hcju/isabelleps huggingface dataset:
abs_summable_on_norm_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "((\<lambda>x. norm (f x)) abs_summable_on A) = (f abs_summable_on A)"
nonzero_minus_divide_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "divide (uminus a) (uminus b) = divide a b"
dom_lhs: shows "\<D>\<^sub>\<circ> \<NN> = 5\<^sub>\<nat>"
map_le_refl: fixes f :: "'g \<Rightarrow> 'h option" shows "f \<subseteq>\<^sub>m f"
one_le_Fract_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(1 \<le> Fract a b) = (b \<le> a)"
u2: shows "\<parallel>r\<parallel>\<^sup>2 \<le> \<parallel>(1 / 2) *\<^sub>R r + (1 / 2) *\<^sub>R s\<parallel>\<^sup>2"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>nat z. P z (TVar nat)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 \<rightarrow> ty2)" shows "P z ty"
Sigma_def: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" shows "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {(x, y)}"
wf_subcls_nCls: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list"   and D' :: "char list" assumes "wf_prog wf_md P"   and "\<not> is_class P C"   and "P \<turnstile> D \<preceq>\<^sup>* D'"   and "D \<noteq> C" shows "D' \<noteq> C"
rtrancl_path_mono: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and p :: "'a list"   and y :: "'a"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "rtrancl_path R x p y"   and "\<And>x y. R x y \<Longrightarrow> S x y" shows "rtrancl_path S x p y"
a: shows "\<forall>x\<in>H'. h' x \<le> p x"
point_equations_4: fixes p :: "'a" assumes "is_point p" shows "1 ; p\<^sup>T = p\<^sup>T"
2: shows "{seller} \<times> ({\<Omega> - \<Union> (Range a)} - {{}}) \<in> allocationsUniverse"
construct_basis: fixes B :: "'a set"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "cindependent B"   and "b \<in> B" shows "cconstruct B f b = f b"
comp_fun_commute_apply: fixes x :: "('b, 'c) comp_fun_commute" shows "comp_fun_commute_apply x \<in> {f. comp_fun_commute f}"
collapse: fixes option :: "'b option" assumes "option \<noteq> None" shows "Some (the option) = option"
mn_2: shows "fds_nth f m \<noteq> (0::'a)"
resultant_const_1: fixes a :: "'a"   and q :: "'a poly" shows "resultant [:a:] q = a ^ degree q"
one_less_log_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(1 < log a x) = (a < x)"
eval_pmod_var_1: fixes L :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) ring"   and i :: "'d"   and q :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) list" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> Hom.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = Hom.indexed_const \<zero>\<^bsub>image_ring (eval_pmod q) (poly_ring L)\<^esub>"   and "Hom.indexed_const \<in> ring_hom (image_ring (eval_pmod q) (poly_ring L)) L"   and "q \<in> carrier (poly_ring L)"   and "1 < degree q" shows "Hom.eval_pmod L i q X\<^bsub>L\<^esub> = Hom.indexed_var i"
intro: assumes "var_infinite undefined"   and "var_regular undefined"   and "varSort_lt_var undefined undefined" shows "FixVars TYPE('a) TYPE('b)"
ereal_less_eq_1: fixes x :: "ereal" shows "x \<le> \<infinity>"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'c set" assumes "n \<le> card S" obtains T :: "'c set" where "T \<subseteq> S" and "card T = n" and "finite T"
kF: fixes x :: "'a" shows "norm (f' x) \<le> norm x * kF"
rq_lq_reassoc_suf: fixes u :: "'b list"   and z :: "'b list"   and w :: "'b list" assumes "u \<le>s z"   and "u \<le>p w" shows "z\<^sup><\<inverse>u  \<cdot> w = z \<cdot> u\<inverse>\<^sup>>w"
sum_finite_imp_summands_finite_2: fixes a :: "eint"   and b :: "eint" assumes "a + b \<noteq> \<infinity>" shows "b \<noteq> \<infinity>"
diff_less_Suc: fixes m :: "nat"   and n :: "nat" shows "m - n < Suc m"
obtain_subset_with_card_n: fixes n :: "nat"   and S :: "'q set" assumes "n \<le> card S" obtains T :: "'q set" where "T \<subseteq> S" and "card T = n" and "finite T"
