hcju/isabelleps huggingface dataset:
shext_objD: fixes sh :: "sheap"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state" assumes "sh \<unlhd>\<^sub>s sh'"   and "sh C = \<lfloor>(sfs, i)\<rfloor>" shows "\<exists>sfs' i'. sh' C = \<lfloor>(sfs', i')\<rfloor> \<and> i \<le>\<^sub>i i'"
B'_def: shows "B' = (\<lambda>x. if x \<in> A then B x else {})"
submonoid_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "submonoid H G \<equiv> H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H) \<and> \<one>\<^bsub>G\<^esub> \<in> H"
\<R>\<^sub>2_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "closed_glob_consistent (\<R>\<^sub>2 \<Gamma>')"
isomorphic_src: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (src\<^sub>D (G a)) (G a)"
vbrelation_vrrestriction: fixes r :: "V"   and A :: "V" shows "vbrelation (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
pseudominimal_ne_bottom: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ordering less_eq less"   and "ordering.pseudominimal_in less_eq less P x" shows "x \<noteq> ordering.bottom less_eq P"
exhaust_sel: fixes stateT :: "('d, 'e) stateT" obtains "stateT = StateT (run_state stateT)"
FactRing_def: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, monoid.mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
cs: fixes s :: "'a set" assumes "s \<in> S" shows "connected s"
I2: fixes m :: "nat"   and e' :: "pexpr"   and n :: "nat"   and e :: "pexpr" assumes "m \<noteq> 0 \<Longrightarrow> P e' (m * n) e" shows "P (PExpr2 (PPow e' m)) n e"
v_le_1: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "v cfg \<le> 1"
dsj: fixes A :: "fmla"   and B :: "fmla"   and i :: "name"   and x :: "trm" shows "(A OR B)(i::=x) = A(i::=x) OR B(i::=x)"
hseq_leg\<^sub>0: shows "hseq (tab\<^sub>0 (cod \<mu>)) \<mu>.chine"
residualFreshChainSimp_3: fixes xvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* (\<tau> \<prec> P) = xvec \<sharp>* P"
intros_1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "rtrancl_path r x [] x"
subst_App: fixes t :: "dB"   and u :: "dB"   and s :: "dB"   and k :: "nat" shows "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"
sup_left_unit: fixes x :: "'b" shows "bot \<squnion> - x = - x"
aff_dim_zero: fixes S :: "'a set" assumes "(0::'a) \<in> affine hull S" shows "aff_dim S = int (dim S)"
INF_superset_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A"   and "\<And>x. x \<in> B \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
chainI: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a" assumes "class.po below"   and "\<And>i. below (Y i) (Y (Suc i))" shows "po.chain below Y"
mn: shows "(length fs \<le> n) = (card (set fs) \<le> local.dim)"
assms_3: shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
simplex_like_map: assumes "domain.simplex_like P" shows "codomain.simplex_like (f ` P)"
II2: shows "II2 \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
d_sub_identity: fixes x :: "'a" shows "- - x \<le> top"
eqvtI: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "perm \<bullet> P \<leadsto>\<guillemotleft>Rel'\<guillemotright> perm \<bullet> Q"
complete_lattice_partial_function_definitions: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
induct: fixes x1 :: "Expr"   and x2 :: "TP"   and P :: "Expr \<Rightarrow> TP \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_expr"   and "\<And>x t. CONTEXT x = t \<Longrightarrow> P (varE x) t"   and "\<And>v. P (valE v) low"   and "\<And>e1 t e2 f. \<lbrakk>(e1, t) \<in> VS_expr; P e1 t; (e2, t) \<in> VS_expr; P e2 t\<rbrakk> \<Longrightarrow> P (opE f e1 e2) t"   and "\<And>e. P e high" shows "P x1 x2"
