hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "nat"   and a3 :: "('fun, 'var) Term.term" assumes "\<langle>a1; a2\<rangle> \<turnstile>\<^sub>n a3" obtains t :: "('fun, 'var) Term.term" and M :: "('fun, 'var) Term.term set" where "a1 = Ma__" and "a2 = 0" and "a3 = ta__" and "ta__ \<in> Ma__"   | T :: "('fun, 'var) Term.term list" and f :: "'fun" and M :: "('fun, 'var) Term.term set" and steps :: "('fun, 'var) Term.term \<Rightarrow> nat" where "a1 = Mb__" and "a2 = Suc (Max (insert 0 (steps__ ` set T__)))" and "a3 = Fun f T__" and "length T__ = arity f" and "public f" and "\<And>t. t \<in> set T__ \<Longrightarrow> \<langle>Mb__; steps__ t\<rangle> \<turnstile>\<^sub>n t"   | M :: "('fun, 'var) Term.term set" and n :: "nat" and t :: "('fun, 'var) Term.term" and K :: "('fun, 'var) Term.term list" and T :: "('fun, 'var) Term.term list" and steps :: "('fun, 'var) Term.term \<Rightarrow> nat" and t\<^sub>i :: "('fun, 'var) Term.term" where "a1 = M" and "a2 = Suc (Max (insert n (steps ` set K)))" and "a3 = t\<^sub>i" and "\<langle>M; n\<rangle> \<turnstile>\<^sub>n t" and "Ana t = (K, T)" and "\<And>k. k \<in> set K \<Longrightarrow> \<langle>M; steps k\<rangle> \<turnstile>\<^sub>n k" and "t\<^sub>i \<in> set T"
Fun_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "S.Fun (g \<cdot> f) = restrict (S.Fun g \<circ> S.Fun f) (S.set (S.dom f))"
arr_iff_in_hom: fixes f :: "'a2" shows "A2.arr f = \<guillemotleft>f : A2.dom f \<rightarrow>\<^sub>A\<^sub>2 A2.cod f\<guillemotright>"
prefs': shows "u \<le>p s' \<^sup>@ l"
i_take_drop_append: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "f \<Down> m @ f \<Down> n \<up> m = f \<Down> n"
NC2Sep: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll1 p" shows "NetsCollected2 (separate p)"
trm_strand_subst_cong_2: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t \<in> trms\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)" shows "(\<exists>t'. t = t' \<cdot> \<delta> \<and> t' \<in> trms\<^sub>s\<^sub>t S) \<or> (\<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> (\<exists>t'\<in>trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. t = t' \<cdot> rm_vars (set X) \<delta>))"
rQuots_simps_1: fixes A :: "'b list set" shows "rQuots [] A = A"
widen_asym: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "ty"   and D :: "ty" assumes "wf_prog wf_md P"   and "P \<turnstile> C \<le> D"   and "C \<noteq> D" shows "\<not> P \<turnstile> D \<le> C"
rbt_lookup_rbt_delete: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and k :: "'c" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_delete less k t) = ord.rbt_lookup less t |` (- {k})"
inject_2: fixes x2 :: "char list"   and y2 :: "char list" shows "(Var x2 = Var y2) = (x2 = y2)"
r2p_surj: shows "\<forall>P. \<exists>R\<subseteq>Id. P = r2p R"
big_stepD: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state" assumes "\<sigma>' \<in> big P \<sigma>"   and "\<sigma> \<notin> endset" shows "\<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>' \<in> big P \<sigma>1"
\<R>\<^sub>L_comp: fixes \<beta> :: "'d \<Rightarrow> 'd"   and \<beta>' :: "'d \<Rightarrow> 'd"   and \<alpha>' :: "'e \<Rightarrow> 'e"   and \<alpha> :: "'e \<Rightarrow> 'e"   and ls :: "('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr" assumes "bij \<beta>"   and "bij \<beta>'" shows "\<R>\<^sub>L \<alpha>' \<beta>' (\<R>\<^sub>L \<alpha> \<beta> ls) = \<R>\<^sub>L (\<alpha>' \<circ> \<alpha>) (\<beta>' \<circ> \<beta>) ls"