dist_minus_Busemann_max_ineq: fixes x :: "'b"   and z :: "'b"   and xi :: "'b Gromov_completion"   and y :: "'b" shows "dist x z - Busemann_function_at xi z x \<le> max (dist x y - Busemann_function_at xi y x) (dist y z - Busemann_function_at xi z y - 2 * Busemann_function_at xi y x) + 8 * deltaG TYPE('b)"
lappend_llist_of_LCons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b llist" shows "lappend (llist_of xs) (LCons y ys) = lappend (llist_of (xs @ [y])) ys"
poly_power: fixes p :: "'b poly"   and n :: "nat"   and x :: "'b" shows "poly (p ^ n) x = poly p x ^ n"
Pow_top: fixes A :: "'b set" shows "(A::'b set) \<in> Pow A"
apath_sd_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and e :: "'a \<times> 'a"   and w :: "'a" assumes "S.apath u p v"   and "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)" shows "pre_digraph.apath (with_proj (subdivide (subdivide G e w) e w)) u (sd_path e w p) v"
module_def: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" shows "module scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'d) x = x)"
diagonalize_is_functor: shows "functor (\<cdot>) (\<cdot>) \<F>C.D"
Smith_2x2_JNF_eucl_works: fixes A :: "'b mat"   and P :: "'b mat"   and S :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat 2 2"   and "(P, S, Q) = Smith_2x2_JNF_eucl A" shows "is_SNF A (P, S, Q)"
inv_allI: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat" assumes "\<And>i j. \<lbrakk>i < n; j < n\<rbrakk> \<Longrightarrow> p A i j" shows "??.Jordan_Normal_Form_Existence.inv_all n p A"
hermitian_H_k: fixes k :: "nat" assumes "k < n" shows "hermitian (H_k k)"
finite_representation: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set"   and v :: "'e" assumes "module scale" shows "finite {b. module.representation scale basis v b \<noteq> (0::'d)}"
mprefix_singl: fixes a :: "'b"   and P :: "'b \<Rightarrow> 'b process" shows "Mprefix {a} P = (a \<rightarrow> P a)"
And\<^sub>n_dnf: fixes \<Phi> :: "'b ltln set" assumes "finite \<Phi>" shows "dnf (And\<^sub>n \<Phi>) = Finite_Set.fold (\<lambda>\<phi>. (\<otimes>) (dnf \<phi>)) {{||}} \<Phi>"
assms: shows "\<forall>x. f x \<noteq> (0::'b)"
assms_3: shows "default_action = Accept \<or> default_action = Drop"
UnE: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
padd_Nil: fixes l2 :: "'a list" shows "[] +++ l2 = l2"
scale_one: fixes x :: "'c" shows "(1::'c) * x = x"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and kv :: "'a \<times> 'b"   and xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.update_by_raw rep_key_order kv (xs, ko) = (comparator.update_by_pair (key_compare (rep_key_order ko)) kv xs, ko)"
consis_sb: shows "sharing_consistent \<S> \<O>\<^sub>t (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t)"
domA_map_of_Some_the: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list" assumes "x \<in> domA \<Gamma>" shows "map_of \<Gamma> x = Some (the (map_of \<Gamma> x))"
4: shows "\<forall>x. x |\<in>| |dom|f \<longrightarrow> g |@| (f |@| x) |\<in>| |cod|g"
irreducible_altdef: fixes x :: "'a" shows "irreducible x = (x \<noteq> (0::'a) \<and> \<not> x dvd (1::'a) \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> b dvd (1::'a)))"
that: fixes l :: "'n" assumes "(\<pi> m, l) \<in> pdg"   and "(l, \<pi> n) \<in> pdg" shows "thesis"
is_equality_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "is_equality R = (R = (=))"
alw_mp: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream"   and \<psi> :: "'d stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs" shows "alw \<psi> xs"
ball_one_point2: fixes A :: "'d set"   and a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>A. a = x \<longrightarrow> P x) = (a \<in> A \<longrightarrow> P a)"
selection_completion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M K \<phi> \<subseteq> * \<gamma>\<^sub>A\<^sub>G\<^sub>M K \<phi>"
spn_hcomp: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and g :: "'a"   and f :: "'a" assumes "seq \<tau> \<mu>"   and "g \<cong> spn \<tau>"   and "f \<cong> spn \<mu>" shows "spn (\<tau> \<cdot> \<mu>) \<cong> g \<star> f"
