hcju/isabelleps huggingface dataset:
simps_7: fixes f11 :: "'b \<Rightarrow> 'a"   and f12 :: "'b bexp \<Rightarrow> 'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f2 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'c"   and x11 :: "'b" shows "rec_aexp f11 f12 f2 (Number x11) = f11 x11"
substEnv_def2: fixes ys :: "'varSort"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho &[Y / y]_ys = (\<lambda>xs x. case rho xs x of None \<Rightarrow> if xs = ys \<and> x = y then Some Y else None | Some X \<Rightarrow> Some (X #[Y / y]_ys))"
iter_Suc2: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and n :: "nat"   and s :: "'a" shows "loop_spmf.iter guard body (Suc n) s = loop_spmf.iter guard body n s \<bind> (\<lambda>s. if guard s then body s else return_spmf s)"
\<omega>_simps_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c" assumes "uw\<theta>\<omega> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<omega>" shows "partial_magma.arr V \<omega>"
exhaust: fixes y :: "bool" obtains "y = True"   | "y = False"
feq_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "feq a b = (fcompare a b = ccode.Eq)"
SUP_upper: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and i :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.upd_cs kind cs [] = cs"
isom_gch_units_transpTr5_2: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "Gchain n h"   and "i \<le> n"   and "j \<le> n"   and "i < j"   and "isom_Gchains n (transpos i j) g h" shows "g j \<cong> h i"
assms_1: assumes "\<paragraph>''invar-initial''" shows "I s\<^sub>0"
conv_abscissa_PInfI: fixes f :: "'a fds" assumes "\<And>s. \<not> fds_converges f s" shows "conv_abscissa f = \<infinity>"
split_pairD: fixes x :: "'a"   and y :: "'b"   and p :: "'a \<times> 'b" assumes "(x, y) = p" shows "x = fst p \<and> y = snd p"
u_period: fixes n :: "nat"   and l :: "nat" shows "unity_root k (- int (n + k) * int l) = unity_root k (- int n * int l)"
quietEqvt: fixes P :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "quiet P" shows "quiet (p \<bullet> P)"
leftmost_Derives1_leftderives: fixes i :: "nat"   and a :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list"   and c :: "symbol list" assumes "leftmost i a"   and "Derives1 a i r b"   and "leftderives b c" shows "leftderives a c"
fixp_lowerbound: fixes f :: "'b \<Rightarrow> 'b"   and z :: "'b" assumes "monotone ordb ordb f"   and "ordb (f z) z" shows "ordb (b.fixp f) z"
restrict_bind_spmf: fixes p :: "'b spmf"   and g :: "'b \<Rightarrow> 'a spmf"   and A :: "'a set" shows "(p \<bind> g) \<upharpoonleft> A = p \<bind> (\<lambda>x. g x \<upharpoonleft> A)"
PP': fixes t :: "'trm" assumes "t \<in> trm" shows "PP' t \<in> fmla"
power_numeral: fixes k :: "num"   and l :: "num" shows "numeral k ^ numeral l = numeral (Num.pow k l)"
Cantor_\<omega>\<omega>_def: shows "Cantor_\<omega>\<omega> \<equiv> \<lambda>x. SOME ns. x = omega_sum ns \<and> normal ns"
unat_split: fixes P :: "nat \<Rightarrow> bool"   and x :: "'b word" shows "P (unat x) = (\<forall>n. word_of_nat n = x \<and> n < 2 ^ LENGTH('b) \<longrightarrow> P n)"
th2: shows "(\<Oplus>p\<in>{p. p permutes S}. f (inv p)) = finsum R (f \<circ> inv) {p. p permutes S}"
l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<equiv> type_wf = ShadowRootClass.type_wf \<and> get_shadow_root = get_shadow_root \<and> get_shadow_root_locs = get_shadow_root_locs"
eq: fixes ops :: "'a arith_ops_record"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> (=)) (=) (=)"
interpret_slp_append: fixes slp1 :: "floatarith list"   and slp2 :: "floatarith list"   and xs :: "real list" shows "interpret_slp (slp1 @ slp2) xs = interpret_slp slp2 (interpret_slp slp1 xs)"
