hcju/isabelleps huggingface dataset:
disjoint_insert_1: fixes B :: "'c set"   and a :: "'c"   and A :: "'c set" shows "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
assms_3: shows "\<forall>\<^sub>F x in net. f x \<bullet> i \<le> b"
star_left_unfold_equal: fixes x :: "'a" shows "(1::'a) \<squnion> x\<^sup>+ = x\<^sup>\<star>"
rep_eq: fixes x :: "('a, 'b) mapping" shows "Mapping.keys x = dom (mapping.rep x)"
point_index_distrib: fixes B1 :: "'b set multiset"   and B2 :: "'b set multiset"   and ps :: "'b set" shows "(B1 + B2) index ps = B1 index ps + B2 index ps"
lookup_env_restr_eq: fixes S :: "'b set"   and m :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(m f|` S) x = (if x \<in> S then m x else \<bottom>)"
ord_equiv: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "ord_equiv D E"
discI: fixes option :: "'d option"   and x2 :: "'d" assumes "option = \<lfloor>x2\<rfloor>" shows "option \<noteq> None"
dom: shows "\<D>\<^sub>\<circ> vreal_uminus = \<real>\<^sub>\<circ>"
Gcd_minors_dvd_diagonal: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n m"   and "Smith_normal_form_mat A"   and "k \<le> min n m" shows "Gcd (minors A k) dvd (\<Prod>i = 0..<k. A $$ (i, i))"
bred: shows "b[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* b'"
locks_conv: fixes ls :: "'a"   and tsmws :: "('b \<times> 'c) \<times> 'd \<times> 'e" shows "locks (ls, tsmws) = ls"
subset_plus_left: fixes G :: "'a set"   and A :: "'a set" assumes "AbGroup G" shows "A \<subseteq> A + G"
fps_of_poly_linear': fixes a :: "'a" shows "fps_of_poly [:1::'a, a:] = 1 + fps_const a * fps_X"
finite_rows: fixes A :: "(('b, 'c) vec, 'd) vec" shows "finite (rows A)"
completeness': fixes T :: "tree"   and Cs :: "fterm literal set set" assumes "unification"   and "closed_tree T Cs"   and "\<forall>C\<in>Cs. finite C" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
methd_inheritedD: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "class G C = Some c"   and "ws_prog G"   and "methd G C sig = Some m" shows "declclass m \<noteq> C \<longrightarrow> G \<turnstile> C inherits method sig m"
no_path: fixes v :: "'a" assumes "v \<in> verts G" shows "(dist v = \<infinity>) = (num v = \<infinity>)"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexMorphism X X f"   and "ChamberComplexEndomorphism_axioms X f" shows "ChamberComplexEndomorphism X f"
simps: fixes n :: "nat" shows "eval_primfun' (LnChain n) = ln ^^ n"
int_induct_abs: fixes P :: "int \<Rightarrow> bool"   and j :: "int" assumes "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n" shows "P j"
LENGTH_state_set: fixes e :: "'c list"   and X :: "'c list" assumes "e \<in> state_set X" shows "length e \<le> length X"
assms: fixes i :: "'a" assumes "i \<in> A" shows "B i \<lesssim> C i"
injectivity_6: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "k = k'"
integrable_on_subcbox: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "f integrable_on S"   and "cbox a b \<subseteq> S" shows "f integrable_on cbox a b"
rw: shows "addrow_mat (dim_row H) (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 * P_init * H = addrow (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 K_init"
0: shows "list_all2 eintT (arOf f) (map2 ntsem (arOf f) al)"
NetModel: shows "SecurityInvariant sinvar_spec (nm_default m) (nm_receiver_violation m)"
exec_meth_take: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and ins' :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P (ins @ ins') xt t h (stk, loc, pc, xcp) ta h' s'"   and "pc < length ins" shows "exec_meth ci P ins xt t h (stk, loc, pc, xcp) ta h' s'"
prime: fixes p :: "nat" assumes "prime p" shows "g p = g_prime p"
winning_region_def: fixes p :: "Player" shows "winning_region p \<equiv> {v \<in> V. \<exists>\<sigma>. strategy p \<sigma> \<and> winning_strategy p \<sigma> v}"