myUNIV_char: fixes x :: "nat"   and y :: "nat" shows "lang (myUNIV x y) = lang (Times (Star (L_lasthasxx x y)) (Plus rexp.One (nodouble x y)))"
ld'_red: shows "reduce [C] D (L # D') \<noteq> L # D'"
LPD_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Futu Strict (from_mregex mr \<phi>s)"   and "ms \<in> LPD mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
longest_Lyndon_suf_sing: fixes a :: "'a" shows "LynSuf [a] = [a]"
Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and s' :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Seq c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
sign_r_pos_mod: fixes p :: "real poly"   and x :: "real"   and q :: "real poly" assumes "poly p x = 0"   and "poly q x \<noteq> 0" shows "sign_r_pos (q mod p) x = sign_r_pos q x"
collapse_3: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Equality stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_lhs stateful_strand_step \<doteq> the_rhs stateful_strand_step\<rangle> = stateful_strand_step"
dvd_prod: fixes n :: "nat"   and ns :: "nat list" shows "n dvd (\<Prod>m\<in>#mset (n # ns). m)"
nonneg_lincomb_def: fixes c :: "'a vec \<Rightarrow> 'a"   and Vs :: "'a vec set"   and b :: "'a vec" shows "next_dim.nonneg_lincomb c Vs b = (next_dim.lincomb c Vs = b \<and> c ` Vs \<subseteq> {x. (0::'a) \<le> x})"
properfactor_mult_lI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor G a b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (c \<otimes> a) (c \<otimes> b)"
ldfix: shows "LeftDerivationFix (ladder_stepdown_\<alpha>_0 \<alpha> D L) (ladder_i L' 0) (take (ladder_n L' 0) (drop (ladder_stepdown_diff L) D)) (ladder_j L' 0) (ladder_\<gamma> (ladder_stepdown_\<alpha>_0 \<alpha> D L) (drop (ladder_stepdown_diff L) D) L' 0)"
pred_inject_2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and aa :: "'c list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
pause: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f"   and s' :: "'f" assumes "g s = Skip s'"   and "s' \<in> terminates_on g" shows "s \<in> terminates_on g"
NoCardinalityAnalysis_axioms: shows "NoCardinalityAnalysis Aexp Aheap"
op_coll_is_empty_def: shows "op_coll_is_empty \<equiv> is_empty"
fds_const_minus: fixes a :: "'b"   and b :: "'b" shows "fds_const (a - b) = fds_const a - fds_const b"
Keys_minus: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "Keys (A - B) \<subseteq> Keys A"
eval_runiq_in_Range: fixes R :: "('c \<times> 'd) set"   and a :: "'c" assumes "runiq R"   and "a \<in> Domain R" shows "R ,, a \<in> Range R"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<Rightarrow>\<^sub>S a\<guillemotright>"   and "S.iso (L f)" shows "S.iso f"
iso_class_memb_is_ide: fixes F :: "('a, 'a set) Maps.arr set"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.is_iso_class F"   and "f \<in> F" shows "Maps.ide f"
simps_3: fixes vb :: "variable"   and vc :: "typ"   and va :: "Core.term list" shows "strip_comb_imp (Fv vb vc, va) = (Fv vb vc, va)"
basic_trans_rules_32: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a = b"   and "b \<in> A" shows "a \<in> A"
model_to_dimacs_model_distinct: fixes n_vars :: "nat"   and M :: "nat \<Rightarrow> bool" assumes "1 < n_vars" shows "distinct (map dimacs_lit_to_var (model_to_dimacs_model M [1..<n_vars]))"
fresh_7: fixes a :: "vrs"   and x2 :: "tyvrs"   and x3 :: "ty"   and x1 :: "ty" shows "a \<sharp> (\<forall>x2<:x1. x3) = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
assms_1: shows "\<forall>t1<t. i1 t1 = []"
unat_sub_if': fixes x :: "'b word"   and y :: "'b word" shows "unat (x - y) = (if unat y \<le> unat x then unat x - unat y else unat x + 2 ^ LENGTH('b) - unat y)"
path_components_of_def: fixes X :: "'b topology" shows "path_components_of X \<equiv> path_component_of_set X ` topspace X"
