hcju/isabelleps huggingface dataset:
in_spanE: fixes v :: "'a vec"   and n :: "nat"   and W :: "'a vec set" assumes "v \<in> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) W" shows "\<exists>a A. v = module.lincomb (module_vec TYPE('a) n) a A \<and> finite A \<and> A \<subseteq> W"
implication_implies_pointstamp: fixes t :: "'t"   and c :: "('loc, 't) configuration"   and loc :: "'loc" assumes "t \<in>\<^sub>A frontier (c_imp c loc)"   and "inv_imps_work_sum c" shows "\<exists>t' loc' s. s \<in>\<^sub>A path_summary loc' loc \<and> results_in t' s \<le> t \<and> (t' \<in>\<^sub>A frontier (c_pts c loc') \<or> zcount (c_work c loc') t' < 0)"
mset_merge_all: fixes xss :: "'b list list" shows "mset (merge_all xss) = mset_mset xss"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<sqsubseteq> a) = (a = a \<squnion> b)"
ereal_cong: fixes x :: "real"   and y :: "real" assumes "x = y" shows "ereal x = ereal y"
delete_point_wf: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b" assumes "incidence_system point_set block_collection" shows "incidence_system (incidence_system.del_point point_set p) (incidence_system.del_point_blocks block_collection p)"
leI: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
that_1: shows "real_of_float (lower X) \<le> x"
simps: fixes a1 :: "'f set"   and a2 :: "'f set" shows "(a1 \<rhd> a2) = (\<exists>M N. a1 = M \<and> a2 = N \<and> M - N \<subseteq> Red_F N)"
fsymsC_def: fixes c :: "('c, 'd) lit list" shows "fsymsC c = \<Union> (set (map fsymsL c))"
intrinsic_borderI': fixes ys :: "'b list" assumes "ys \<noteq> []" shows "strict_border (intrinsic_border ys) ys"
list_nonMt_append: fixes s :: "'a list" assumes "s \<noteq> []" shows "\<exists>a t. s = t @ [a]"
LAss: fixes A :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "assn"   and x :: "vname" assumes "A \<turnstile>\<^sub>e {P} e {\<lambda>v s. Q (lupd(x\<mapsto>v) s)}" shows "A \<turnstile> {P} x :== e {Q}"
a2: shows " M2 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
sWlsNE_def: fixes SEM :: "('b, 'c, 'd, 'e, 'f, 'g, 'h) semDom_scheme" shows "sWlsNE SEM \<equiv> \<forall>s. \<exists>sX. sWls SEM s sX"
to_nat_from_nat_k: shows "mod_type_class.to_nat (mod_type_class.from_nat (Suc k)) = Suc k"
pt_atom1: fixes x :: "'a" shows "[] \<bullet> x = x"
Diff_insert_absorb: fixes x :: "'g"   and A :: "'g set" assumes "x \<notin> A" shows "insert x A - {x} = A"
mult_left_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "times bot x = bot"
induct: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and prod :: "'a \<times> 'b" assumes "\<And>a b. P (a, b)" shows "P prod"
unlabel_subst: fixes S :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "unlabel S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = unlabel (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
meet_compl_bot_left2: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> (\<^bold>-\<^bsub>L\<^esub> x \<sqinter>\<^bsub>L\<^esub> y) = \<bottom>\<^bsub>L\<^esub>"
external_invariant_translation: fixes Sa :: "('f, 'e) spec"   and Q :: "'e set"   and pi :: "'g \<Rightarrow> 'e"   and P :: "'g set"   and Sc :: "('h, 'g) spec" assumes "oreach Sa \<subseteq> Q"   and "pi -` Q \<subseteq> P"   and "implements pi Sa Sc" shows "oreach Sc \<subseteq> P"
count_diff: fixes M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" shows "count (M - N) a = count M a - count N a"
hasRead': shows "\<forall>d\<in>D. hasRead s' p d q = hasRead s p d q"
