows "l_get_host_wf_axioms heap_is_wellformed known_ptrs type_wf get_host \<equiv> \<forall>h shadow_root_ptr. heap_is_wellformed h \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_host shadow_root_ptr"
comp_ide_self: fixes a :: "'a1" assumes "A1.ide a" shows "a \<cdot>\<^sub>A\<^sub>1 a = a"
case_ifexici_def: fixes fti :: "'a Heap"   and ffi :: "'a Heap"   and fii :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a Heap"   and ni :: "nat"   and bddi :: "bddi" shows "case_ifexici fti ffi fii ni bddi \<equiv> destrci ni bddi \<bind> (\<lambda>dest. case dest of TD \<Rightarrow> fti | FD \<Rightarrow> ffi | IFD v ti ei \<Rightarrow> fii v ti ei)"
emeasure_PiM_emb: fixes J :: "'a set"   and X :: "'a \<Rightarrow> 'b set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M'.events i" shows "emeasure (Pi\<^sub>M I (\<lambda>i. if i \<in> I then M i else count_space {undefined})) (P.emb I J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (if i \<in> I then M i else count_space {undefined}) (X i))"
eq_\<I>_gpv_mono': fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I>' :: "('c, 'd) \<I>" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "\<And>x y. A x y \<Longrightarrow> A' x y"   and "\<I> \<le> \<I>'" shows "eq_\<I>_gpv A' \<I>' gpv gpv'"
freshBoundDerivative_2: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "y \<sharp> P"   and "y \<noteq> x" shows "y \<sharp> P'"
sorted_is_sorted_list_of_set_2: fixes A :: "'b set"   and xs :: "'b list" assumes "is_sorted_list_of_set A xs" shows "distinct xs"
relation_of_Let: fixes v :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and A :: "('c, 'd) action" shows "action.relation_of (Denotational_Semantics.Let v A) = (action.relation_of A ;; R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = decrease v (alpha_rp.more A)))"
im_uminus_add: fixes g :: "'g"   and g' :: "'g" assumes "g \<in> G"   and "g' \<in> G" shows "T (- g + g') = - T g + T g'"
is_interval_existence_ivl: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X" shows "is_interval (ll_on_open.existence_ivl T f X t0 x0)"
\<kappa>_def: shows "\<kappa> = (\<lambda>x. if x < l then \<kappa>\<^sub>i x - \<kappa>\<^sub>l x * \<mu> i l else - \<mu> i l)"
sq_mtx_minus_eq: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" shows "A - B = to_mtx (\<chi>i j. A $$ i $ j - B $$ i $ j)"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>ys. P [] ys"   and "\<And>x xs ys. P ys xs \<Longrightarrow> P (x # xs) ys" shows "P a0 a1"
assms_1: shows "t \<rightarrow>\<^sub>v\<^sup>* v"
connected_component_eq_UNIV: fixes s :: "'a set"   and x :: "'a" shows "(connected_component_set s x = UNIV) = (s = UNIV)"
assms_2: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A' :: "'a trm" shows "A \<noteq> Fn x T A'"
mderivation_sound: fixes Cs :: "fterm literal set set"   and Cs' :: "fterm literal set set"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "mresolution_deriv Cs Cs'"   and "eval\<^sub>c\<^sub>s F G Cs" shows "eval\<^sub>c\<^sub>s F G Cs'"
foldi_fold_disj: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT.rbt"   and val :: "bool" shows "RBT.foldi (\<lambda>s. s = False) (\<lambda>k v s. s \<or> P k) t val = fold_keys (\<lambda>k s. s \<or> P k) t val"
safe_only_owner_enter_normal_aux: fixes s :: "state"   and r :: "room"   and k' :: "key"   and g :: "guest" assumes "s \<in> reach"   and "safe s r"   and "(k', roomk s r) \<in> cards s g" shows "owns s r = \<lfloor>g\<rfloor>"
countable_UN: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and A :: "'c \<Rightarrow> 'b set"   and X :: "'c set" assumes "sigma_algebra \<Omega> M"   and "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"
finsertI1: fixes a :: "'m"   and B :: "'m fset" shows "a |\<in>| finsert a B"
compact_Times: fixes s :: "'e set"   and t :: "'f set" assumes "compact s"   and "compact t" shows "compact (s \<times> t)"