generator: shows "carrier (H\<lparr>carrier := generate H {b}\<rparr>) = generate (H\<lparr>carrier := generate H {b}\<rparr>) {b}"
reduce_basis_def: shows "reduce_basis = basis_reduction_main (True, 0, fs_init)"
iso_relative_homology_group_lower_hemisphere: fixes p :: "int"   and n :: "nat"   and k :: "nat" shows "hom_induced p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0} (nsphere n) {x. 0 \<le> x k} id \<in> Group.iso (relative_homology_group p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0}) (relative_homology_group p (nsphere n) {x. 0 \<le> x k})"
assms_3: shows "Var x \<in> ik\<^sub>s\<^sub>t A"
assms_2: shows "maximal S (\<lambda>T. T \<subseteq> set (cols A) \<and> lin_indpt T)"
aform_err_addI: fixes y :: "real"   and b :: "real"   and e :: "nat \<Rightarrow> real"   and a :: "real"   and xs :: "real pdevs"   and d :: "real" assumes "y - b \<in> aform_err e ((a, xs), d)" shows "y \<in> aform_err e ((a + b, xs), d)"
prefix_map_inj: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (ran\<^sub>l xs \<union> ran\<^sub>l ys)"   and "prefix (map f xs) (map f ys)" shows "prefix xs ys"
singleton_sub_trivial_uniq: fixes x :: "'a"   and X :: "'a set" assumes "{x} \<subseteq> X"   and "trivial X" shows "x = the_elem X"
mult_right_sub_dist_sup_right: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<le> (x \<squnion> y) \<sqinter> z"
restrict_apply: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and x :: "'f" shows "restrict f A x = (if x \<in> A then f x else undefined)"
accept\<^sub>G\<^sub>R_I: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and M_fin :: "('b ltl \<Rightarrow> nat option) \<Rightarrow> (('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_to_rabin_base \<delta> \<delta>\<^sub>M q\<^sub>0 q\<^sub>0\<^sub>M M_fin \<Sigma> w"   and "accept\<^sub>G\<^sub>R (ltl_to_rabin_base_def.ltl_to_generalized_rabin \<delta> \<delta>\<^sub>M q\<^sub>0 q\<^sub>0\<^sub>M M_fin \<Sigma> \<phi>) w" obtains \<pi> :: "'b ltl \<Rightarrow> nat option" where "dom \<pi> \<subseteq> \<^bold>G \<phi>" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < ltl_to_rabin_base_def.max_rank_of \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> \<chi>" and "accepting_pair\<^sub>R (ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma>) (ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi>) (M_fin \<pi>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma>) (ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi>) (ltl_to_rabin_base_def.Acc_fin \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> \<pi> \<chi>, ltl_to_rabin_base_def.Acc_inf \<delta>\<^sub>M q\<^sub>0\<^sub>M \<pi> \<chi>) w"
fun_scale_def: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" shows "fun_scale c f \<equiv> \<lambda>v. c * f v"
ord_isom_mem: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "f a \<in> carrier E"
top_action: shows "top \<equiv> action_of (R (true \<turnstile> false))"
dickson_grading_minus: fixes d :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "dickson_grading d"   and "s adds t" shows "d (t - s) \<le> d t"
open_segment_bound_2: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> {a<--<b}" shows "norm (x - b) < norm (b - a)"
is_associative_weak_composition: shows "associative_weak_composition (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C"
interchange_rows_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and a :: "'f"   and b :: "'f"   and i :: "'f" shows "interchange_rows_row A a b i = (\<chi>j. if i = a then A $ b $ j else if i = b then A $ a $ j else A $ i $ j)"