acquired_all_acquired: fixes pending_write :: "bool"   and xs :: "'b memref list"   and A :: "nat set" shows "acquired pending_write xs A \<subseteq> A \<union> all_acquired xs"
a_Prj2: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Prj2 e, Prj2 eP, Prj2 eV : \<tau>\<^sub>2"
arr_mkIde: fixes A :: "'s set" shows "S.arr (S.mkIde A) = (A \<subseteq> S.Univ)"
list_lcm_least: fixes xs :: "'a list"   and y :: "'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x dvd y" shows "lcm_list xs dvd y"
map_add_dom_app_simps_1: fixes m :: "'c"   and l2 :: "'c \<Rightarrow> 'd option"   and l1 :: "'c \<Rightarrow> 'd option" assumes "m \<in> dom l2" shows "(l2 \<Oplus> l1) m = l2 m"
Opair_def: fixes a :: "ZF"   and b :: "ZF" shows "Opair a b \<equiv> Upair (Upair a a) (Upair a b)"
less_eq_document_ptr_def: fixes x :: "(_) document_ptr"   and y :: "(_) document_ptr" shows "x \<le> y \<equiv> case x of document_ptr.Ref i \<Rightarrow> case y of document_ptr.Ref j \<Rightarrow> i \<le> j | document_ptr.Ext x \<Rightarrow> True | document_ptr.Ext i \<Rightarrow> case y of document_ptr.Ref x \<Rightarrow> False | document_ptr.Ext j \<Rightarrow> i \<le> j"
update_convs_5: fixes thread' :: "(thread_id_t \<Rightarrow> thread_t) \<Rightarrow> thread_id_t \<Rightarrow> thread_t"   and sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "thread_update thread' \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread' thread, \<dots> = more\<rparr>"
max_Var_floatariths_Max: fixes xs :: "floatarith list" shows "max_Var_floatariths xs = (if set xs = {} then 0 else Max (max_Var_floatarith ` set xs))"
if_eval_elim': fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and e :: "'d"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "(((Stmt.If e c\<^sub>1 c\<^sub>2, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "(c' = c\<^sub>1 \<and> ev\<^sub>B mem e \<or> c' = c\<^sub>2 \<and> \<not> ev\<^sub>B mem e) \<and> mds' = mds \<and> mem' = mem"
sorted_list_of_set_empty: shows "pp_pm.ordered_powerprod_lin.sorted_list_of_set {} = []"
x0_hb_bound2: fixes x :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> x"   and "b \<in> set bs" shows "1 < x * (1 - b - hb / ln x powr (1 + e))"
simps_6: fixes V :: "'a set"   and uu :: "'a list"   and uv :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and S :: "('b, 'a) strand_step list" shows "wf\<^sub>s\<^sub>t V (\<forall>uu\<langle>\<or>\<noteq>: uv\<rangle>\<^sub>s\<^sub>t # S) = wf\<^sub>s\<^sub>t V S"
concl_lt_d: shows "concl_of \<iota> < D"
coeff_Poly_list_of_vec_nth': fixes i :: "nat"   and x :: "'b vec" assumes "i < dim_vec x" shows "coeff (Poly (list_of_vec x)) i = x $ i"
sup_inf_implies: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<squnion> (y - x) = y"
matrix_vector_mul_lid: fixes x :: "('b, 'c) vec" shows "mat (1::'b) *v x = x"
Category_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "Category C \<equiv> ExtCategory C \<and> Category_axioms C"
\<tau>move_heap: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x' :: "'x2"   and m' :: "'m2" assumes "r2.r_syntax t x m ta x' m'"   and "\<tau>move2 (x, m) ta (x', m')" shows "m = m'"
splitAt_split: fixes ram :: "'c"   and vs :: "'c list"   and a :: "'c list"   and b :: "'c list" assumes "ram \<in> set vs"   and "(a, b) = splitAt ram vs" shows "vs = a @ ram # b"
that: fixes a0 :: "'d" assumes "a = iso_arcs hom a0"   and "a0 \<in> arcs G" shows "thesis"
C0_in: shows "(C0, active) \<in> lnth Ns (Suc n)"
sign_simps_2: fixes af :: "'a"   and bf :: "'a" shows "af + bf = bf + af"
ctr: fixes s1 :: "'d stream"   and s2 :: "'d stream" shows "sinterleave s1 s2 = shd s1 ## sinterleave s2 (stl s1)"
lookup1_update1: fixes t :: "'b tree"   and n :: "nat"   and x :: "'b"   and m :: "nat" assumes "braun t"   and "n \<in> {1..size t}" shows "lookup1 (update1 n x t) m = (if n = m then x else lookup1 t m)"
dominates_shortestPath_order: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "old.dominates g n m"   and "m \<in> set (\<alpha>n g)"   and "invar g" shows "old.shortestPath g n \<le> old.shortestPath g m"