Un_left_commute: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
i_set0_ind_def: shows "i_set0_ind \<equiv> {x. i_set0_indp x}"
fst_Gauss_Jordan_det_P_not_0: fixes A :: "(('b, 'c) vec, 'c) vec" shows "fst (Gauss_Jordan_det_P A) \<noteq> (0::'b)"
ennreal_le_1: fixes x :: "real" shows "(ennreal x \<le> 1) = (x \<le> 1)"
sel_1: fixes a :: "real" shows "Re (cis a) = cos a"
dom_map_option: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and m :: "'a \<Rightarrow> 'c option" shows "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
id_eq_obj_2: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[x\<^sup>P \<^bold>= y\<^sup>P \<^bold>\<rightarrow> y\<^sup>P \<^bold>= x\<^sup>P in v]"
digraph_isomorphism_inj_on_verts: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "inj_on (iso_verts hom) (verts G)"
dim_tangent_space: fixes p :: "'a" assumes "p \<in> carrier"   and "k = \<infinity>" shows "dim (tangent_space p) = DIM('b)"
subresultant_prs_locale2_axioms: shows "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"
simps_37: fixes f1 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat list \<Rightarrow> 'a"   and f3 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f4 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f5 :: "form \<Rightarrow> 'a"   and f6 :: "form \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat list" shows "(case PAtom x11 x12 of PAtom x xa \<Rightarrow> f1 x xa | NAtom x xa \<Rightarrow> f2 x xa | FConj x xa \<Rightarrow> f3 x xa | FDisj x xa \<Rightarrow> f4 x xa | FAll x \<Rightarrow> f5 x | FEx x \<Rightarrow> f6 x) = f1 x11 x12"
two_pow_sub: fixes m :: "nat" assumes "0 < m" shows "2 ^ m - 2 ^ (m - 1) = 2 ^ (m - 1)"
inj_on_inverseI: fixes A :: "'a set"   and g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. s2p A x \<Longrightarrow> g (f x) = x" shows "inj_on f A"
T_hiding: fixes P :: "'a process"   and A :: "'a set" shows "T (P \ A) = {trace_hide t (ev ` A) |t. (t, ev ` A) \<in> F P} \<union> div_hide P A"
wf_tuple_tabulate_Some: fixes n :: "nat"   and A :: "nat set"   and f :: "nat \<Rightarrow> 'a option"   and x :: "nat" assumes "wf_tuple n A (tabulate f 0 n)"   and "x \<in> A"   and "x < n" shows "\<exists>y. f x = Some y"
groupF_empty: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "(groupF f xs \<noteq> []) = (xs \<noteq> [])"
rep_eq: fixes x :: "'b word \<Rightarrow> 'c word" shows "is_down x = (LENGTH('c) \<le> LENGTH('b))"
sc_hom_meet: fixes x :: "'a"   and y :: "'a" shows "(d x \<sqinter> d y) \<cdot> 1\<^sub>\<pi> = d x \<cdot> 1\<^sub>\<pi> \<sqinter> d y \<cdot> 1\<^sub>\<pi>"
bisims: fixes i :: "nat" assumes "i < length cms" shows "abs.strong_low_bisim_mm (\<R>\<^sub>A_rel (cms ! i))"
nDAeqSet: fixes p :: "('a, 'b) Combinators list"   and s :: "('a, 'b) Combinators list" assumes "set p = set s" shows "noDenyAll p = noDenyAll s"
measurable_Pair_compose_split: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and M1 :: "'a measure"   and M2 :: "'b measure"   and N :: "'c measure"   and g :: "'d \<Rightarrow> 'a"   and M :: "'d measure"   and h :: "'d \<Rightarrow> 'b" assumes "(\<lambda>(x, y). f x y) \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M M1"   and "h \<in> M \<rightarrow>\<^sub>M M2" shows "(\<lambda>x. f (g x) (h x)) \<in> M \<rightarrow>\<^sub>M N"