widens_Cons: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and x :: "ty"   and xs :: "ty list"   and ys :: "ty list" shows "P \<turnstile> (x # xs) [\<le>] ys = (\<exists>z zs. ys = z # zs \<and> subtype P x z \<and> P \<turnstile> xs [\<le>] zs)"
Iff_MP_left': fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A IFF B"   and "insert B H \<turnstile> C" shows "insert A H \<turnstile> C"
matrix_vector_mul_linear_gen: fixes A :: "(('d, 'e) vec, 'f) vec" shows "Vector_Spaces.linear (*s) (*s) ((*v) A)"
inverse_transformations_axioms: shows "inverse_transformations (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<tau> \<tau>'"
bdd_belowI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m x" shows "preorder.bdd_below less_eq A"
compact_translation: fixes s :: "'c set"   and a :: "'c" assumes "compact s" shows "compact ((+) a ` s)"
dghm_comp_components_2: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr> = \<GG>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>"
set_intros: fixes x2 :: "'b" shows "x2 \<in> set_message_af (Msg x2)"
simps_1: fixes z :: "quat" shows "quat.Re (Quaternions.cnj z) = quat.Re z"
type_less_right_simps_2: fixes \<tau> :: "'a OCL_Types.type"   and \<upsilon> :: "'a basic_type" shows "(\<tau> < \<upsilon>[1]) = (\<exists>\<rho>. \<tau> = \<rho>[1] \<and> \<rho> < \<upsilon>)"
mult_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and a :: "complex"   and y :: "'c" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times x (scaleC a y) = scaleC a (times x y)"
fun_evaluate_relation_def: shows "fun_evaluate_relation = inv_image (less_than <*lex*> less_than) (\<lambda>x. case x of Inl (s, x, xa, pes, xb) \<Rightarrow> (clock s, size_list (size_prod size size_exp') pes) | Inr (s, x, es) \<Rightarrow> (clock s, size_list size_exp' es))"
delay_bisim_locale: fixes t :: "'t" shows "delay_bisimulation_final_base (r1 t) (r2 t) (bisim t) \<tau>move1 \<tau>move2 (\<lambda>(x1, m). final1 x1) (\<lambda>(x2, m). final2 x2)"
elims: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and x :: "'a"   and xa :: "'b list"   and xb :: "'c list"   and y :: "nat" assumes "On_Off.T step t x xa xb = y" obtains s :: "'a" where "x = s__" and "xa = []" and "xb = []" and "y = 0"   | s :: "'a" and r :: "'b" and rs :: "'b list" and a :: "'c" and as :: "'c list" where "x = s" and "xa = r # rs" and "xb = a__ # as" and "y = t s r a__ + On_Off.T step t (step s r a__) rs as"   | a :: "'a" and v :: "'b" and va :: "'b list" where "x = aa__" and "xa = v__ # va__" and "xb = []" and "y = undefined"   | a :: "'a" and v :: "'c" and va :: "'c list" where "x = a" and "xa = []" and "xb = v # va" and "y = undefined"
make_tr_nth: fixes j :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "j < n" shows "make_tr f n i ! j = f (i + j)"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c ms" shows "map_ms g (map_ms f v) = map_ms (g \<circ> f) v"
Min_le_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(Min A \<le> x) = (\<exists>a\<in>A. a \<le> x)"
subset'_code_2: fixes ys :: "'a list"   and B :: "'a set" shows "subset' (set ys) B = (\<forall>y\<in>set ys. y \<in> B)"
stack_discovered: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "set (stack s) \<subseteq> dom (discovered s)"
conj_cancel_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<sim> x = \<one>"
t: shows "ltakeWhile (Not \<circ> P) (ltake (enat (lfind P lst)) lst) = ltake (enat (lfind P lst)) lst"
non_empty_candidate_results: fixes js :: "nat list"   and n :: "nat" assumes "\<And>f xs. \<lbrakk>associative f; xs \<noteq> []\<rbrakk> \<Longrightarrow> candidate f xs = scanl1 f xs"   and "js \<in> set (candidate (@) (map wrap [0..<n + 1]))" shows "js \<noteq> []"