subst_atoms: fixes k :: "'b"   and F :: "'b formula"   and G :: "'b formula" assumes "k \<in> atoms F" shows "atoms (F[G/k]) = atoms F - {k} \<union> atoms G"
basis_combination_unique: fixes X :: "(('a, 'b) vec, 'b) vec"   and g :: "'b \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a" assumes "is_basis (set_of_vector X)"   and "(\<Sum>i\<in>UNIV. g i *s X $ i) = (\<Sum>i\<in>UNIV. f i *s X $ i)" shows "f = g"
doctor_separable_onD: fixes a :: "'x"   and f :: "'x set \<Rightarrow> 'x set"   and b :: "'x"   and B :: "'x set"   and A :: "'x set"   and c :: "'x" assumes "Xd a \<notin> Xd ` f (insert a (insert b B))"   and "doctor_separable_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "b \<in> A"   and "c \<in> A"   and "Xd a \<noteq> Xd b"   and "Xd b = Xd c" shows "Xd a \<notin> Xd ` f (insert a (insert b (insert c B)))"
collect_interrupts_def: fixes ias :: "'a interrupt_action list" shows "collect_interrupts ias = foldr collect_interrupt ias {}"
rtrancl3p_step: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and bs :: "'b list"   and a' :: "'a"   and b :: "'b"   and a'' :: "'a" assumes "rtrancl3p r a bs a'"   and "r a' b a''" shows "rtrancl3p r a (bs @ [b]) a''"
is_poincare_line_poincare_line: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "is_poincare_line (poincare_line z w)"
nfresh_isfresh: fixes fresh :: "'a set \<Rightarrow> 'a"   and S :: "'a set"   and n :: "nat" assumes "freshstuff fresh"   and "finite S" shows "set (freshstuff.nfresh fresh S n) \<inter> S = {}"
unitor_coincidence_1: shows "MC.lunit MC.unity = \<iota>"
Image_within_domain': fixes x :: "'a"   and R :: "('a \<times> 'b) set" shows "(x \<in> Domain R) = (R `` {x} \<noteq> {})"
hyp: shows "card (evnodd t 0) = card (evnodd t 1)"
mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'b list list" shows "mat nr nc m = (length m = nc \<and> Ball (set m) (vec nr))"
irred_idom: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.idom times one plus zero minus uminus" shows "Divisibility.irreducible \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr> x = (x = zero \<or> comm_semiring_1.irreducible times one zero x)"
bind_update: fixes f :: "('v \<Rightarrow> int option) \<Rightarrow> 'v \<Rightarrow> int option"   and m :: "'m"   and g :: "int \<Rightarrow> 'm" shows "bind (update f m) g = update f (bind m g)"
CompleteLatticeI: fixes po :: "'b potype" assumes "po \<in> PartialOrder"   and "\<forall>S\<subseteq>pset po. \<exists>L. isLub S po L"   and "\<forall>S\<subseteq>pset po. \<exists>G. isGlb S po G" shows "po \<in> CompleteLattice"
elims: fixes x :: "'b"   and xa :: "'b tree"   and xb :: "'b tree"   and y :: "'b tree" assumes "mkt_bal_l x xa xb = y" obtains n :: "'b" and l :: "'b tree" and r :: "'b tree" where "x = n" and "xa = l" and "xb = r" and "y = (if ht l = ht r + 2 then case l of MKT ln ll lr x \<Rightarrow> if ht ll < ht lr then case lr of MKT lrn lrl lrr x \<Rightarrow> mkt lrn (mkt ln ll lrl) (mkt n lrr r) else mkt ln ll (mkt n lr r) else mkt n l r)"
Domainp_rel: fixes R1 :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "Domainp (rel_prod R1 R2) = pred_prod (Domainp R1) (Domainp R2)"
