hcju/isabelleps huggingface dataset:
oghoare_imp_ann_matches: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and a :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a c Q, A" shows "ann_matches \<Gamma> \<Theta> a c"
simps_2: fixes x :: "'b"   and xs :: "'b negation_type list" shows "getNeg (Neg x # xs) = x # getNeg xs"
card_order_on_well_order_on: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "card_order_on A r" shows "well_order_on A r"
fbrelationI: fixes r :: "V" assumes "fpairs r = r" shows "fbrelation r"
induct: fixes P1 :: "('d, 'e, 'f) val \<Rightarrow> bool"   and P2 :: "('d, 'e, 'f) expr \<Rightarrow> bool"   and val :: "('d, 'e, 'f) val"   and expr :: "('d, 'e, 'f) expr" assumes "\<And>x. P1 (CV x)"   and "\<And>x. P1 (Var x)"   and "\<And>x. P1 (Loc x)"   and "\<And>x. P1 (Rid x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Lambda x1 x2)"   and "\<And>x. P1 x \<Longrightarrow> P2 (VE x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (Apply x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (Ite x1 x2 x3)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Ref x)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Read x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (Assign x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Rfork x)"   and "\<And>x. P2 x \<Longrightarrow> P2 (Rjoin x)" shows "P1 val \<and> P2 expr"
mono_Mprefix3: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and Q :: "'b \<Rightarrow> 'b process" assumes "\<forall>x\<in>A. P x \<sqsubseteq> Q x" shows "min_elems (D (Mprefix A P)) \<subseteq> T (Mprefix A Q)"
map_def: shows "local.map = (\<lambda>f. if C.arr f then S.mkArr (\<iota> ` D.cones (C.cod f)) (\<iota> ` D.cones (C.dom f)) (\<iota> \<circ> D.cones_map f \<circ> \<o>) else S.null)"
IH: fixes gpv :: "('a, 'b, 'c) gpv" assumes "results_gpv \<I> gpv = {}"   and "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv' gpv = 0"
List_Ref: fixes a :: "'c"   and h :: "'c \<Rightarrow> 'c"   and as :: "'c list" assumes "a \<noteq> Null" shows "List h a as = (\<exists>bs. as = a # bs \<and> List h (h a) bs)"
inj_on_wrap: shows "inj_on (wrap_iso_arcs hom f) (iso_arcs hom ` A)"
add_diff_cancel_eint: fixes x :: "eint"   and y :: "eint" assumes "x \<noteq> \<infinity>" shows "x + y - x = y"
assoc'_simps\<^sub>A\<^sub>W\<^sub>C_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "associative_weak_composition V H \<a>"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "H f g \<noteq> partial_magma.null V"   and "H g h \<noteq> partial_magma.null V" shows "partial_magma.arr V (category.inv V (\<a> f g h))"
dim_mono: fixes V :: "'c set"   and W :: "'c set" assumes "V \<subseteq> vs2.span W" shows "vs2.dim V \<le> vs2.dim W"
ord_isom_surj_forall: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "\<forall>b\<in>carrier E. \<exists>a\<in>carrier D. b = f a"
stutter_extend_edges_def: fixes V :: "'c set"   and E :: "('c \<times> 'c) set" shows "stutter_extend_edges V E \<equiv> E \<union> {(v, v) |v. v \<in> V \<and> v \<notin> Domain E}"
subgame_E': fixes V' :: "'a set" assumes "V' \<subseteq> V\<^bsub>G'\<^esub>" shows "E\<^bsub>G'.subgame V'\<^esub> = Restr E\<^bsub>G'\<^esub> V\<^bsub>G'.subgame V'\<^esub>"
remove: fixes A :: "'g set"   and x :: "'g"   and g :: "'g \<Rightarrow> 'h" assumes "finite A"   and "x \<in> A" shows "sum g A = g x + sum g (A - {x})"
Abs_inj_on: shows "inj_on of_bl {bl. length bl = LENGTH('b)}"
seq_char: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxV.seq g f = (seq (fst g) (fst f) \<and> seq (snd g) (snd f))"
