hcju/isabelleps huggingface dataset:
not_integrable_integral_eq: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<not> integrable M f" shows "integral\<^sup>L M f = (0::'c)"
wfMultLess: fixes M :: "(Literal \<times> bool) list" shows "wf (multLess M)"
less_s: shows "(\<guillemotleft> G \<guillemotright> && (wp a Q \<circ> \<phi>)) s < (\<guillemotleft> G \<guillemotright> && wp b (Q \<circ> \<phi>)) s"
compose_assoc: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and h :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "f \<in> A \<rightarrow> B" shows "compose A h (compose A g f) = compose A (compose B h g) f"
finite_has_maximal2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and a :: "'a" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq a m \<and> (\<forall>b\<in>A. less_eq m b \<longrightarrow> m = b)"
bounded_clinear_mult_left: fixes y :: "'a" shows "bounded_clinear (\<lambda>x. x * y)"
polynate: fixes bs :: "'a list"   and p :: "poly" shows "\<lparr>polynate p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
of_real_divide: fixes x :: "real"   and y :: "real" shows "of_real (x / y) = of_real x / of_real y"
inj_on_inv_of_rat_rat: shows "inj_on (inv of_rat) \<rat>"
learn_lim_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_lim \<psi> U s" shows "\<exists>t. learn_lim \<chi> U t"
h1: shows "\<forall>i<deg + 1. MPoly_Type.degree (f i) var \<le> 1"
no_\<alpha>\<omega>: fixes x :: "\<Pi>\<^sub>1 set"   and y :: "\<omega>" shows "\<nu>\<upsilon> (\<alpha>\<nu> x) \<noteq> \<omega>\<upsilon> y"
lideal_axioms_def: fixes I :: "'c set"   and R :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "lideal_axioms I R multiplication \<equiv> \<forall>r a. r \<in> R \<longrightarrow> a \<in> I \<longrightarrow> multiplication r a \<in> I"
in_set_rev_path: fixes v' :: "'a"   and w :: "'b"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list" shows "((v', w, v) \<in> set (rev_path ps)) = ((v, w, v') \<in> set ps)"
someinit_def: fixes i :: "nat" shows "someinit i \<equiv> SOME x. x \<in> (fst \<circ> sr) ` init (np i)"
inject_5: fixes x1 :: "ccs"   and y1 :: "ccs" shows "(!x1 = !y1) = (x1 = y1)"
set_valued_transformation_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" shows "set_valued_transformation C S F G \<tau> \<equiv> (category C \<and> replete_set_category S) \<and> set_valued_functor C S F \<and> set_valued_functor C S G \<and> natural_transformation C S F G \<tau>"
matrix_matrix_mult_def: fixes m :: "(('c, 'f) vec, 'e) vec"   and m' :: "(('c, 'd) vec, 'f) vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $ i $ k * m' $ k $ j"
map_add_Some_iff: fixes m :: "'e \<Rightarrow> 'd option"   and n :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and x :: "'d" shows "((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
simps_2: fixes x :: "'e"   and xs :: "'e list"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli (x # xs) c f \<sigma> = (if c \<sigma> then foldli xs c f (f x \<sigma>) else \<sigma>)"
mat_plus_assoc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and m3 :: "'c list list" assumes "class.semigroup_add plus"   and "mat nr nc m1"   and "mat nr nc m2"   and "mat nr nc m3" shows "mat_plusI plus m1 (mat_plusI plus m2 m3) = mat_plusI plus (mat_plusI plus m1 m2) m3"
cobounded1: fixes a :: "'g multiset"   and b :: "'g multiset" shows "a \<inter># b \<subseteq># a"
sym_digraph_axioms: shows "sym_digraph (app_iso hom G)"
repeatable_f_nxt_upd: fixes f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and T :: "'b set"   and x' :: "'b" assumes "repeatable (f x)" shows "repeatable (f_nxt f T x' x)"
assms_1: shows "is_heap_list (priority t1) (children t1)"