find_lambda_def: shows "find_lambda \<equiv> (\<lambda>s. s\<lparr>l := 1, hare := f (tortoise s)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>hare := f (hare s), l := l s + 1\<rparr>)"
fine_simplicial_subdivision_of_cell_complex: fixes e :: "real"   and \<M> :: "'b set set" assumes "0 < e"   and "finite \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"   and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1" obtains \<T> :: "'b set set" where "simplicial_complex \<T>" and "\<And>K. K \<in> \<T> \<Longrightarrow> diameter K < e" and "\<Union> \<T> = \<Union> \<M>" and "\<And>C. C \<in> \<M> \<Longrightarrow> \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F" and "\<And>K. K \<in> \<T> \<Longrightarrow> \<exists>C. C \<in> \<M> \<and> K \<subseteq> C"
JVM_states_unfold: fixes P :: "'a prog"   and mxs :: "nat"   and mxl :: "nat" shows "states P mxs mxl \<equiv> err (opt (\<Union> {list n (types P) |n. n \<le> mxs} \<times> list mxl (err (types P))))"
three: shows "(\<lambda>f\<in>Hom A B. set_func (F \<^bsub>\<a>\<^esub> f \<odot> u A) (Id A)) = (\<lambda>f\<in>Hom A B. set_func (u B \<odot> Hom(A,_) \<^bsub>\<a>\<^esub> f) (Id A))"
vimage_vlrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A) `\<^sub>\<circ> B = r `\<^sub>\<circ> (A \<inter>\<^sub>\<circ> B)"
iMODb_finite: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "finite [ r, mod m, c ]"
check_unfold_snoc: fixes \<C> :: "t_context"   and es :: "b_e list"   and ts :: "checker_type"   and e :: "b_e" assumes "check \<C> es ts \<noteq> Bot" shows "check \<C> (es @ [e]) ts = check_single \<C> e (check \<C> es ts)"
construct_fun_from_rel_def: fixes R :: "('b \<times> 'a) set"   and x :: "'b" shows "construct_fun_from_rel R x = (THE y. (x, y) \<in> R)"
EM_remainder_eqI: fixes n :: "nat"   and f :: "real \<Rightarrow> 'a"   and b :: "int"   and L :: "'a" assumes "((\<lambda>x. EM_remainder' n f (real_of_int b) (real_of_int x)) \<longlongrightarrow> L) at_top" shows "EM_remainder n f b = L"
may_join_cond_action_oks: fixes cas :: "'a conditional_action list"   and s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set"   and t :: "'a" assumes "\<And>t'. Join t' \<in> set cas \<Longrightarrow> \<not> not_final_thread s t' \<and> t \<noteq> t'" shows "cond_action_oks s t cas"
IDirProds_incl: fixes i :: "'d"   and I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "i \<in> I" shows "S i \<subseteq> IDirProds G S I"
p_le_1: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "p x y n \<le> 1"
simps_2: fixes ao :: "'a option"   and xs :: "'a list" shows "set_less_eq_aux_Compl ao xs [] = True"
sup_bot_right: fixes x :: "'c multiset" shows "x \<union># {#} = x"
comparator_list_simps_3: fixes comp\<^sub>'\<^sub>a :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and xa :: "'b list" shows "comparator_list comp\<^sub>'\<^sub>a (x # xa) [] = Gt"
lcd: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
valid_plan_valid_tail: fixes h :: "'d"   and as :: "'d list"   and PROB :: "'d set" assumes "h # as \<in> valid_plans PROB" shows "as \<in> valid_plans PROB"
init_red_init: fixes e\<^sub>0 :: "char list exp"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list" assumes "init_exp_of e\<^sub>0 = \<lfloor>e\<rfloor>"   and "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>" shows "init_exp_of e\<^sub>1 = \<lfloor>e\<rfloor> \<and> (INIT_class e\<^sub>0 = \<lfloor>C\<rfloor> \<longrightarrow> INIT_class e\<^sub>1 = \<lfloor>C\<rfloor>) \<or> e\<^sub>1 = e \<and> b\<^sub>1 = icheck P (the (INIT_class e\<^sub>0)) e \<or> (\<exists>a. e\<^sub>1 = throw a)"