fixp_preserves_mono1: fixes F :: "('c \<Rightarrow> 'd llist) \<Rightarrow> 'c \<Rightarrow> 'd llist"   and f :: "'c \<Rightarrow> 'd llist"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. mono_llist (\<lambda>f. F f x)"   and "f \<equiv> llist.fixp_fun F"   and "\<And>f. monotone ord (\<sqsubseteq>) f \<Longrightarrow> monotone ord (\<sqsubseteq>) (F f)" shows "monotone ord (\<sqsubseteq>) f"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Psi>.map f) = Cop.arr f"
not_eventuallyD: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<not> eventually P F" shows "\<exists>x. \<not> P x"
lang_final: fixes r :: "'b rexp"   and n :: "nat" shows "final r = ([] \<in> lang n r)"
catch_splits_1: fixes P :: "'a + 'b \<Rightarrow> bool"   and m :: "'c + 'b"   and f :: "'c \<Rightarrow> 'a + 'b" shows "P (try m catch f) = ((\<forall>e. m = Inl e \<longrightarrow> P (f e)) \<and> (\<forall>x. m = Inr x \<longrightarrow> P (Inr x)))"
b_1: fixes x :: "'a" assumes "x \<in> A" shows "dist x0 x \<le> b"
simps_3: fixes f1 :: "'b"   and f2 :: "'b" shows "(case FinalAllow of FinalAllow \<Rightarrow> f1 | FinalDeny \<Rightarrow> f2) = f1"
simps_1: fixes a :: "'b" shows "literal_formula_to_literal (Atom a) = a\<^sup>+"
FOREACHi_def: fixes \<Phi> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'c set" shows "FOREACH\<^bsup>\<Phi>\<^esup> S \<equiv> FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S (\<lambda>_. True)"
one_eq_prime_power_iff: fixes p :: "'a"   and n :: "nat" assumes "prime p" shows "((1::'a) = p ^ n) = (n = 0)"
parallel_decomposition: fixes V :: "'a list"   and V' :: "'a list"   and SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" assumes "(V, V') \<in> Strong_Security.USdB SR DA d" shows "\<forall>i<length V. ([V ! i], [V' ! i]) \<in> Strong_Security.USdB SR DA d"
assms: fixes w :: "'a" shows "w \<in> space M =simp=> f w = g w"
set_mp: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
proj2_line_rep_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "\<exists>k. k \<noteq> 0 \<and> proj2_line_rep (proj2_line_abs v) = k *\<^sub>R v"
h1: shows "all_list_constr (get_subsets (reduce_system p (qs, matrix, subsets, signs))) (length qs)"
is_derivation_derivation: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "is_derivation X p"   and "f \<in> diff_fun_space"   and "g \<in> diff_fun_space" shows "X (f * g) = f p * X g + g p * X f"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b = b \<cdot> a"
prime_elem_imp_power_coprime: fixes p :: "'a"   and a :: "'a"   and m :: "nat" assumes "prime_elem p"   and "\<not> p dvd a" shows "coprime a (p ^ m)"
bs_delete_correct: fixes i :: "nat"   and s :: "integer" shows "bs_\<alpha> (bs_delete i s) = bs_\<alpha> s - {i}"
Q_def: shows "Q = matrix_sum d (\<lambda>l. tensor_P (proj_psi_l l) (proj_k l)) R"
pp_a_def: fixes X :: "'b ppath set" shows "pp_a X = {Node u |u. \<not> (\<exists>v\<in>X. u = pp_first v)}"
f_props_4: shows "\<forall>x\<in>one_chain1. finite (f x)"
least_method_implies_path_unique: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C has least M = (Ts, T, m) via Cs"   and "wf_prog wf_md P" shows "P \<turnstile> Path C to last Cs unique"
simps_1: fixes x21 :: "'c"   and x22 :: "'c llist"   and y21 :: "'c"   and y22 :: "'c llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
projs_3: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "\<pi>\<^sub>4 (dsn, dsk, flag, hops, nhip) = flag"
integrable_mult_indicator: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
ball_cong: fixes A :: "'e set"   and B :: "'e set"   and P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<forall>x\<in>A. P x) = (\<forall>x\<in>B. Q x)"
basis_reduction_add_rows_cost_2: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs" shows "cost (basis_reduction_add_rows_cost upw i state) \<le> (2 * n + 2 * i + 7) * i"