ide_right: shows "ide (tab\<^sub>0 r)\<^sup>*"
WPO_NS_ctxt: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and bef :: "('f, 'b) Term.term list"   and aft :: "('f, 'b) Term.term list" assumes "(s, t) \<in> WPO_NS" shows "(Fun f (bef @ s # aft), Fun f (bef @ t # aft)) \<in> WPO_NS"
assms_1: shows "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'"
ereal_indicator: fixes A :: "'b set"   and x :: "'b" shows "ereal (indicat_real A x) = indicator A x"
sum_three_elements: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "\<exists>x y z. comm_monoid_add.sum plus zero (\<lambda>i. times (f i) i) {a, b, c} = plus (plus (times x a) (times y b)) (times z c)"
borel_measurable_enn2real: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. enn2real (f x)) \<in> borel_measurable M"
leadsto_exists: fixes F :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "\<turnstile> ((\<exists>x. F x) \<leadsto> G) = (\<forall>x. F x \<leadsto> G)"
esssup_const: fixes M :: "'b measure"   and c :: "'c" assumes "emeasure M (space M) \<noteq> 0" shows "esssup M (\<lambda>x. c) = c"
MGF_nested_Methd: fixes pn :: "'b"   and A :: "'a set" assumes "\<forall>b\<in>bdy pn. P (insert (mgf (f pn)) A) {mgf b}" shows "P A {mgf (f pn)}"
id_nat_ereal_tendsto_PInf: shows "(\<lambda>x. ereal (real x)) \<longlonglongrightarrow> \<infinity>"
remdups_rev_code_ps_append: fixes ps1 :: "'a list"   and ps2 :: "'a list"   and rsa :: "'a list" shows "remdups_rev_code (ps1 @ ps2) rsa = remdups_rev_code ps2 (filter (\<lambda>r. r \<notin> set ps1) rsa)"
that: fixes \<alpha> :: "nat \<Rightarrow> int" assumes "f \<bullet> \<alpha> = \<zero>"   and "\<alpha> \<in> carrier Zp"   and "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"   and "val_Zp (a \<ominus> \<alpha>) = val_Zp (local.divide (f \<bullet> a) (pderiv f \<bullet> a))"   and "val_Zp (pderiv f \<bullet> \<alpha>) = val_Zp (pderiv f \<bullet> a)" shows "thesis"
Agree: fixes cfg :: "('p, 'v, 's) configuration" shows "reachable (last exec) cfg \<longrightarrow> agreement cfg"
that: fixes N :: "nat set"   and j :: "nat \<Rightarrow> nat" assumes "infinite N"   and "\<And>k u. \<lbrakk>0 < k; u \<in> [WW]\<^bsup>2\<^esup>; Form k u; [enum N k] < inter_scheme k u; list.set (inter_scheme k u) \<subseteq> N\<rbrakk> \<Longrightarrow> g u = j k" shows "thesis"
allDefs_var_disjoint: fixes n :: "'node"   and g :: "'g"   and v :: "'val"   and v' :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> allDefs g n"   and "v' \<in> allDefs g n"   and "v \<noteq> v'" shows "var g v' \<noteq> var g v"
i_Exec_Stream_nth_gr0_calc: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" assumes "0 < n" shows "i_Exec_Comp_Stream trans_fun input c n = trans_fun (input n) (i_Exec_Comp_Stream trans_fun input c (n - 1))"
cross7_zero_left: fixes x :: "(real, 7) vec" shows "0 \<times>\<^sub>7 x = 0"
nat_pow_closed: fixes x :: "nat"   and n :: "nat" assumes "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<in> carrier G"
3: shows "g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n = g [^]\<^bsub>UP R\<^esub> n"
fold_commute_apply: fixes xs :: "'c list"   and h :: "'d \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'d" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h (fold g xs s) = fold f xs (h s)"