list_assn_aux_ineq_len: fixes l :: "'c list"   and li :: "'d list"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "length l \<noteq> length li" shows "list_assn A l li = false"
spmf_of_set: fixes A :: "'b set"   and x :: "'b" shows "spmf (spmf_of_set A) x = indicat_real A x / real (card A)"
comp_fun_idem: fixes x :: "'c" shows "Mapping.update x p \<circ> Mapping.update x p = Mapping.update x p"
eq_ket_vec: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" assumes "u = v" shows "|u\<rangle> = |v\<rangle>"
slice_kind_Pred_obs_not_nearer: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and m :: "'node"   and x :: "nat" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = (Q)\<^sub>\<surd>"   and "m \<in> obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "distance (sourcenode a) m (x + 1)"   and "\<not> distance (targetnode a) m x" shows "slice_kind S a = (\<lambda>s. False)\<^sub>\<surd>"
isoE: fixes f :: "'a \<times> 'a" assumes "S.VV.iso f" obtains g :: "'a \<times> 'a" where "S.VV.inverse_arrows f g"
index_append_vec_2: fixes v :: "'a vec"   and w :: "'a vec" shows "dim_vec (v @\<^sub>v w) = dim_vec v + dim_vec w"
infinite_Complement_set_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B"   and "infinite UNIV" shows "\<not> ord.set_less less_eq (- A) B"
fmeasurableD: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> fmeasurable M" shows "A \<in> sets M"
inner_prod_smult_left: fixes u :: "complex vec"   and n :: "nat"   and v :: "complex vec"   and a :: "complex" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n" shows "inner_prod u (a \<cdot>\<^sub>v v) = a * inner_prod u v"
chi1: fixes s :: "'a" shows "chi (stval s) = characteristic_formula A s"
\<iota>\<chi>': shows "\<iota> \<chi>' \<in> Cones.SET a'"
map_add_upt: fixes n :: "nat"   and m :: "nat" shows "map (\<lambda>i. i + n) [0..<m] = [n..<m + n]"
infinite: fixes A :: "'g set" assumes "infinite A" shows "card A = 0"
fundfold_comp_pgallery_ex_funpow: fixes ss :: "'a permutation list"   and Cs :: "'a set list"   and C :: "'a set" assumes "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "pgallery (C0 # Cs @ [C])" shows "\<exists>n. (fold fst (map Spair ss) ^^ n) ` C = C0"
set_of_mul_inc_right: fixes B :: "'a interval"   and B' :: "'a interval"   and A :: "'a interval" assumes "set_of B \<subseteq> set_of B'" shows "set_of (A * B) \<subseteq> set_of (A * B')"
andalso_eq_FF_iff: fixes x :: "tr"   and y :: "tr" shows "((x andalso y) = FF) = (x = FF \<or> x = TT \<and> y = FF)"
permute_minus_cancel_2: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "perm"   and x :: "'e" assumes "class.pt permute" shows "permute (- p) (permute p x) = x"
M_bounded: shows "AE M' in M. emeasure M' (space M') \<le> ennreal B'"
rev_path_choose_iapath: fixes u :: "'a"   and v :: "'a" assumes "u \<noteq> v" shows "rev_path (choose_iapath u v) = choose_iapath v u"
is_array_def: fixes l :: "'a list"   and p :: "'a array" shows "is_array l p \<equiv> p \<mapsto>\<^sub>a l"
assms: shows "check_single \<C> e (Type tn) = Type tm"
type_definition_bit0: shows "type_definition Rep_bit0 Abs_bit0 {0..<2 * int CARD('a)}"