map_add_find_right: fixes n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and xx :: "'a"   and m :: "'b \<Rightarrow> 'a option" assumes "n k = \<lfloor>xx\<rfloor>" shows "(n \<Oplus> m) k = \<lfloor>xx\<rfloor>"
c3: assumes "b \<le> - 1" shows "\<exists>w. \<forall>y<w. b * y < y\<^sup>2"
X1: shows "distinct (map fst (map (\<lambda>(uu_, x, uu_, y). (x, y)) (filter (eff_enabled s) effs)))"
simps_1: shows "local.coeff [] = (\<lambda>_. \<zero>)"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('j \<Rightarrow> 'j) \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme"   and v :: "'j" assumes "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
n\<^sub>\<UU>_def: fixes m :: "nat"   and x :: "nat" assumes "n\<^sub>\<UU> \<le> m" shows "\<UU>.token_succeeds x = (m < x \<or> (\<exists>j\<ge>i\<^sub>\<UU>. \<UU>.rank x m = Some j) \<or> \<UU>.token_run x m \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
verts3_slim_in_verts3: fixes v :: "'a" assumes "v \<in> verts3 (with_proj slim)" shows "v \<in> verts3 (with_proj G)"
Cons: fixes ys :: "(real \<times> real) list"   and xa :: "real \<times> real"   and x :: "real \<times> real" assumes "\<forall>y\<in>set ys. ccw xa x y"   and "ccw.sortedP xa ys" shows "ccw.sortedP xa (x # ys)"
aligned_add_offset_mod: fixes x :: "'b word"   and sz :: "nat"   and k :: "'b word" assumes "is_aligned x sz"   and "k < 2 ^ sz" shows "(x + k) mod 2 ^ sz = k"
P5_a: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x l\<rightarrow> y = (1::'a)"   and "y l\<rightarrow> z = (1::'a)" shows "x l\<rightarrow> z = (1::'a)"
sup_continuous_lfp: fixes F :: "'a \<Rightarrow> 'a" assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
\<R>p_\<R>_eq: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "\<R>p \<Gamma> \<S> P = (\<lambda>x. \<R>_abv x \<Gamma> \<S> P)"
rtranclp_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
lift_state_oracle_def: fixes F :: "('o \<Rightarrow> 'm \<Rightarrow> (('n \<times> 'k) \<times> 'o) spmf) \<Rightarrow> 'l \<Rightarrow> 'm \<Rightarrow> (('n \<times> 'k) \<times> 'l) spmf"   and "oracle" :: "'k \<times> 'o \<Rightarrow> 'm \<Rightarrow> ('n \<times> 'k \<times> 'o) spmf" shows "lift_state_oracle F oracle = (\<lambda>(t, s') a. map_spmf rprodl (F ((Pair t ---> id ---> map_spmf lprodr) oracle) s' a))"
norm_le_lminus: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x - n \<le> y" shows "x \<le> y + n"
chain_boundary_chain_map: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and g :: "'b \<Rightarrow> 'c" assumes "singular_chain p X c" shows "chain_boundary p (chain_map p g c) = chain_map (p - Suc 0) g (chain_boundary p c)"
T_on__as_sum: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s0 :: "'state"   and rs :: "'request list" shows "T_on'' A s0 rs = sum (T_on_n A s0 rs) {..<length rs}"
sorted_filter: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)" shows "linorder.sorted less_eq (map f (filter P xs))"
map_spmf_conv_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
maximal_Extend: fixes f :: "nat \<Rightarrow> ('a, 'b) fm"   and S :: "('a, 'b) fm set" assumes "surj f" shows "maximal (Extend S f)"
finter_fempty_right: fixes A :: "'a fset" shows "A |\<inter>| {||} = {||}"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) cntxt" shows "map_cntxt g1 g2 g3 (map_cntxt f1 f2 f3 v) = map_cntxt (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
finfun_rec_wf_aux_def: fixes cnst :: "'a \<Rightarrow> 'b"   and upd :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "finfun_rec_wf_aux cnst upd \<equiv> (\<forall>a b. upd a b (cnst b) = cnst b) \<and> (\<forall>a a' b b' c. a \<noteq> a' \<longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)) \<and> (\<forall>b b' a b''. b \<noteq> b' \<longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b))"