double_SN_aux: fixes a :: "lam"   and b :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes "SN a"   and "SN b"   and "\<And>x z. \<lbrakk>\<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> SN y; \<And>y.  x \<longrightarrow>\<^sub>\<beta> y \<Longrightarrow> P y z; \<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> SN u; \<And>u.  z \<longrightarrow>\<^sub>\<beta> u \<Longrightarrow> P x u\<rbrakk> \<Longrightarrow> P x z" shows "P a b"
dom_char: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.dom f = (if A1.arr (fst f) \<and> A2.arr (snd f) then (A1.dom (fst f), A2.dom (snd f)) else (A1.null, A2.null))"
no_better_subset_pref: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation"
limproc_Seq_D2: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "D (S `;` lim_proc (range Y)) = D (lim_proc (range (\<lambda>i. S `;` Y i)))"
progress_simps_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and e :: "String.literal"   and ts :: "Formula.trm list"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Pred e ts) j = (case P e of None \<Rightarrow> j | Some k \<Rightarrow> k)"
finite_UN: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Union> (A ` I) \<in> sets M"
fls_regpart_fps_trivial: fixes f :: "'a fps" shows "fls_regpart (fps_to_fls f) = f"
mult2_s_locally_one_step': fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "ns O s \<subseteq> s"   and "refl ns"   and "locally_irrefl s X"   and "locally_irrefl s Y"   and "trans s" shows "((X, Y) \<in> mult2_s ns s) = (\<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (X2, Y2) \<in> mult s)"
gauge_def: fixes E :: "real set"   and g :: "real \<Rightarrow> real" shows "gauge E g = (\<forall>x\<in>E. 0 < g x)"
simps_4: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list" shows "well_com (Measure n M S) = (measurement d n M \<and> length S = n \<and> list_all well_com S)"
kernel_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" shows "kernel G H h = {x \<in> carrier G. h x = \<one>\<^bsub>H\<^esub>}"
isPath_append_edge: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat"   and p :: "(nat \<times> nat) list"   and v' :: "nat"   and v'' :: "nat" assumes "Graph.isPath c v p v'"   and "(v', v'') \<in> Graph.E c" shows "Graph.isPath c v (p @ [(v', v'')]) v''"
simps_1: fixes t :: "trie" shows "prefix_trie [] t = t"
SUP_bot_conv_1: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and B :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
Rep_num1_inject: fixes x :: "1"   and y :: "1" shows "(Rep_num1 x = Rep_num1 y) = (x = y)"
intros_3: shows "(\<lambda>x. x) \<in> PrimRec1"
1: shows "llength (lselect s u) = llength (lselect t v)"
Greatest_plus_one_def: shows "Greatest_plus_one = (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)"
order_def: fixes a :: "'b"   and p :: "'b poly" shows "order a p = (LEAST n. \<not> [:- a, 1::'b:] ^ Suc n dvd p)"
suffix_Cons: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "suffix xs (y # ys) = (xs = y # ys \<or> suffix xs ys)"
powr_numeral: assumes "0 \<le> x" shows "x powr numeral y = x ^ numeral y"
thin: fixes G' :: "'a triple set"   and ts :: "'a triple set"   and G :: "'a triple set" assumes "G'||-ts"   and "G' \<subseteq> G" shows "G||-ts"
f1: shows "\<b> = int (\<Sum>\<^sub># (image_mset size \<P>))"
Subject_reduction: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"   and "P,E,s \<turnstile> e : T \<surd>" shows "\<exists>T'. P,E,s' \<turnstile> e' : T' \<surd> \<and> P \<turnstile> T' \<le> T"
some_H'h'2: fixes M :: "('b \<times> real) set set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real"   and c :: "('b \<times> real) set set"   and H :: "'b set"   and h :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "graph H h = \<Union> c"   and "x \<in> H"   and "y \<in> H" shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h \<and> linearform H' h' \<and> Subspace.subspace H' E \<and> Subspace.subspace F H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x\<in>H'. h' x \<le> p x)"