decs_add_clock: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and ds :: "dec list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s ds res" shows "\<forall>s' r. res = (s', r) \<and> ck = False \<longrightarrow> (\<exists>c. evaluate_decs True mn env (update_clock (\<lambda>_. c) s) ds (update_clock (\<lambda>_. 0) s', r))"
bilinear_continuous_compose: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "continuous F f"   and "continuous F g"   and "bilinear h" shows "continuous F (\<lambda>x. h (f x) (g x))"
generated_ringE: fixes a :: "'a set" assumes "a \<in> generated_ring" obtains C :: "'a set set" where "finite C" and "disjoint C" and "C \<subseteq> M" and "a = \<Union> C"
basis_max_indep: fixes X :: "'a set"   and x :: "'a" assumes "\<E>.basis X"   and "x \<in> \<E> - X" shows "\<not> indep_in \<E> (insert x X)"
assms_2: shows "n\<^sub>1 \<in> s"
semilattice_order_set_axioms: shows "semilattice_order_set sup (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
elims: fixes x :: "'b eclause"   and xa :: "('b \<times> 'b trm) list"   and y :: "'b eclause" assumes "subst_ecl x xa = y" obtains C :: "'b literal set" and S :: "'b trm set" and \<sigma> :: "('b \<times> 'b trm) list" where "x = Ecl C S" and "xa = \<sigma>" and "y = Ecl (subst_cl C \<sigma>) {t \<lhd> \<sigma> |t. t \<in> S}"
fmerge_def: fixes A :: "'a fset"   and f :: "'a \<Rightarrow> 'm" shows "local.fmerge A f = fUnionM (f |`| A)"
asIMOD_igSwapAbsIPresIGWlsAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapAbsIPresIGWlsAbs (asIMOD SEM)"
sup_option_idem: fixes x :: "'c option" shows "sup_option x x = x"
drop_0: fixes xs :: "'b list" shows "drop 0 xs = xs"
induct: fixes x1 :: "'b"   and x2 :: "nat"   and x3 :: "'b fset"   and P :: "'b \<Rightarrow> nat \<Rightarrow> 'b fset \<Rightarrow> bool" assumes "x1 |\<in>|\<^bsub>x2\<^esub> x3"   and "\<And>i s. i < size s \<Longrightarrow> P (list_of_fset s ! i) i s" shows "P x1 x2 x3"
prob_space: fixes n :: "nat" shows "Mn.prob n (space (M n)) = 1"
walk_tl: fixes xs :: "'c list" assumes "T.walk xs" shows "T.walk (tl xs)"
prj_char_2: shows "Maps.PRJ\<^sub>1 \<lbrakk>\<lbrakk>tab\<^sub>0 r\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 s\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>rs.p\<^sub>1\<rbrakk>\<rbrakk>"
simps_1: fixes kv :: "'c \<times> 'd"   and t :: "('c, 'd) rbt"   and kts :: "(('c \<times> 'd) \<times> ('c, 'd) rbt) list" shows "rbt_entries_next ((kv, t) # kts, rbt.Empty) = (kv, kts, t)"
monofun_cfun_arg: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<rightarrow> 'd" assumes "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> f\<cdot>y"
intro: fixes is_list :: "'a list \<Rightarrow> 'b \<Rightarrow> assn" assumes "precise is_list" shows "imp_list is_list"
IH: fixes b' :: "nat" assumes "local.prefixes t j \<down>= b"   and "local.prefixes (t + d) j \<down>= b'" shows "e_length b \<le> e_length b'"
ignore_one: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "P 1 n"   and "1 < n" shows "card {a \<in> {1..<n}. P a n} = 1 + card {a. 2 \<le> a \<and> a < n \<and> P a n}"
nonzero_pow_nonzero: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "domain R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "a [^]\<^bsub>R\<^esub> n \<noteq> \<zero>\<^bsub>R\<^esub>"
count_of_empty: fixes x :: "'a"   and xs :: "('a \<times> nat) list" assumes "x \<notin> fst ` set xs" shows "count_of xs x = 0"
standard_liouville_is_liouville: fixes q :: "int"   and p :: "nat \<Rightarrow> int" assumes "1 < q"   and "range p \<subseteq> {0..<q}"   and "\<exists>\<^sub>F n in sequentially. p n \<noteq> 0" shows "liouville (standard_liouville p q) (\<lambda>n. \<Sum>k\<le>n. p k * q ^ (fact (Suc n) - fact (Suc k))) (\<lambda>n. q ^ fact (Suc n))"