connected_monotone_quotient_preimage: fixes T :: "'c set"   and S :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "connected T"   and "continuous_on S f"   and "f ` S = T"   and "\<And>U. U \<subseteq> T \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"   and "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})" shows "connected S"
map_snd_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
intersection_numbers_def: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" assumes "incidence_system point_set block_collection" shows "incidence_system.intersection_numbers block_collection \<equiv> {b1 |\<inter>| b2 |b1 b2. b1 \<in># block_collection \<and> b2 \<in># remove1_mset b1 block_collection}"
infm: shows "\<forall>i\<ge>N. \<exists>j>i. P i j"
v_TS_defs_2: fixes Quorum :: "process set set" assumes "quorum_process Quorum" shows "v_init = {\<lparr>next_round = 0, votes = \<lambda>r. Map.empty, decisions = Map.empty\<rparr>}"
order_trans: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm x y"   and "lex_pm y z" shows "lex_pm x z"
C: shows "UNIV = (\<Union>i. {..- inverse (real (Suc i))}) \<union> (\<Union>i. {inverse (real (Suc i))..}) \<union> {0}"
simps_5: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and f :: "form" shows "FEval MI e (FAll f) = (\<forall>m\<in>fst MI. FEval MI (\<lambda>y. case y of 0 \<Rightarrow> m | Suc n \<Rightarrow> e n) f)"
unit_simps_3: shows "cod \<eta> = g \<star> f"
z_gt: shows "\<forall>y. y \<in># Ya \<longrightarrow> y \<in> A \<and> z \<in> A \<and> gt z y"
matrix_mul_lid: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.mat (1::'b) ** A = A"
\<phi>_def: fixes y :: "'d"   and h :: "'c" shows "\<phi> y h = G h \<cdot>\<^sub>D transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map \<eta>o y"
assms_3: shows "xs all_in\<^sub>i range_tms e env"
unallocatedBlocksInvariant_def: fixes state :: "CFile" shows "unallocatedBlocksInvariant state \<equiv> \<forall>blockNum i. \<not> blockNum < nextFreeBlock state \<and> blockNum < numBlocks \<and> i < blockSize \<longrightarrow> data state blockNum i = fillByte"
bit_word_reverse_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (word_reverse w) n = (n < LENGTH('b) \<and> bit w (LENGTH('b) - Suc n))"
cong_face_def: fixes f\<^sub>1 :: "face"   and f\<^sub>2 :: "face" shows "f\<^sub>1 \<cong> f\<^sub>2 \<equiv> vertices f\<^sub>1 \<cong> vertices f\<^sub>2"
natural_isomorphism_axioms: shows "natural_isomorphism Left_a.comp Left_a.comp Left_a.L (H\<^sub>L a') (H\<^sub>L \<phi>)"
Fun: fixes arity\<^sub>f :: "'a \<Rightarrow> nat"   and public\<^sub>f :: "'a \<Rightarrow> bool"   and Ana\<^sub>f :: "'a \<Rightarrow> (('a, 'b, 'c) prot_fun, nat) Term.term list \<times> nat list"   and \<Gamma>\<^sub>f :: "'a \<Rightarrow> 'b option"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and f :: "('e, 'f, 'g) prot_fun"   and T :: "(('e, 'f, 'g) prot_fun, 'h) Term.term list"   and S :: "(('e, 'f, 'g) prot_fun, 'h) Term.term list"   and A :: "'g set set" assumes "stateful_protocol_model arity\<^sub>f public\<^sub>f Ana\<^sub>f \<Gamma>\<^sub>f label_witness1 label_witness2"   and "\<not> is_Abs f"   and "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> timpls_transformable_to_pred A (T ! i) (S ! i)" shows "timpls_transformable_to_pred A (Fun f T) (Fun f S)"
step: fixes x :: "name"   and xs :: "name list"   and F :: "'d frame" shows "\<lparr>\<nu>*(x # xs)\<rparr>F = \<lparr>\<nu>x\<rparr>\<lparr>\<nu>*xs\<rparr>F"