elims: fixes x :: "'b set"   and xa :: "'b interrupt_action list"   and y :: "'b set" assumes "redT_updIs x xa = y" obtains "is" :: "'b set" where "x = isa__" and "xa = []" and "y = isa__"   | "is" :: "'b set" and ia :: "'b interrupt_action" and ias :: "'b interrupt_action list" where "x = is" and "xa = ia # ias" and "y = redT_updIs (redT_updI is ia) ias"
simps_48: fixes \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "safe_formula (Since \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
sort_fun_OK: shows "sorted_wrt R (sort_fun l) \<and> mset (sort_fun l) = mset l"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_same a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> chkchop_same (>\<^sub>t) t s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)))"
proj_eq_imp_arg_eq: fixes u :: "nat"   and v :: "nat" assumes "c_fst u = c_fst v"   and "c_snd u = c_snd v" shows "u = v"
tvar: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> x = Some \<tau>" shows "\<Gamma> \<turnstile> Var x : \<tau>"
diff_inv_chart: fixes c :: "('b, 'b) chart" assumes "c \<in> a.dest.atlas" shows "diff k (manifold_eucl.dest.charts_submanifold (codomain c)) (manifold_eucl.dest.charts_submanifold (codomain c__)) (inv_chart c)"
mem_context_val_w32_9_unchanged: fixes addr1 :: "word36"   and s1 :: "('b, 'c) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" shows "mem_context_val_w32 9 addr1 (mem s1) = mem_context_val_w32 9 addr1 ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None)))"
countably_compact_imp_acc_point: fixes s :: "'b set"   and t :: "'b set" assumes "countably_compact s"   and "countable t"   and "infinite t"   and "t \<subseteq> s" shows "\<exists>x\<in>s. \<forall>U. x \<in> U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
update_vec_def: fixes v :: "'b vec"   and i :: "nat"   and a :: "'b" shows "v |\<^sub>v i \<mapsto> a = vec (dim_vec v) (\<lambda>i'. if i' = i then a else v $ i')"
valid_1: shows "recfn (length (x # xs)) (Pr n f g)"
uncountable_def: fixes A :: "'c set" shows "uncountable A = (A \<noteq> {} \<and> (\<nexists>f. range f = A))"
simps_6: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (Le (CN 0 c e)) = c"
intros: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "r a b" shows "r\<inverse>\<inverse> b a"
shadow_root_eq_h: fixes character_data_ptr :: "(_) element_ptr"   and shadow_root_opt :: "(_) shadow_root_ptr option" shows "h \<turnstile> get_shadow_root character_data_ptr \<rightarrow>\<^sub>r shadow_root_opt = h2 \<turnstile> get_shadow_root character_data_ptr \<rightarrow>\<^sub>r shadow_root_opt"
equiintegrable_closed_interval_restrictions: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "f integrable_on cbox a b" shows "(\<Union>c d. {\<lambda>x. if x \<in> cbox c d then f x else (0::'b)}) equiintegrable_on cbox a b"
assms_2: shows "M \<le> ESPEC (\<lambda>_. False) (\<lambda>c. (c, a) \<in> R)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(tInit x1 = tInit y1) = (x1 = y1)"
ideal_Inter: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set set" assumes "class.ring plus zero minus uminus times"   and "\<forall>I\<in>S. ring.ideal plus zero uminus times I" shows "ring.ideal plus zero uminus times (\<Inter> S)"
hyps_2: fixes s :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> u" shows "\<exists>t'. s => t' \<and> t' \<rightarrow>\<^sub>\<eta>\<^sup>* u'"