wens_Union: fixes T :: "'b set"   and B :: "'b set"   and F :: "'b program"   and G :: "'b program"   and X :: "'b set" assumes "T - B \<subseteq> awp F T"   and "\<And>X. X \<in> wens_set F B \<Longrightarrow> T \<inter> X - B \<subseteq> awp G (T \<inter> X)"   and "X \<in> wens_set F B" shows "\<exists>Y\<in>wens_set (F \<squnion> G) B. Y \<subseteq> X \<and> T \<inter> X = T \<inter> Y"
sel_1: fixes x :: "quat"   and y :: "quat" shows "quat.Re (x + y) = quat.Re x + quat.Re y"
filter_tab_SomeD: fixes c :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and x :: "'c" assumes "filter_tab c t k = Some x" shows "t k = Some x \<and> c k x"
d2: fixes u :: "'a"   and v :: "'b" assumes "u \<in> S"   and "dist u x < d2"   and "v \<in> C" shows "f u \<inter> T \<inter> ball v (e / 2) \<noteq> {}"
disc_nodes_h': shows "h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h'"
conj_subst: fixes P :: "'a upred"   and R :: "'a upred"   and Q :: "'a upred" assumes "P = R" shows "(P \<and> Q) = (R \<and> Q)"
that: fixes \<tau> :: "nat \<Rightarrow> 'a" assumes "\<tau> \<simeq>A\<simeq> \<sigma>"   and "canonical_sequence A \<tau>" shows "thesis"
norm_ACI_NFAND: fixes \<phi> :: "('a, 'b) aformula" assumes "nf_ACI \<phi>" shows "\<phi> = NFAND (norm_ACI ` conjuncts \<phi>)"
order_trans: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "less_eq_prod x y"   and "less_eq_prod y z" shows "less_eq_prod x z"
Rep_coreflexive_inverse: fixes x :: "'b coreflexive" shows "Abs_coreflexive (Rep_coreflexive x) = x"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "I.wtE \<xi> \<equiv> \<forall>x. I_intT (tpOfV x) (\<xi> x)"
eq: fixes X :: "'a set" assumes "X \<in> E" shows "emeasure M X = emeasure N X"
in_P2R_pair: fixes x :: "'c"   and y :: "'d"   and \<phi> :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "((x, y) \<in> P2R \<phi>) = \<phi> x y"
last_snoc: fixes xs :: "'a list"   and x :: "'a" shows "last (xs @ [x]) = x"
game_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "ind_cca'.game \<A> = key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (ind_cca'.oracle_encrypt key b \<oplus>\<^sub>O ind_cca'.oracle_decrypt key) \<A> {} \<bind> (\<lambda>(b', L'). return_spmf (b = b'))))"
set_pmf_join_pmf: fixes f :: "'c pmf pmf" shows "set_pmf (join_pmf f) = \<Union> (set_pmf ` set_pmf f)"
set_drop_subset: fixes n :: "nat"   and xs :: "'c list" shows "set (drop n xs) \<subseteq> set xs"
induct: fixes P :: "(nat \<times> nat) list \<Rightarrow> bool"   and a0 :: "(nat \<times> nat) list" assumes "P []"   and "\<And>x y ps. \<lbrakk>\<And>xa. P ps; \<And>xa. P (deleteAround xa x ps)\<rbrakk> \<Longrightarrow> P ((x, y) # ps)" shows "P a0"
indifferent_prefI: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<preceq> x" shows "x \<^bsub>r\<^esub>\<approx> y"
the_default_eqvt: fixes P :: "'a \<Rightarrow> bool"   and p :: "perm"   and d :: "'a" assumes "\<exists>!x. P x" shows "p \<bullet> THE_default d P = THE_default (p \<bullet> d) (p \<bullet> P)"
trm_prm_simp_4: fixes \<pi> :: "'b prm"   and x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm" shows "\<pi> \<cdot> Fn x T A = Fn (\<pi> $ x) T (\<pi> \<cdot> A)"
assms_1: shows "fmdom' f1 \<subseteq> vs"
PSZ_1: shows "space P = space (density M (\<lambda>x. 1 + h x))"
norm_Pair1: fixes x :: "'c" shows "\<parallel>(0::'b, x)\<parallel> = \<parallel>x\<parallel>"
AA26: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and w :: "'a \<Rightarrow> 'c" assumes "|~ \<langle>A\<rangle>_v = \<langle>B\<rangle>_w" shows "\<turnstile> \<diamond>\<langle>A\<rangle>_v = \<diamond>\<langle>B\<rangle>_w"