map_cong0: fixes x :: "'b stream"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in> sset x \<Longrightarrow> f z = g z" shows "smap f x = smap g x"
continuity: fixes \<phi> :: "complex_homo \<Rightarrow> bool"   and \<psi> :: "complex_homo \<Rightarrow> bool" assumes "\<exists>A\<in>unit_disc. set_order A \<phi> \<psi>" shows "\<exists>B\<in>unit_disc. point_between_sets \<phi> B \<psi>"
cong_r_filter: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_r F a b = (a l\<rightarrow> b \<in> F \<and> b l\<rightarrow> a \<in> F)"
not_manipulable: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt" shows "(\<not> manipulable f) = (\<forall>P\<in>Prof. \<forall>i. \<forall>L\<in>Lin. f P \<noteq> f (P(i := L)) \<longrightarrow> f (P(i := L)) <\<^bsub>P i\<^esub> f P \<and> f P <\<^bsub>L\<^esub> f (P(i := L)))"
vv: shows "vv = map gso [Suc i..<m]"
continuous_on_compose_frac: fixes f :: "real \<Rightarrow> real" assumes "continuous_on {0..1} f"   and "f 1 = f 0" shows "continuous_on UNIV (f \<circ> frac)"
invs_def: shows "invs = (\<lambda>s. handshake_phase_inv s \<and> phase_rel_inv s \<and> strong_tricolour_inv s \<and> sys_phase_inv_aux (sys_ghost_hs_phase s) s \<and> tso_store_inv s \<and> valid_refs_inv s \<and> valid_W_inv s \<and> (\<forall>x. mut_m.mutator_phase_inv_aux x (ghost_hs_phase (s (mutator x))) s) \<and> fA_rel_inv s \<and> fM_rel_inv s)"
transfer_sum: fixes fl :: "'e \<Rightarrow> 'c"   and Fl :: "'e \<Rightarrow> 'a"   and fr :: "'f \<Rightarrow> 'c"   and Fr :: "'f \<Rightarrow> 'a"   and x :: "'e + 'f" assumes "\<And>l. \<alpha> (fl l) \<le> Fl l"   and "\<And>r. \<alpha> (fr r) \<le> Fr r" shows "\<alpha> (case x of Inl x \<Rightarrow> fl x | Inr x \<Rightarrow> fr x) \<le> (case x of Inl x \<Rightarrow> Fl x | Inr x \<Rightarrow> Fr x)"
VSLuckiestBlocks_def: shows "VSLuckiestBlocks = opt_group \<circ> QE_dnf' opt_group (the_real_step_augment luckiestFind) \<circ> opt_group"
pseudonatural_transformation_whisker_left_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>H :: "'d \<times> 'd \<Rightarrow> 'e"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<and> bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> (pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G) \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D H \<Phi>\<^sub>H \<and> pseudonatural_transformation V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
hseqI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>C c\<guillemotright>" shows "C.hseq \<nu> \<mu>"
denseE: fixes A :: "'a set" assumes "A dense" obtains "carrier \<subseteq> closure A"
nrows_iarray_def: fixes A :: "'b iarray iarray" shows "nrows_iarray A = IArray.length A"
g'_def: shows "g' \<equiv> fold (\<lambda>(u, v, d). add_edge (u, v) d) l g"
nonempty_some: fixes A :: "'e set" assumes "A \<noteq> {}" shows "(SOME x. x \<in> A) \<in> A"
conjunction_sup_nonaborting: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (c::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (d::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice) = c \<squnion> d"
step: fixes m0 :: "'b nres"   and cond :: "'b \<Rightarrow> bool"   and step :: "'b \<Rightarrow> 'b nres"   and x :: "'b"   and Y :: "'b set"   and y :: "'b" assumes "rwof m0 cond step x"   and "cond x"   and "step x = RES Y"   and "y \<in> Y" shows "rwof m0 cond step y"
ord_of_nat_\<omega>: fixes n :: "nat" shows "n\<^sub>\<nat> \<in>\<^sub>\<circ> \<omega>"
AssignHigh: fixes x :: "Var"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "CONTEXT x = high"   and "\<forall>s. noLowDPs s \<longrightarrow> Expr_good e s" shows "G \<rhd> Assign x e : HighSec"