row_of_dagger: fixes i :: "nat"   and M :: "complex mat" assumes "i < dim_col M" shows "row M\<^sup>\<dagger> i = vec (dim_row M) (\<lambda>j. cnj (M $$ (j, i)))"
ord_adds: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s adds t" shows "lex_pm s t"
D_def: shows "D \<equiv> \<Prod>\<^sub># (factors_of_factor_tree l)"
dconvfactor_times: fixes c :: "'c \<Rightarrow>\<^sub>U 'd"   and x :: "(int, sdim) dimvec"   and y :: "(int, sdim) dimvec" shows "dconvfactor c (x \<cdot> y) = dconvfactor c x \<cdot> dconvfactor c y"
fact2: fixes V :: "variable set"   and W :: "variable set" assumes "snd (usubstappp \<sigma> V \<alpha>) \<noteq> undefg"   and "snd (usubstappp \<sigma> W \<alpha>) \<noteq> undefg" shows "BVG (the (snd (usubstappp \<sigma> W \<alpha>))) \<subseteq> fst (usubstappp \<sigma> V \<alpha>)"
hom_minus: fixes x :: "'a poly"   and y :: "'a poly"   and a :: "'a" shows "poly (x - y) a = poly x a - poly y a"
carr: shows "{add_col_sub_row a k l A, A, addrow_mat n (- a) k l, addrow_mat n a k l} \<subseteq> carrier_mat n n"
Int_Collect: fixes x :: "'g"   and A :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
1: shows "\<forall>i<k. b ((c ^^ i) s)"
ffun_plus_commute: fixes f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" assumes "fdom f \<inter> fdom g = {}" shows "f + g = g + f"
lookup_wfenv: fixes \<rho> :: "val list"   and \<eta> :: "val set list"   and \<Gamma> :: "(ty \<times> nat) list \<times> nat"   and n :: "nat"   and \<tau> :: "ty" assumes "\<turnstile> \<rho>,\<eta> : \<Gamma>"   and "lookup \<Gamma> n = Some \<tau>" shows "\<exists>v. \<rho> ! n = v \<and> v \<in> T \<tau> \<eta>"
as3: fixes x :: "'a" shows "ad (ad x) + ad x = (1::'a)"
OUTfromChCorrect_data3: shows "OUTfromChCorrect data3"
skelAbs_def: fixes A :: "('d, 'e, 'f, 'g, 'h) qAbs set" shows "skelAbs A \<equiv> qSkelAbs (pick A)"
inf_commute: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
strict_implies_order: fixes b :: "'d list"   and a :: "'d list" assumes "strict_prefix b a" shows "prefix b a"
class_add_types: fixes P :: "jvm_prog"   and C :: "char list"   and cdec :: "jvm_method class" shows "types P \<subseteq> types (class_add P (C, cdec))"
in_insert_hd: fixes f :: "'a"   and X :: "'a set" shows "f \<in> insert f X"
lm070: fixes f :: "('a \<times> 'b) set"   and X :: "'a set" assumes "runiq (f || X)" shows "sum (toFunction (f || X)) (X \<inter> Domain f) = sum (toFunction f) (X \<inter> Domain f)"
comp_converter_parallel2: fixes conv1l :: "('i, 'k, 'q, 's) converter"   and conv1r :: "('j, 'l, 'r, 't) converter"   and conv2l :: "('q, 's, 'm, 'o) converter"   and conv2r :: "('r, 't, 'n, 'p) converter" shows "(conv1l |\<^sub>= conv1r) \<odot> (conv2l |\<^sub>= conv2r) = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r"
NegNeg_E: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A H \<turnstile> B" shows "insert (Neg (Neg A)) H \<turnstile> B"
count_poly_roots_mset: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "count (poly_roots_mset p) a = order a p"