bi_unique_rel_12_23: shows "bi_unique rel_12_23"
map_sel: fixes f :: "'e \<Rightarrow> 'd"   and a :: "('c, 'e) resource" shows "run_resource (map_resource' f a) = map_spmf (map_prod f (map_resource' f)) \<circ> run_resource a"
simps_2: fixes x :: "brick"   and xs :: "brick list"   and ys :: "brick list" shows "(x # xs) \<otimes> ys = x # xs \<otimes> ys"
submodule_def: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme" shows "LinearCombinations.submodule R N M \<equiv> (module R M \<and> N \<subseteq> carrier M) \<and> (\<forall>v w. v \<in> N \<longrightarrow> w \<in> N \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<in> N) \<and> \<zero>\<^bsub>M\<^esub> \<in> N \<and> (\<forall>c v. c \<in> carrier R \<longrightarrow> v \<in> N \<longrightarrow> c \<odot>\<^bsub>M\<^esub> v \<in> N)"
ftip_cons_falseI: fixes cld :: "cld"   and fds :: "fd list"   and P :: "cld list"   and ctx :: "ctx"   and f :: "char list"   and ctxcld_list :: "(ctx \<times> cld) list"   and ty_opt :: "ty option" assumes "class_fields cld fds"   and "ftype_in_fds P ctx fds f (ty_opt_bot_opt None)"   and "ftype_in_path P ctxcld_list f ty_opt" shows "ftype_in_path P ([(ctx, cld)] @ ctxcld_list) f ty_opt"
rev_image_eqI: fixes x :: "'k"   and A :: "'k set"   and b :: "'l"   and f :: "'k \<Rightarrow> 'l" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
asset_ii_unique: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s"   and "(Asset n, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s" shows "m = n"
FtorA: shows "Ftor (Hom\<^bsub>C\<^esub>[\<midarrow>,A]) : Op C \<longrightarrow> SET"
respects_cut_def: fixes A :: "('a \<times> 'a) set"   and C :: "'a set" shows "respects_cut A C \<equiv> A \<subseteq> C \<times> C \<union> (- C) \<times> - C"
kth_obs_unique: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "is_kth_obs \<pi> k i"   and "is_kth_obs \<pi> k j" shows "i = j"
pdivides_imp_is_root: fixes p :: "'a list list"   and x :: "'a list" assumes "p \<noteq> []"   and "x \<in> carrier (poly_ring R)"   and "[\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] pdivides\<^bsub>poly_ring R\<^esub> p" shows "UP.is_root p x"
finprod_empty: fixes f :: "'b \<Rightarrow> 'a" shows "sum f {} = (0::'a)"
odd_two_times_div_two_succ: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "plus (times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One)))) one = a"
backlinks_set: fixes E :: "('b \<times> 'b) list" shows "set (backlinks E) = {(e2, e1). (e1, e2) \<in> set E}"
new_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> new_shadow_root_ptr" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
weakTransCases: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Prop :: "act \<Rightarrow> ccs \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "\<lbrakk>\<alpha> = \<tau>; P = P'\<rbrakk> \<Longrightarrow> Prop (\<tau>) P"   and "P \<Longrightarrow>\<alpha> \<prec> P' \<Longrightarrow> Prop \<alpha> P'" shows "Prop \<alpha> P'"
WhileT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and c :: "expr"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>3 :: "expr"   and s\<^sub>3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P,E \<turnstile> \<langle>while (e) c,s\<^sub>2\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>" shows "P,E \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>"
matrix_vector_mult_0_right: fixes A :: "(('b, 'd) vec, 'c) vec" shows "A *v 0 = 0"
\<V>_def: shows "\<V> \<equiv> (\<lambda>U. {x \<in> topspace X. f x \<in> U}) ` \<U>"
iter_aux: fixes c :: "'b com"   and I :: "'b set"   and b :: "'b set"   and s :: "'b"   and n :: "nat"   and s' :: "'b" assumes "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I"   and "s \<in> I"   and "iter n b (Sem c) s s'" shows "s' \<in> I \<and> s' \<notin> b"