tm_wf_t_wcode: shows "tm_wf0 t_wcode"
\<theta>_simps_3: shows "trg \<pi>.the_\<theta> = src t"
axioms_1: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" assumes "Limit.cone J C D a \<chi>" shows "category C"
t1_in_higher_levels: shows "t1 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high"
indRelRTPO_impl_TRel_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_respects_binary_pred (indRelRTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_respects_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
rep_eq: fixes x :: "('f, 'e) generator" shows "generator (take_trans x) = take_raw (generator x)"
prime_factorization_cong: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalize x = normalize y" shows "factorial_semiring.prime_factorization zero times one normalize x = factorial_semiring.prime_factorization zero times one normalize y"
assms: shows "\<forall>f\<in>set fs. is_cnf f"
is_arg_min_linorderI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and m :: "'b \<Rightarrow> 'c" assumes "P x"   and "\<And>y. P y \<Longrightarrow> order.greater_eq (m y) (m x)" shows "is_arg_min m P x"
preserves_inv: fixes f :: "'c \<times> 'c" assumes "C.CC.iso f" shows "C.CC.map (C.CC.inv f) = C.CC.inv (C.CC.map f)"
indep_vars_def: fixes M' :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and I :: "'c set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
inject_17: fixes x18 :: "float"   and y18 :: "float" shows "(Num x18 = Num y18) = (x18 = y18)"
ci_le_bj: fixes j :: "nat"   and i :: "nat"   and m :: "nat"   and n :: "nat" assumes "odd j"   and "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 1 / 2 ^ n" shows "c (real i / 2 ^ m) \<le> b (real j / 2 ^ n)"
induct_matrix_row_operations: fixes P :: "((real, 'a) vec, 'a) vec \<Rightarrow> bool"   and A :: "((real, 'a) vec, 'a) vec" assumes "\<And>A i. row i A = 0 \<Longrightarrow> P A"   and "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0) \<Longrightarrow> P A"   and "\<And>A m n. \<lbrakk>P A; m \<noteq> n\<rbrakk> \<Longrightarrow> P (\<chi>i j. A $ i $ Fun.swap m n id j)"   and "\<And>A m n c. \<lbrakk>P A; m \<noteq> n\<rbrakk> \<Longrightarrow> P (\<chi>i. if i = m then row m A + c *\<^sub>R row n A else row i A)" shows "P A"
uminus_mpoly_def: shows "uminus \<equiv> map_fun mapping_of MPoly uminus"
hyps: fixes ys :: "'b list" shows "set (fold (\<lambda>x. f (g x)) xs ys) = (\<Union>x\<in>set xs. set (g x)) \<union> set ys"
cases: fixes a1 :: "'b sequent list \<times> 'b sequent"   and a2 :: "'b form" assumes "rightPrincipal a1 a2" obtains C :: "'b sequent" and F :: "'b" and Fs :: "'b form list" and Ps :: "'b sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>)"
fin': shows "finite (insert 1 ((\<lambda>(xb, xa). \<bar>real_of_int xb - real xa * x\<bar>) ` {(t, u). 0 < u \<and> coprime (int u) t \<and> \<bar>real_of_int t - real u * x\<bar> < 1 / real u}))"
rt_graph_def: fixes \<sigma> :: "nat \<Rightarrow> state" shows "rt_graph \<sigma> = (\<lambda>dip. {uu_. \<exists>ip ip' dsn dsk hops. uu_ = (ip, ip') \<and> ip \<noteq> dip \<and> rt (\<sigma> ip) dip = Some (dsn, dsk, val, hops, ip')})"
less_numeral_extra_3: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
Lim_bounded_PInfty: fixes f :: "nat \<Rightarrow> ereal"   and l :: "ereal"   and B :: "real" assumes "f \<longlonglongrightarrow> l"   and "\<And>n. f n \<le> ereal B" shows "l \<noteq> \<infinity>"
atLeastPlusOneAtMost_greaterThanAtMost_integer: fixes l :: "integer"   and u :: "integer" shows "{l + 1..u} = {l<..u}"
Lcm_eq_0_I: fixes A :: "'a set" assumes "(0::'a) \<in> A" shows "Lcm A = (0::'a)"
privileged_ASR_def: fixes r :: "word5" shows "privileged_ASR r \<equiv> False"
steps_appendD2: fixes xs :: "'a list"   and ys :: "'a list" assumes "steps (xs @ ys)"   and "ys \<noteq> []" shows "steps ys"