iT_subset_conv_1: fixes n' :: "nat"   and n :: "nat" shows "([n'\<dots>] \<subseteq> [n\<dots>]) = (n \<le> n')"
B_authenticates_and_keydist_to_A: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and B :: "agent"   and Ts :: "nat" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "A Issues B with Crypt servK \<lbrace>Agent A, Number T3\<rbrace> on evs"
\<R>\<^sub>2p_\<R>\<^sub>2_eq: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "\<R>\<^sub>2p \<Gamma>' = (\<lambda>x xa xb xc xd xe. \<langle>x, xa, xb\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>xc, xd, xe\<rangle>)"
rel_fun_eq_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
inpt_def: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d" shows "inpt r x = (\<exists>y. r x y)"
simps: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms, mem) = (list_all (\<lambda>cm. sifum_security_init.locally_sound_mode_use dma \<C>_vars eval (cm, mem)) cms \<and> sifum_security_init.globally_sound_mode_use eval (cms, mem))"
finite_dimensional_vector_space_with_def: fixes plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>V\<^sub>S :: "'c"   and minus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale basis \<equiv> vector_space_with plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale \<and> finite_dimensional_vector_space_with_axioms plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis"
summable_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in sequentially. f x = g x" shows "summable f = summable g"
eqpoll_iff_card: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(A \<approx> B) = (card A = card B)"
cond_a_def: fixes b :: "nat list"   and xs :: "nat list"   and ys :: "nat list" shows "cond_a b xs ys = (\<forall>x\<in>set xs. x \<le> maxne0 ys b)"
mult_cancel_left2: fixes c :: "'a"   and a :: "'a" shows "(c * a = c) = (c = (0::'a) \<or> a = (1::'a))"
finprod_def: fixes G :: "('e, 'f) monoid_scheme"   and f :: "'g \<Rightarrow> 'e"   and A :: "'g set" shows "finprod G f A = (if finite A then foldD (carrier G) ((\<otimes>\<^bsub>G\<^esub>) \<circ> f) \<one>\<^bsub>G\<^esub> A else \<one>\<^bsub>G\<^esub>)"
mkIde_set: fixes a :: "'s" assumes "S.ide a" shows "S.mkIde (S.set a) = a"
rp: shows "rp.random_permutation = (\<lambda>\<sigma> x. case \<sigma> x of None \<Rightarrow> spmf_of_set A \<bind> (\<lambda>y. if y \<in> ran \<sigma> then spmf_of_set (A - ran \<sigma>) else return_spmf y) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
assms: shows "z \<in> set (polychain_of Pc Ps)"
rep_eq: shows "bounds_of_interval 0 = (0::'a, 0::'a)"
all_aligned_on_semifin_chain: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set"   and y :: "'a"   and z :: "'a" assumes "[f[x ..]X]"   and "y \<in> X"   and "z \<in> X"   and "x \<noteq> y"   and "x \<noteq> z"   and "y \<noteq> z" shows "[[x y z]] \<or> [[x z y]]"
ide_in_hom: fixes a :: "(bool, unit) X.arr" shows "X.ide a = X.in_hom a a a"
existence_ivl_cballs: fixes x0 :: "'a" assumes "0 \<in> UNIV"   and "x0 \<in> X" obtains t :: "real" and u :: "real" and L :: "real" where "\<And>y. y \<in> cball x0 u \<Longrightarrow> cball 0 t \<subseteq> existence_ivl0 y" and "\<And>s y. \<lbrakk>y \<in> cball x0 u; s \<in> cball 0 t\<rbrakk> \<Longrightarrow> flow0 y s \<in> cball y u" and "L-lipschitz_on (cball 0 t \<times> cball x0 u) (\<lambda>(t, x). flow0 x t)" and "\<And>y. y \<in> cball x0 u \<Longrightarrow> cball y u \<subseteq> X" and "0 < t" and "0 < u"
stop_seq_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Stop ;; c\<^sub>A_tail"   and "c\<^sub>C = Stop ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "(\<langle>c\<^sub>A_tail, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