intros: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and ts\<^sub>d :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> 'f \<times> 'g) list" assumes "length ts = length ts\<^sub>d"   and "\<forall>i<length ts. \<exists>\<O>' \<D>' \<R>'. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>d ! i in ts ! i = (p, is, \<theta>, [], \<D>', \<O>', \<R>')" shows "ts \<sim>\<^sub>d ts\<^sub>d "
prems: shows "fv (RI (sh'',e) ; [] \<leftarrow> e_) \<subseteq> W"
sup_ge2: fixes y :: "'c \<times> 'd"   and x :: "'c \<times> 'd" shows "less_eq_prod y (sup_prod x y)"
aux3lose: fixes k :: "nat"   and d :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and j :: "nat" assumes "\<forall>j\<le>k + d + 3. lose (t + j) = [False]"   and "j \<le> k" shows "lose (Suc (Suc (t + j))) = [False]"
xtrans_7: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y < x"   and "z \<le> y" shows "z < x"
pseudo_divmod_list_def: fixes p :: "'b list"   and q :: "'b list" shows "pseudo_divmod_list p q = (if q = [] then ([], p) else let rq = rev q; (qu, re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in (qu, rev re))"
AU: shows "A \<in> Units (ring_mat TYPE('a) n b)"
abs_eq: fixes x :: "'c \<Rightarrow> 'd option" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmdom' (Abs_fmap x) = dom x"
cod_simp: fixes f :: "'c local.term set" assumes "arr f" shows "cod f = mkarr (COD f)"
fNext_not_member: fixes "next" :: "'b \<Rightarrow> 'b"   and arb :: "'b"   and S :: "'b fset" assumes "fresh next" shows "fresh.fNext next arb S |\<notin>| S"
completeness_def: shows "chaum_ped_sigma.completeness \<equiv> \<forall>h w e. (h, w) \<in> R \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (chaum_ped_sigma.completeness_game h w e) True = 1"
zcount_inI: fixes M :: "'a zmultiset"   and x :: "'a" assumes "zcount M x = 0 \<Longrightarrow> False" shows "x \<in>#\<^sub>z M"
fresh_star_bij_1: fixes pi :: "(name \<times> name) list"   and a :: "name set"   and x :: "'d" shows "(pi \<bullet> a) \<sharp>* (pi \<bullet> x) = a \<sharp>* x"
preserves_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.in_hom f a b" shows "\<guillemotleft>\<F>C.D f : \<F>C.D a \<rightarrow> \<F>C.D b\<guillemotright>"
imageI: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> A" shows "f x \<in> \<P> f A"
select_convs_3: fixes mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a" shows "inp \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = inp"
not_eout_eoutM: fixes P :: "specID"   and E :: "Expression"   and M :: "chanID set" assumes "\<not> eout P E" shows "\<not> eoutM P M E"
f_def: fixes n :: "nat"   and xo :: "'v option" shows "f n xo = (case xo of None \<Rightarrow> enn2real (s n - sum (weight \<Gamma>) (A_n n)) | Some x \<Rightarrow> if x \<in> A_n n then enn2real (weight \<Gamma> x) else 0)"
spmf_of_set: fixes A :: "'c set"   and x :: "'c" shows "spmf (spmf_of_set A) x = indicat_real A x / real (card A)"
disc_iff_2: fixes p :: "'d \<Rightarrow> bool"   and g21 :: "'d \<Rightarrow> 'c"   and g22 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "(\<not> lnull (unfold_llist p g21 g22 a)) = (\<not> p a)"
ind: shows "indep_var borel (\<lambda>x. - \<mu> + X x) borel (\<lambda>x. - \<nu> + Y x)"
inv_inv: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (homology_group p X)" shows "inv\<^bsub>homology_group p X\<^esub> (inv\<^bsub>homology_group p X\<^esub> x) = x"
LVar: fixes Env :: "env"   and B :: "lname set"   and vn :: "lname" shows "Env\<turnstile> B \<guillemotright>\<langle>LVar vn\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