1: shows "integral {0..1} (\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) = integral {0..1} (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)))"
ts1t: shows "ts1 t = \<lfloor>xln\<rfloor>"
assms_3: shows "round_robin_invar \<sigma> (dom (thr s))"
group_axioms: shows "Group_Theory.group G (\<cdot>) \<one>"
assms_2: shows "j \<in>\<^sub>\<circ> Vset \<alpha>"
nullable_empty_mrexp: fixes r :: "'b rexp" shows "nullable (empty_mrexp r) = nullable r"
e: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev u @ h) [y, y] = 2"
represents_uminus: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "poly_uminus p represents - x"
simps: fixes V :: "'d list set"   and M :: "('d, 'e, 'f) FSM" shows "Perm V M = {f ` V |f. \<forall>v\<in>V. f v \<in> language_state_for_input M (initial M) v}"
fresh_Var_simp: fixes ys :: "'varSort"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" shows "fresh ys y (Var xs x) = (ys \<noteq> xs \<or> y \<noteq> x)"
gcd_factorial_commute: fixes a :: "'a"   and b :: "'a" shows "gcd_factorial a b = gcd_factorial b a"
induct: fixes P :: "'a aexp \<Rightarrow> bool"   and aexp :: "'a aexp" assumes "\<And>x. P (L x)"   and "\<And>x. P (V x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Plus x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Minus x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Times x1a x2a)" shows "P aexp"
Ap_reduce2: fixes x :: "comb"   and y :: "comb"   and z :: "comb" assumes "x \<rightarrow> y" shows "z \<bullet> x \<rightarrow> z \<bullet> y"
minus_closed: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot J)"   and "y \<in> carrier (R Quot J)" shows "x \<ominus>\<^bsub>R Quot J\<^esub> y \<in> carrier (R Quot J)"
subgroup_img_is_subgroup: fixes I :: "'a set" assumes "I \<le> G" shows "h ` I \<le> H"
verts_in_translation_finite_1: fixes X :: "'b allegorical_term" shows "finite (vertices (translation X))"
pr0_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.comp (pr0 (cod f) (cod g)) (tuple f g) = g"
absorb_iff1: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" shows "(b \<preceq> a) = (a \<oplus> b = a)"
Un_carrier_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('c, 'e) aGroup_scheme" shows "Un_carrier I A = \<Union> {X. \<exists>i\<in>I. X = carrier (A i)}"
between_swap: fixes A :: "'a"   and B :: "'a"   and X :: "'a"   and Y :: "'a" assumes "between (A, B) X"   and "between (A, B) Y" shows "between (X, B) Y = between (A, Y) X"
scc_ofI_reachable: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "sym_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<in> pre_digraph.scc_of G v"
subject_reduction2: fixes 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 :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E \<turnstile> (h, l) \<surd>"   and "P,E,h \<turnstile> e : T" shows "\<exists>T'. P,E,h' \<turnstile> e' : T' \<and> P \<turnstile> T' \<le> T"
integer_equivalent_def: fixes n :: "nat"   and fs :: "int vec list"   and gs :: "int vec list" shows "integer_equivalent n fs gs = (let fs' = of_int_hom.mat_hom (mat_of_cols n fs); gs' = of_int_hom.mat_hom (mat_of_cols n gs); I = 1\<^sub>m n in gauss_jordan_integer_inverse n fs' gs' I \<and> gauss_jordan_integer_inverse n gs' fs' I)"
median_in_set: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "median xs \<in> set xs"
empty_iff: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "(t = []) = (BinomialHeapStruc.queue_to_multiset t = {#})"
uminus_uint32_def: shows "uminus \<equiv> map_fun Rep_uint32 Abs_uint32 uminus"
hull_same: fixes S :: "'a set \<Rightarrow> bool"   and s :: "'a set" assumes "S s" shows "S hull s = s"
S: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "\<phi> c d" shows "\<phi> d c"
generate_img: fixes K :: "'a set" assumes "K \<subseteq> carrier B" shows "generate C (g ` K) = g ` generate B K"