assms_1: shows "\<lparr>[$C v1, $C v2, $C vn, $e]\<rparr> \<leadsto> \<lparr>[$C v3]\<rparr>"
possible_sym: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color"   and w' :: "'person \<Rightarrow> 'color" shows "local.possible n p w w' = local.possible n p w' w"
traces_failures: fixes xs :: "'a list"   and P :: "'a process" assumes "xs \<in> traces P" shows "(xs, {}) \<in> failures P"
enat_the_enat_iden: fixes n :: "enat" assumes "n \<noteq> \<infinity>" shows "enat (the_enat n) = n"
v_not_in: shows "v \<notin> fv\<^sub>s\<^sub>t (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
cycle_Cons: fixes x :: "'b"   and xs :: "'b list" shows "cycle (x # xs) = x ## cycle (xs @ [x])"
distinct_insort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "distinct (linorder.insort_key less_eq f x xs) = (x \<notin> set xs \<and> distinct xs)"
has_codomain_iff_arr: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "(Cop_S.codomains f \<noteq> {}) = Cop_S.arr f"
iT_Plus_neg_cut_ge2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus>- k) \<down>\<ge> (t - k) = I \<down>\<ge> t \<oplus>- k"
sets_bind: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and N :: "'d 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"
Enc_def: fixes SK :: "nat"   and K :: "nat"   and EK :: "nat"   and M :: "msg" shows "Enc SK K EK M = \<lbrace>Crypt K (sign SK M), Crypt EK (Key K)\<rbrace>"
inj_on_inverseI: fixes A :: "'h set"   and g :: "'i \<Rightarrow> 'h"   and f :: "'h \<Rightarrow> 'i" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
section_def: fixes s :: "'a \<Rightarrow> real"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real"   and S :: "'a set" shows "section s Ds S = ((\<forall>x. (s has_derivative blinfun_apply (Ds x)) (at x)) \<and> (\<forall>x. isCont Ds x) \<and> (\<forall>x\<in>S. s x = 0 \<longrightarrow> blinfun_apply (Ds x) (f x) \<noteq> 0) \<and> closed S \<and> S \<subseteq> X)"
FAssRedsNull: fixes e\<^sub>1 :: "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>2 :: "char list exp"   and v :: "val"   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 F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
mem: fixes x :: "'c"   and xs :: "'c list" shows "mem x xs = (x \<in> set xs)"
v_1: shows "v \<in> sphere (0::'a) 1"
assms_2: shows "inj_on f (\<Union>x\<in>set xs. x)"
valid_plans_def: fixes prob :: "'d set" shows "valid_plans prob \<equiv> {as. set as \<subseteq> prob}"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "poly.coeff (Mp f) i = M (poly.coeff f i)"
rep_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.rep \<l>[a] = \<^bold>\<parallel>\<^bold>\<l>\<^bold>[\<F>C.rep a\<^bold>]\<^bold>\<parallel>"
fmap_bind: fixes f :: "'f \<rightarrow> 'd"   and xs :: "'g\<cdot>'e"   and k :: "'g \<rightarrow> 'f\<cdot>'e" shows "fmap\<cdot>f\<cdot>(xs \<bind> k) = xs \<bind> (\<Lambda> x. fmap\<cdot>f\<cdot>(k\<cdot>x))"
qtable2: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<psi>) rel2"
assms: shows "Diag (t \<^bold>\<otimes> u)"
assms_2: shows "R2 \<subseteq> nprovRules"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list" assumes "\<And>x y p z. \<lbrakk>DenyAllFromTo x y \<in> set z \<Longrightarrow> P z; DenyAllFromTo x y \<notin> set z \<Longrightarrow> P z\<rbrakk> \<Longrightarrow> P (AllowPortFromTo x y p # z)"   and "\<And>y. P y \<Longrightarrow> P (DenyAll # y)"   and "\<And>v va y. P y \<Longrightarrow> P (DenyAllFromTo v va # y)"   and "\<And>v va y. P y \<Longrightarrow> P (v \<oplus> va # y)"   and "P []" shows "P a0"
a2: shows "B = Matrix.mat 2 1 (\<lambda>(i, j). if i = 0 then b0 else b1)"