af_subsequence_W_GF_advice: fixes i :: "nat"   and n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltln"   and X :: "'b ltln set"   and \<phi> :: "'b ltln" assumes "i \<le> n"   and "suffix n w \<Turnstile>\<^sub>n af \<psi> (w [i \<rightarrow> n])[X]\<^sub>\<nu>"   and "\<And>j. j < i \<Longrightarrow> suffix n w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>" shows "suffix (Suc n) w \<Turnstile>\<^sub>n af (\<phi> W\<^sub>n \<psi>) (prefix (Suc n) w)[X]\<^sub>\<nu>"
that: fixes X :: "nat \<Rightarrow> 'a set" assumes "decseq X"   and "\<And>n. open (X n)"   and "\<And>n. x \<in> X n"   and "nhds x = (INF n. principal (X n))" shows "thesis"
mk_coeff_def: fixes vs :: "'b list"   and c :: "nat \<Rightarrow> 'a"   and v :: "'b" shows "mk_coeff vs c v \<equiv> R.sumlist (map c (find_indices v vs))"
length_replicate: fixes n :: "nat"   and x :: "'b" shows "length (replicate n x) = n"
gram_schmidt0_result_2: fixes ws :: "'b vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n" shows "distinct (gram_schmidt0 n ws)"
connected_inV_iff: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" assumes "Graph.connected c u v" shows "(v \<in> Graph.V c) = (u \<in> Graph.V c)"
cons_canceled: fixes a :: "bool \<times> 'e"   and x :: "(bool \<times> 'e) list" assumes "canceled (a # x)" shows "canceled x"
a: shows "a \<in> U \<union> U' \<rightarrow> carrier K"
TIP: shows "\<forall>\<phi>. \<phi> \<in> fmla \<and> Fvars \<phi> = {} \<and> isTrue (wrepr.PP \<langle>\<phi>\<rangle>) \<longrightarrow> prv \<phi>"
nth_prime_unique_aux: fixes p :: "nat"   and n :: "nat"   and p' :: "nat" assumes "prime p"   and "card {q. prime q \<and> q < p} = n"   and "prime p'"   and "card {q. prime q \<and> q < p'} = n" shows "p = p'"
finite_S0: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R" shows "finite S0"
uv_sigma_2: shows "(V, y) \<in> Sigma I \<FF>"
B_Issues_A: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Says B A (Crypt K (Number Ta)) \<in> set evs"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "B Issues A with Crypt K (Number Ta) on evs"
in_hom_def: fixes f :: "'c"   and a :: "'c"   and b :: "'c" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<equiv> a \<in> C.domains f \<and> b \<in> C.codomains f"
True_in_funpow_harry_step: fixes n :: "nat"   and xs :: "bool list" assumes "n < harry_meas xs" shows "True \<in> set ((harry_step ^^ n) xs)"
r_dettime_1: fixes x :: "nat"   and i :: "nat" assumes "determined i x" shows "dettime i x \<down>= (LEAST t. x < e_length (block i t) \<and> x \<noteq> gap i t)"
cont: fixes i :: "'a" assumes "i \<in> I" shows "continuous_map (subtopology X (T i)) Y (f i)"
butlast_snoc: fixes xs :: "'g list"   and x :: "'g" shows "butlast (xs @ [x]) = xs"
uniformly_continuous_onE: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and e :: "real" assumes "uniformly_continuous_on s f"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>x x'. \<lbrakk>x \<in> s; x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
cod_char: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "A2_B.cod f = (if A2_B.arr f then A2_B.MkIde (A2_B.Cod f) else A2_B.null)"
exists_trace_for_any_i_j: 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 c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and j :: "nat" assumes "distributed_system channel"   and "\<exists>c'. distributed_system.trace channel trans send recv c t c'"   and "i \<le> j" shows "distributed_system.trace channel trans send recv (distributed_system.s channel trans send recv c t i) (take (j - i) (drop i t)) (distributed_system.s channel trans send recv c t j)"
Smith_normal_form_upt_k_condition2: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and a :: "'d"   and b :: "'c" assumes "Smith_normal_form_upt_k A k"   and "mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b"   and "mod_type_class.to_nat a < k \<or> mod_type_class.to_nat b < k" shows "A $ a $ b = (0::'b)"