recursive_compile_to_tm_correct1: fixes recf :: "recf"   and ap :: "abc_inst list"   and ary :: "nat"   and fp :: "nat"   and args :: "nat list"   and tp :: "(action \<times> nat) list"   and ires :: "cell list"   and rn :: "nat" assumes "rec_ci recf = (ap, ary, fp)"   and "terminate recf args"   and "tp = tm_of (ap [+] dummy_abc (length args))" shows "\<exists>stp m l. steps0 (Suc 0, Bk # Bk # ires, <args> @ Bk \<up> rn) (tp @ shift (mopup (length args)) (length tp div 2)) stp = (0, Bk \<up> m @ Bk # Bk # ires, Oc \<up> Suc (rec_exec recf args) @ Bk \<up> l)"
zmde: fixes b :: "'a"   and a :: "'a" shows "b * (a div b) = a - a mod b"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "B.dom (A_B.Dom (fst Fg) f) = A_B.Dom (fst Fg) (A.dom f)"
SFAssInitThrowReds: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.\<rr> f) = B.R (B.dom f)"
left_inverse_matrix_def: fixes A :: "'b matrix"   and X :: "'b matrix" shows "left_inverse_matrix A X \<equiv> X * A = one_matrix (max (nrows X) (ncols A)) \<and> ncols X \<le> nrows A"
norm_cauchy_schwarz_equal: fixes x :: "'a"   and y :: "'a" shows "(\<bar>x \<bullet> y\<bar> = norm x * norm y) = collinear {0::'a, x, y}"
exhaust: fixes y :: "event" obtains x11 :: "agent" and x12 :: "agent" and x13 :: "msg" where "y = Says x11 x12 x13"   | x21 :: "agent" and x22 :: "msg" where "y = Gets x21 x22"   | x31 :: "agent" and x32 :: "msg" where "y = Notes x31 x32"
reversepath_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "reversepath (subpath u v g) = subpath v u g"
real_of_rat_sgn: fixes x :: "rat" shows "sgn (real_of_rat x) = real_of_rat (sgn x)"
inf_continuous_def: fixes F :: "'c \<Rightarrow> 'd" shows "inf_continuous F = (\<forall>M. decseq M \<longrightarrow> F (Inf (range M)) = (INF i. F (M i)))"
encode_operator_effect_valid_1: fixes t :: "nat"   and op :: "(nat \<times> nat) strips_operator"   and sv :: "sat_plan_variable"   and vs :: "'a list"   and asses :: "'a list" assumes "t < h"   and "op \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)"   and "sv \<in> atoms \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>) op))) \<^bold>\<or> Atom (State (Suc t) (index vs v))) asses" shows "valid_state_var sv"
cp0: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" assumes "profile_mono_schemeV f g" shows "f X \<tau> = f (\<lambda>_. X \<tau>) \<tau>"
rpc_expl_aux: fixes x :: "'b"   and y :: "'b" shows "\<Squnion> {z. x \<sqinter> \<nu> z \<le> y} = \<Squnion> {z. x \<sqinter> z \<le> y \<and> \<nu> z = z}"
key_map_def: fixes U :: "'a set set" shows "local.key_map U \<equiv> \<lambda>s\<in>\<O> U. s \<pp>"
iso_char: fixes f :: "'a" shows "hom'.iso f = (local.iso f \<and> hom'.arr f \<and> hom'.arr (local.inv f))"
aa_def: shows "aa \<equiv> \<lambda>n. real_of_int (a n)"
vrrestriction_atI: fixes r :: "V"   and a :: "V"   and b :: "V"   and A :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
le_imp_inverse_le: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less zero a" shows "less_eq (inverse b) (inverse a)"
non_adj_def: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unSimpGraph G" shows "valid_unSimpGraph.non_adj G v v' \<equiv> v \<in> nodes G \<and> v' \<in> nodes G \<and> v \<noteq> v' \<and> \<not> valid_unMultigraph.adjacent G v v'"
ot_eq: fixes X :: "V set" assumes "X \<subseteq> elts \<beta>" shows "ordertype (cv X) r = tp X"
ConI: fixes p :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! p"   and "\<turnstile>\<^sub>! q" shows "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)"