simps_16: fixes x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list" shows "Label x41 x42 x43 \<noteq> Trap"
atm_of_subst_lit: fixes L :: "('a, 'b) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "atm_of (subst_lit L \<sigma>) = subst_atm_abbrev (atm_of L) \<sigma>"
derives1_eq_derivations1: fixes x :: "symbol list"   and y :: "symbol list" shows "derives1 x y = ((x, y) \<in> derivations1)"
dom_def: fixes f :: "'c local.term set" shows "local.dom f = (if domains f \<noteq> {} then SOME a. a \<in> domains f else null)"
assms_1: fixes c :: "'a option acom" shows "strip (f c) = strip c"
measurable_in_subalg': fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "subalgebra M F"   and "f \<in> restr_to_subalg M F \<rightarrow>\<^sub>M N" shows "f \<in> F \<rightarrow>\<^sub>M N"
ipurge_tr_rev_t_invariance_aux: fixes Y :: "('c, 'd) ipurge_rec"   and X :: "('c, 'd) ipurge_rec" assumes "Y \<in> ipurge_tr_rev_t_set X" shows "Pol Y = Pol X \<and> Map Y = Map X \<and> Dom Y = Dom X"
choose_component_input_condition_def: fixes e :: "'a"   and v :: "'a" shows "choose_component_input_condition e v \<equiv> regular e \<and> equivalence e \<and> vector v \<and> regular v \<and> e * v = v"
s_mul_ext_singleton: fixes a :: "'a"   and b :: "'a"   and s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "(a, b) \<in> s" shows "({#a#}, {#b#}) \<in> s_mul_ext ns s"
min_set_min_union_2: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<union>\<^sub>m min_set B = A \<union>\<^sub>m B"
symmetric_conv: fixes G :: "('b, 'c) pre_digraph" shows "symmetric G = (\<forall>e1\<in>arcs G. \<exists>e2\<in>arcs G. tail G e1 = head G e2 \<and> head G e1 = tail G e2)"
uses_finite: fixes g :: "'g"   and n :: "'node" shows "finite (uses g n)"
temporaries_inv_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "temporaries_inv ts \<equiv> tmps_distinct ts \<and> load_tmps_fresh ts"
irreflexive_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "irreflexive x" shows "irreflexive (x \<sqinter> y)"
HInv5_inner_def: fixes s :: "state"   and p :: "Proc" shows "HInv5_inner s p = (phase s p = 2 \<longrightarrow> HInv5_inner_R s p)"
map'_try_gpv: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" shows "map_gpv' f g h (TRY gpv ELSE gpv') = TRY map_gpv' f g h gpv ELSE map_gpv' f g h gpv'"
mem_Ch_CH: fixes x :: "'x"   and h :: "'h"   and X :: "'x set" assumes "x \<in> Ch h X" shows "x \<in> CH X"
Qg_top_closed: fixes N :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "G \<triangleright> N"   and "A \<in> set_rcs G N"   and "B \<in> set_rcs G N" shows "c_top G N A B \<in> set_rcs G N"
HUnion_iff: fixes x :: "hf"   and A :: "hf" shows "(x \<^bold>\<in> \<Squnion>A) = (\<exists>y. y \<^bold>\<in> A \<and> x \<^bold>\<in> y)"
asymp_equiv_plus_const_left: fixes c :: "real" shows "(\<lambda>n. c + real n) \<sim>[sequentially] real"
bv_uminus_type: fixes w :: "bit list" shows "bv_uminus (norm_signed w) = bv_uminus w"
Pow_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A \<in> Pow B) = (A \<subseteq> B)"
mono_Sup: fixes A :: "('b \<Rightarrow> 'c) set" assumes "A \<subseteq> {f. mono f}" shows "mono (\<Squnion>A)"
simps_2: fixes a :: "'a"   and as :: "'a list" shows "Poly (a # as) = pCons a (Poly as)"
els: shows "n = 0 \<longrightarrow> v' \<in> E e3 \<rho>"
transD: fixes r :: "('k \<times> 'k) set"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
NS_S_compat: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" assumes "NS s t"   and "S t u" shows "S s u"
map_snd_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
REACH: shows "(src, v) \<in> E\<^sup>*"