unit_closed: shows "L.Class \<one> \<in> L.Partition"
root: shows "a * r\<^sup>2 - 2 * a * r * r + c = 0"
has_path_trans: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "has_path n S i j"   and "has_path n S j k" shows "has_path n S i k"
beta_abs: fixes a :: "name"   and t :: "lam"   and t' :: "lam" assumes " Lam [a].t \<longrightarrow>\<^sub>\<beta> t'" shows "\<exists>t''. t' = Lam [a].t'' \<and>  t \<longrightarrow>\<^sub>\<beta> t''"
valid_prefix_alt: fixes p :: "'b prefix_match" shows "valid_prefix p = (pfxm_prefix p && 2 ^ (LENGTH('b) - pfxm_length p) - 1 = 0)"
unop_f: fixes t :: "t"   and \<C> :: "t_context"   and uv :: "unop_f" assumes "is_float_t t" shows "\<C> \<turnstile> [Unop_f t uv] : [t] _> [t]"
rep_eq: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'d monom" shows "eval_monom x xa = eval_monom_list x (Rep_monom xa)"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism T T f g"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"   and "bounded {x. \<not> (f x = x \<and> g x = x)}"   and "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" shows "thesis"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>S)"
sympartp_sympartp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sympartp (sympartp r) = sympartp r"
simps_4: fixes e :: "expr"   and r :: "ref_ty" shows "assignsE (e InstOf r) = assignsE e"
fset_equiv: fixes f1 :: "'b fset"   and f2 :: "'b fset" shows "(f1 = f2) = (fset f1 = fset f2)"
sets_bind: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets N"
mk_mat_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b" shows "local.mk_mat f = (\<chi>i j. f i j)"
le_cases3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" obtains "x \<le> y" and "y \<le> z"   | "y \<le> x" and "x \<le> z"   | "x \<le> z" and "z \<le> y"   | "z \<le> y" and "y \<le> x"   | "y \<le> z" and "z \<le> x"   | "z \<le> x" and "x \<le> y"
IH_1: shows "State_Monad.ap (State_Monad.return dlabels) (label_tree l) = symbols (leaves l)"
NUM_imp_learn_total: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set" assumes "\<psi> \<in> \<R>\<^sup>2"   and "U \<in> NUM_wrt \<psi>" shows "learn_total \<psi> U (min_cons_hyp \<psi>)"
root_list_lb: fixes ps :: "real poly list"   and b :: "real" assumes "0 \<notin> set ps" obtains lb :: "real" where "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb < x" and "\<forall>x\<le>lb. \<forall>p\<in>set ps. sgn (poly p x) = sgn_neg_inf p" and "lb < b"
respects_own_guarantees_def: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and cm :: "'e \<times> (Mode \<Rightarrow> 'c set)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.respects_own_guarantees dma \<C>_vars eval cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> sifum_security_init.doesnt_read_or_modify \<C>_vars eval (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> sifum_security_init.doesnt_modify dma eval (fst cm) x)"
disjoint_union: fixes C :: "'a set set"   and B :: "'a set set" assumes "disjoint C"   and "disjoint B"   and "\<Union> C \<inter> \<Union> B = {}" shows "disjoint (C \<union> B)"
succ_works_2: fixes va :: "'a" assumes "va \<in> VV p" shows "va \<in> W1 \<or> succ va \<notin> W1"
card_image_le: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "order.greater_eq (card A) (card (f ` A))"
median_5_in_set: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and e :: "'b" shows "median_5 a b c d e \<in> {a, b, c, d, e}"
generalizes_trans: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset" assumes "generalizes C D"   and "generalizes D E" shows "generalizes C E"
iwlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "iwlsFSw MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSwapCls MOD \<and> igAbsCongS MOD"
zero_eq_add_iff_both_eq_0: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(zero = plus x y) = (x = zero \<and> y = zero)"