3: shows "(f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)\<^sup>T * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
WTNil: 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" shows "P,E \<turnstile> [] [::] []"
assms_1: shows "poly (complex_of_int_poly p) x = 0"
Abs_bit_inverse: fixes y :: "bool" assumes "y \<in> UNIV" shows "Rep_bit (Abs_bit y) = y"
inv_in_hhom: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "local.iso \<mu>"   and "src \<mu> = a"   and "trg \<mu> = b" shows "\<guillemotleft>local.inv \<mu> : a \<rightarrow> b\<guillemotright>"
comm_groupE_5: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "comm_group G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
inext_in_iff: fixes n :: "nat"   and I :: "nat set" shows "(inext n I \<in> I) = (n \<in> I)"
Keys_minus_zero: fixes A :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "Keys (A - {0}) = Keys A"
finite_dimensional_vector_space_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Basis :: "'d set" shows "finite_dimensional_vector_space scale Basis \<equiv> vector_space scale \<and> finite_dimensional_vector_space_axioms scale Basis"
mat_of_cblinfun_zero: shows "mat_of_cblinfun 0 = 0\<^sub>m (length canonical_basis) (length canonical_basis)"
pivotandupdate_unsat_core_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<U>\<^sub>c (pivot_and_update x\<^sub>i x\<^sub>j c s) = \<U>\<^sub>c s"
Limsup_eqI: fixes F :: "'b filter"   and x :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>P. eventually P F \<Longrightarrow> x \<le> Sup (f ` Collect P)"   and "\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> Sup (f ` Collect P)) \<Longrightarrow> y \<le> x" shows "Limsup F f = x"
hyps_2: shows "liftAll (\<lambda>a. qSkelAbs (a $[[x \<and> y]]_zs) = qSkelAbs a) binp"
lists_succ_not_in: fixes x :: "'b"   and xss :: "'b list list" assumes "x \<notin> \<Union> (sset xss)" shows "lists_succ xss x = x"
body: shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"
nn_integral_indicator_map: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "\<integral>\<^sup>+ x. indicator {x \<in> space N. P x} (f x) \<partial>M = emeasure M {x \<in> space M. P (f x)}"
order_iff_strict: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b) = (a \<subset># b \<or> a = b)"
assms_3: shows "ref \<notin> set (interp_ins xs)"
NQ_def: shows "NQ = newQuery J Q_J_pos (I, zI)"
wp_Seq: fixes P :: "'a \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "P \<tturnstile> wp a Q"   and "Q \<tturnstile> wp b R"   and "well_def a"   and "well_def b"   and "sound Q"   and "sound R" shows "P \<tturnstile> wp (a ;; b) R"
intros_2: fixes t :: "('a, 'b, 'c, 'd) Tree" shows "(t, tNot t) \<in> Tree_wf"
other: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and \<sigma>' :: "nat \<Rightarrow> 'b" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "U \<sigma> \<sigma>'" shows "P (\<sigma>', l)"
maybe_and_invalid: fixes x :: "trilean"   and y :: "trilean" shows "(x \<and>? y = invalid) = (x = invalid \<or> y = invalid)"
hyps: shows "a{b}.Q \<longmapsto> \<alpha> \<prec> Q'"
11: shows "(\<Sum>y\<in>{x. poly p x = 0 \<and> poly q x < 0}. - 1) = - 1 * int (card {x. poly p x = 0 \<and> poly q x < 0})"
one_zeroI: assumes "carrier K = {\<zero>\<^bsub>K\<^esub>}" shows "\<one>\<^bsub>K\<^esub> = \<zero>\<^bsub>K\<^esub>"
d_sinpoly_cospoly: fixes n :: "nat"   and x :: "real" shows "(sinpoly (Suc n) has_real_derivative cospoly n x) (at x)"
CConcEnd1: fixes q :: "((int \<times> int) set set, int) Combinators list"   and p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators" assumes "q @ p \<noteq> []"   and "x \<notin> dom (C a)" shows "C (list2FWpolicy (q @ p @ [a])) x = C (list2FWpolicy (q @ p)) x"