lens_member_override: fixes x :: "'a \<Longrightarrow> 'b"   and A :: "'b scene"   and s\<^sub>1 :: "'b"   and s\<^sub>2 :: "'b"   and s\<^sub>3 :: "'b" assumes "x \<in>\<^sub>S A" shows "s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on A \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<triangleleft>\<^bsub>x\<^esub> s\<^sub>3 on A"
add_point_to_blocks_def: fixes p :: "'a"   and bs :: "'a set set" shows "add_point_to_blocks p bs \<equiv> image_mset (insert p) {#b \<in># \<B>. b \<in> bs#} + {#b \<in># \<B>. b \<notin> bs#}"
n_val_valuation: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "valuation K (n_val K v)"
methd_rec_Some_cases: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and c :: "class" assumes "methd G C sig = Some m"   and "ws_prog G"   and "class G C = Some c"   and "C \<noteq> Object" obtains "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig = Some m"   | "G \<turnstile> C inherits method sig m" and "methd G (super c) sig = Some m"
inf_continuousD: fixes F :: "'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> 'a" assumes "inf_continuous F"   and "decseq M" shows "F (Inf (range M)) = (INF i. F (M i))"
assms_4: shows "\<not> is_zero_row_upt_k ma (Suc k) (Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k))"
awalk_append_iff: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and q :: "'d list"   and v :: "'c" assumes "wf_digraph G" shows "pre_digraph.awalk G u (p @ q) v = (pre_digraph.awalk G u p (last (pre_digraph.awalk_verts G u p)) \<and> pre_digraph.awalk G (last (pre_digraph.awalk_verts G u p)) q v)"
map_option_o_map_upd: fixes f :: "'c \<Rightarrow> 'b"   and m :: "'a \<Rightarrow> 'c option"   and a :: "'a"   and b :: "'c" shows "map_option f \<circ> m(a \<mapsto> b) = (map_option f \<circ> m)(a \<mapsto> f b)"
simps_8: fixes var :: "nat"   and a :: "atom fm"   and b :: "atom fm" shows "freeIn var (Or a b) = (freeIn var a \<and> freeIn var b)"
fr_rename_ext_def: fixes ecnv :: "('d, 'e) graph_rec_scheme \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and G :: "('d, 'e) graph_rec_scheme" shows "fr_rename_ext ecnv f G \<equiv> \<lparr>g_V = f ` g_V G, g_E = rename_E f (g_E G), g_V0 = f ` g_V0 G, \<dots> = ecnv G\<rparr>"
both_empty1: fixes t :: "'b ttree" shows "TTree.empty \<otimes>\<otimes> t = t"
healthy_unitary: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "unitary P" shows "unitary (t P)"
globally_sound_modes_invariant: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "globally_sound_mode_use (cms, mem)"   and "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')" shows "globally_sound_mode_use (cms', mem')"
P1: shows "C B J CongA D E F \<and> \<not> B C OS A J \<and> Coplanar A B C J \<and> A B J CongA G H I"
lvl_update_incr_iff: fixes a :: "'c"   and b :: "'d"   and t :: "(('c \<times> 'd) \<times> nat) tree" shows "(lvl (update a b t) = lvl t + 1) = (\<exists>l x r. update a b t = \<langle>l, (x, lvl t + 1), r\<rangle> \<and> lvl l = lvl r)"
from_digits_altdef: fixes base :: "nat"   and n :: "nat"   and d :: "nat \<Rightarrow> nat" assumes "digits base" shows "digits.from_digits base n d = (\<Sum>i<n. d i * base ^ i)"
ln_upper_7_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_7 x"
Worder_trans: fixes S :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme"   and U :: "('g, 'h) Order_scheme" assumes "Worder S"   and "Worder T"   and "Worder U"   and "ord_equiv S T"   and "ord_equiv T U" shows "ord_equiv S U"
generalized_sfw_append: fixes a :: "('a simple_match \<times> 'b) list"   and b :: "('a simple_match \<times> 'b) list"   and p :: "('a, 'c) simple_packet_scheme" shows "generalized_sfw (a @ b) p = (case generalized_sfw a p of None \<Rightarrow> generalized_sfw b p | Some x \<Rightarrow> Some x)"
greaterThan_iff: fixes i :: "'e word"   and k :: "'e word" shows "(i \<in> signed.greaterThan k) = (k <s i)"