return_time_pos_returns_to: fixes P :: "'a set"   and x :: "'a" assumes "0 < return_time P x" shows "returns_to P x"
Ana_keys_fv: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and K :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and R :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "Ana t = (K, R)" shows "fv\<^sub>s\<^sub>e\<^sub>t (set K) \<subseteq> fv t"
dvd_dvd_imp_unit_mult: fixes x :: "'a"   and y :: "'a" assumes "x dvd y"   and "y dvd x" shows "\<exists>z. z dvd (1::'a) \<and> y = x * z"
eq_iff: fixes x :: "rat"   and y :: "rat" shows "(of_rat x = of_rat y) = (x = y)"
set_iterator_abs_def: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and invar :: "'d \<Rightarrow> bool"   and it :: "('f \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and S0 :: "'e set" shows "set_iterator_abs \<alpha> invar it S0 \<equiv> set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"
strongly_confluent_on_E11: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "strongly_confluent_on r A"   and "x \<in> A"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "\<exists>u. (y, u) \<in> r\<^sup>* \<and> (z, u) \<in> r\<^sup>="
range_composition: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'g" shows "range (\<lambda>x. f (g x)) = f ` range g"
word_sle_eq: fixes a :: "'c word"   and b :: "'c word" shows "(a \<le>s b) = (sint a \<le> sint b)"
lookup_fresh: fixes x :: "name"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and c :: "coname" assumes "x \<sharp> (\<theta>_n, \<theta>_c)" shows "lookup x c \<theta>_n \<theta>_c = lookupa x c \<theta>_c"
funionI1: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A" shows "c |\<in>| A |\<union>| B"
winding_number_unique: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and n :: "complex" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n" shows "winding_number \<gamma> z = n"
elims: fixes x :: "(bool, unit) J.arr"   and y :: "'c" assumes "local.map x = y" obtains "x = J.MkIde False" and "y = a0"   | "x = J.MkIde True" and "y = a1"   | "x = J.Null" and "y = C.null"   | vb :: "unit" where "x = J.MkArr True False vb__" and "y = C.null"   | vb :: "unit" where "x = J.MkArr False True vb" and "y = C.null"
img_mkPoint_2: fixes a :: "'s"   and t :: "'s" assumes "ide a"   and "t \<in> local.set a" shows "img (mkPoint a t) = t"
exec_Normal_elim_cases_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Skip,Normal s\<rangle> \<Rightarrow> t" obtains "t = Normal s"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.keys x = {k. poly_mapping.lookup x k \<noteq> (0::'c)}"
Gromov_extension_isometry_homeomorphism: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry_on UNIV f" shows "homeomorphism_on UNIV (Gromov_extension f)"
proj_def: fixes r :: "('b \<times> 'a) set"   and x :: "'b" shows "proj r x = r `` {x}"
lookup_u: fixes z :: "'b" assumes "z \<le> x" shows "lookup u z = (0::'a)"
Domain_empty_iff: fixes r :: "('b \<times> 'c) set" shows "(Domain r = {}) = (r = {})"
exec_step: shows "exec_step P h stk1 loc1 C1 M1 pc1 ics1 frs1 sh = (xp2, h2, frs2, sh2)"
invar_rel_def: fixes a :: "'e \<Rightarrow> 'd set"   and X :: "('c \<times> 'e) set"   and S :: "('a \<times> 'd set) set" shows "\<langle>X, S\<rangle>invar_rel a = {((x, s'), y). \<exists>s. (s', s) \<in> X \<and> (x, y) \<in> S \<and> y \<subseteq> a s}"
in_zero_stalk: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c"   and V :: "'c set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x"   and "V \<in> I" shows "(V, zero_str V) \<in> stalk.zero_stalk \<FF> \<rho> zero_str I V"
freshBoundOutputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
ccompare_finite_5_def: shows "ccompare \<equiv> None"