measure_pmf_eq_density: fixes p :: "'b pmf" shows "measure_pmf p = density (count_space UNIV) (\<lambda>x. ennreal (pmf p x))"
assms_2: shows "g \<turnstile> m'-ms\<rightarrow>m"
assms_2: fixes h :: "'t \<Rightarrow>\<^sub>0 'b" assumes "h \<in> H" shows "u \<prec>\<^sub>t lt h"
eval_def: fixes iofun :: "'c io"   and world :: "\<^url>" shows "eval iofun world = fst (Rep_io iofun world)"
norm_add_Pythagorean: fixes a :: "'b"   and b :: "'b" assumes "orthogonal a b" shows "(norm (a + b))\<^sup>2 = (norm a)\<^sup>2 + (norm b)\<^sup>2"
lock_lock_ls_Some: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "\<exists>t' n. lock_lock l t = \<lfloor>(t', n)\<rfloor>"
B_eq_Gauss_ij: shows "B = Gauss_Jordan_in_ij A ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)) (mod_type_class.from_nat k)"
emeasure_eq: fixes l :: "'a"   and u :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "emeasure M (box l u) = ennreal (prod ((\<bullet>) (u - l)) Basis)"
assms: shows "\<forall>Y. downclosed_set Y \<longrightarrow> downclosed_set (f -` Y)"
simps_2: fixes ParamDefs :: "'h \<Rightarrow> 'g list" shows "lift_ParamDefs ParamDefs NewEntry = []"
closed_listI: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "closed S f" shows "closed (list n S) (map2 f)"
SC_Chain_def: fixes S :: "complex set" shows "SC_Chain S \<equiv> open S"
unvisitedPath_removeLast: fixes ns :: "'node list"   and un :: "'node list" assumes "unvisitedPath un ns"   and "2 \<le> length ns" shows "unvisitedPath (removeAll (last ns) un) (butlast ns)"
p: fixes p :: "int" assumes "mod_ring_locale TYPE('a) p" shows "p = int CARD('a)"
cases: fixes x :: "'a interval" obtains l :: "'a" and u :: "'a" where "x = Interval l u"
nfa_trans_quantify_nfa: fixes N :: "bool list bdd list \<times> bool list"   and n :: "nat"   and v :: "nat"   and w :: "bool list"   and q :: "bool list" assumes "wf_nfa N (Suc n)"   and "v \<le> n"   and "is_alph n w"   and "nfa_is_node N q" shows "set_of_bv (nfa_trans (quantify_nfa v N) q w) = (\<Union>b. set_of_bv (nfa_trans N q (insertl v b w)))"
HPhase1or2ReadElse_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d r"   and "HInv4a s q"   and "Inv2a s'" shows "HInv4a s' q"
bind_sample2: fixes m :: "'m"   and p :: "'p pmf"   and f :: "'a \<times> 's \<Rightarrow> 'p \<Rightarrow> 'm" shows "bind m (\<lambda>y. sample p (f y)) = sample p (\<lambda>x. bind m (\<lambda>y. f y x))"
induct: fixes P :: "nat \<Rightarrow> real list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "real list" assumes "\<And>m x y xs. P (Suc m) (y # xs) \<Longrightarrow> P m (x # y # xs)"   and "\<And>m. P m []"   and "\<And>m v. P m [v]" shows "P a0 a1"
gen_mult: fixes x :: "'d"   and G :: "('d, 'e) monoid_scheme"   and gens :: "'d set"   and y :: "'d" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"   and "y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
cut_less_Int: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>< t = A \<down>< t \<inter> B \<down>< t"
coprime_1_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one one a"
nonneg_Reals_inverse_I: fixes a :: "'b" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "inverse a \<in> \<real>\<^sub>\<ge>\<^sub>0"
less_eq_Agent_def: fixes x :: "Agent"   and y :: "Agent" shows "x \<le> y \<equiv> x = y \<or> x < y"