distinct_195: fixes x111 :: "'a exp"   and x112 :: "'a exp"   and x14 :: "'a exp" shows "x111;; x112 \<noteq> throw x14"
sub: shows "(\<Union>y\<in>{y. Inl (Inr y) \<in> SIGF x2}. FVT (\<sigma> y)) \<subseteq> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGF (InContext x1 x2)}. FVT (\<sigma> y))"
cindex_poly_mult: fixes p' :: "real poly"   and a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" assumes "p' \<noteq> 0" shows "cindex_poly a b (p' * q) (p' * p) = cindex_poly a b q p"
rel_fset_alt_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fset R = (\<lambda>A B. (\<forall>x. \<exists>y. x |\<in>| A \<longrightarrow> y |\<in>| B \<and> R x y) \<and> (\<forall>y. \<exists>x. y |\<in>| B \<longrightarrow> x |\<in>| A \<and> R x y))"
bangSC: fixes P :: "pi" shows "!P \<sim>\<^sub>l P \<parallel> !P"
cos_mono_less_eq: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> pi"   and "0 \<le> y"   and "y \<le> pi" shows "(cos x < cos y) = (y < x)"
eq: fixes \<U> :: "'a set set" shows "(\<forall>U\<in>\<U>. \<exists>Y. openin X Y \<and> U = Y \<inter> S) = (\<U> \<subseteq> (\<lambda>Y. Y \<inter> S) ` {y. openin X y})"
rref_condition4: fixes A :: "(('d, 'e) vec, 'f) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'d)) = (0::'d))"
emb_LNil1E: fixes ys :: "'b llist" assumes "emb LNil ys" obtains "lfinite ys"
isDiagonal_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "isDiagonal_upt_k A k = (\<forall>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<and> (mod_type_class.to_nat a < k \<or> mod_type_class.to_nat b < k) \<longrightarrow> A $ a $ b = (0::'b))"
disj_if: fixes A :: "'h set"   and B :: "'h set"   and x :: "'h"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" assumes "A \<inter> B = {}"   and "x \<in> B" shows "(if x \<in> A then f x else g x) = g x"
TS_union: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" shows "TS M2 M1 \<Omega> V m i = \<Union> (C M2 M1 \<Omega> V m ` set [0..<Suc i])"
rename_def: fixes ioa :: "('g set \<times> 'g set \<times> 'g set) \<times> 'f set \<times> ('f \<times> 'g \<times> 'f) set"   and ren :: "'e \<Rightarrow> 'g option" shows "rename ioa ren \<equiv> (({b. \<exists>x. Some x = ren b \<and> x \<in> inputs (asig_of ioa)}, {b. \<exists>x. Some x = ren b \<and> x \<in> outputs (asig_of ioa)}, {b. \<exists>x. Some x = ren b \<and> x \<in> internals (asig_of ioa)}), starts_of ioa, {tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr) in \<exists>x. Some x = ren a \<and> (s, x, t) \<in> trans_of ioa})"
\<theta>\<^sub>g': shows "\<guillemotleft>\<theta>\<^sub>g' : tab\<^sub>0 g \<star> w\<^sub>g' \<rightarrow> u\<^sub>g\<guillemotright>"
sup_continuous_mono: fixes F :: "'a \<Rightarrow> 'b" assumes "sup_continuous F" shows "mono F"
source: fixes S1 :: "'a"   and S2 :: "'a"   and SRel :: "('a \<times> 'a) set"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" assumes "(S1, S2) \<in> SRel" shows "(SourceTerm S1, SourceTerm S2) \<in> encoding.indRelRSTPO Enc SRel TRel"
cases: fixes P :: "'a prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" assumes "P \<turnstile> lub(U, V) = T" obtains "P \<turnstile> U \<le> T" and "P \<turnstile> V \<le> T" and "\<And>T'. \<lbrakk>P \<turnstile> U \<le> T'; P \<turnstile> V \<le> T'\<rbrakk> \<Longrightarrow> P \<turnstile> T \<le> T'"
equiv_up_to_while_weak: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com" assumes "P \<Turnstile> b <\<sim>> b'"   and "P \<Turnstile> c \<sim> c'"   and "\<And>s s'. \<lbrakk>(c, s) \<Rightarrow> s'; P s; bval b s\<rbrakk> \<Longrightarrow> P s'" shows "P \<Turnstile> WHILE b DO c \<sim> WHILE b' DO c'"
True_step_conc: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and a :: "'b" shows "((True # p, q) \<in> step (RegExp2NA.conc L R) a) = ((\<exists>r. q = True # r \<and> (p, r) \<in> step L a) \<or> fin L p \<and> (\<exists>r. q = False # r \<and> (start R, r) \<in> step R a))"
continuous_map_from_subtopology_mono: fixes X :: "'c topology"   and T :: "'c set"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_map (subtopology X T) X' f"   and "S \<subseteq> T" shows "continuous_map (subtopology X S) X' f"
eucl_of_list_eqI: fixes xs :: "real list"   and ys :: "real list" assumes "take DIM('b) (xs @ replicate (DIM('b) - length xs) 0) = take DIM('b) (ys @ replicate (DIM('b) - length ys) 0)" shows "eucl_of_list xs = eucl_of_list ys"
igAbs_termMOD: shows "igAbs termMOD = Abs"
linorder_wlog: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "\<And>a b. a \<le> b \<Longrightarrow> P a b"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
contradicting_in_cop: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat"   and k :: "nat" assumes "\<sigma> =\<^sub>L \<sigma>'"   and "(\<sigma>', k') \<cc> (\<sigma>, k)"   and "path \<sigma> k \<in> dom local.att" shows "((\<sigma>, k), \<sigma>', k') \<in> cop"
default_secure_ACS: fixes G :: "'v graph"   and f :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "wf_graph G"   and "f \<in> set_offending_flows G nP" shows "\<forall>i\<in>fst ` f. \<not> sinvar G (nP(i := \<bottom>))"
rank_less_row_i_imp_i_is_zero: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" assumes "rank A \<le> mod_type_class.to_nat i" shows "Gauss_Jordan A $ i = 0"
assms_1: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"
rep_eq: fixes x :: "'a" shows "cblinfun_apply (prod_right x) = (**) x"
FtorCompPreFtor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f) Functor"   and G :: "('b, 'g, 'd, 'h, 'f, 'i) Functor" assumes "F \<approx>>;;; G" shows "PreFunctor (F ;;: G)"
elims_3: fixes x :: "'a Point"   and xa :: "'a Plane" assumes "\<not> inPlane x xa" obtains e :: "'a Point" and pl :: "'a Plane" where "x = e" and "xa = pl" and "\<not> coplanar e (pbasepoint pl) (pbasepoint pl \<leadsto> direction1 pl) (pbasepoint pl \<leadsto> direction2 pl)"
ltl_equivalence_axioms: shows "ltl_equivalence (\<sim>)"
if_false_eval\<^sub>w: fixes mem :: "Eg1.var \<Rightarrow> nat"   and b :: "bexp"   and t :: "(Eg1.var, aexp, bexp) Stmt"   and c' :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set" assumes "\<not> ev\<^sub>B mem b" shows "eval_abv\<^sub>A \<langle>Stmt.If b t c', mds, mem\<rangle>\<^sub>A \<langle>c', mds, mem\<rangle>\<^sub>A"
zeroprimeideal_domainI: assumes "primeideal {\<zero>} R" shows "domain R"
induct: fixes P :: "((variable \<times> String.literal set) \<times> typ) list \<Rightarrow> typ \<Rightarrow> bool"   and a0 :: "((variable \<times> String.literal set) \<times> typ) list"   and a1 :: "typ" assumes "\<And>insts a Ts. (\<And>x. x \<in> set Ts \<Longrightarrow> P insts x) \<Longrightarrow> P insts (Ty a Ts)"   and "\<And>insts idn S. P insts (Tv idn S)" shows "P a0 a1"
sfs: shows "P,h,C' \<turnstile>\<^sub>s sfs \<surd>"
singular_simplex_subtopology: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "singular_simplex p (subtopology X S) f = (singular_simplex p X f \<and> f ` standard_simplex p \<subseteq> S)"
IsZero_2: shows "\<forall>v. \<eta>: \<rho>\<cdot>v \<mapsto> \<rho>'\<cdot>v"
Diff_mono: fixes A :: "'e set"   and C :: "'e set"   and D :: "'e set"   and B :: "'e set" assumes "A \<subseteq> C"   and "D \<subseteq> B" shows "A - B \<subseteq> C - D"