distinct_1: fixes x1 :: "('a, 'b, 'c) val"   and x21 :: "('a, 'b, 'c) expr"   and x22 :: "('a, 'b, 'c) expr" shows "VE x1 \<noteq> Apply x21 x22"
ln_exp_cf4_upper_bound_neg: fixes x :: "real" assumes "x \<le> 0" shows "x \<le> ln (exp_cf4 x)"
bdd_sqrt: fixes X :: "real set" shows "bdd_above X = bdd_above (sqrt ` X)"
tagged_division_Un_interval: fixes p1 :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real"   and p2 :: "('a \<times> 'a set) set" assumes "p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}"   and "p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}"   and "k \<in> Basis" shows "p1 \<union> p2 tagged_division_of cbox a b"
conversep_conversep: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
small_mixed_integer_solution: fixes A\<^sub>1 :: "'a mat"   and nr\<^sub>1 :: "nat"   and A\<^sub>2 :: "'a mat"   and nr\<^sub>2 :: "nat"   and b\<^sub>1 :: "'a vec"   and b\<^sub>2 :: "'a vec"   and Bnd :: "'a"   and x :: "'a vec"   and I :: "nat set" assumes "A\<^sub>1 \<in> carrier_mat nr\<^sub>1 n"   and "A\<^sub>2 \<in> carrier_mat nr\<^sub>2 n"   and "b\<^sub>1 \<in> carrier_vec nr\<^sub>1"   and "b\<^sub>2 \<in> carrier_vec nr\<^sub>2"   and "A\<^sub>1 \<in> \<int>\<^sub>m \<inter> Bounded_mat Bnd"   and "b\<^sub>1 \<in> \<int>\<^sub>v \<inter> Bounded_vec Bnd"   and "A\<^sub>2 \<in> \<int>\<^sub>m \<inter> Bounded_mat Bnd"   and "b\<^sub>2 \<in> \<int>\<^sub>v \<inter> Bounded_vec Bnd"   and "x \<in> carrier_vec n"   and "x \<in> indexed_Ints_vec I"   and "A\<^sub>1 *\<^sub>v x \<le> b\<^sub>1"   and "A\<^sub>2 *\<^sub>v x <\<^sub>v b\<^sub>2"   and "nr\<^sub>1 \<noteq> 0 \<or> nr\<^sub>2 \<noteq> 0 \<or> (0::'a) \<le> Bnd" shows "\<exists>x. x \<in> carrier_vec n \<and> x \<in> indexed_Ints_vec I \<and> A\<^sub>1 *\<^sub>v x \<le> b\<^sub>1 \<and> A\<^sub>2 *\<^sub>v x <\<^sub>v b\<^sub>2 \<and> x \<in> Bounded_vec (fact (n + 1) * Bnd ^ n)"
ct_suffix_extend_ct_list_eq: assumes "ct_suffix [] []"   and "ct_list_eq [TAny] [t]" shows "ct_suffix ([] @ [TAny]) ([] @ [t])"
h3ba: shows "length (filter (\<lambda>i. solve_for_lhs p qs subsets (M_mat signs subsets) $ i \<noteq> 0) [0..<length subsets]) \<le> length subsets"
rows_def: fixes A :: "'c mat" shows "rows A = map (row A) [0..<dim_row A]"
C_subset_FF: fixes x :: "'a set set" assumes "x \<in> C" shows "x \<in> {V \<in> Filters. F \<subseteq> V}"
odd_iff_mod_2_eq_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
type_definition_fps: shows "type_definition ($) Abs_fps {f. True}"
mpoly_base_conv_1: fixes x :: "'a" shows "(0::'a) = poly 0 x"
cut_ge_card: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "card (I \<down>\<ge> t) \<le> card I"
less_linear: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less x y \<or> x = y \<or> less y x"
list_update_overwrite: fixes xs :: "'k list"   and i :: "nat"   and x :: "'k"   and y :: "'k" shows "xs[i := x, i := y] = xs[i := y]"
multiplicity_unit_left: fixes p :: "'a"   and x :: "'a" assumes "is_unit p" shows "multiplicity p x = 0"
Gmult_assoc: fixes g :: "'g"   and h :: "'g"   and v :: "'v" assumes "g \<in> G"   and "h \<in> G"   and "v \<in> V" shows "g *\<cdot> h *\<cdot> v = (g + h) *\<cdot> v"
min_dnf_finite: fixes \<phi> :: "'a ltln" shows "finite (min_dnf \<phi>)"
dvd_0_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times zero a" shows "a = zero"
project_act_Restrict_subset: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'f set"   and act :: "('f \<times> 'f) set" shows "project_act h (Restrict C act) \<subseteq> Restrict (project_set h C) (project_act h act)"
left_plus_circ: fixes x :: "'a" shows "(x * x\<^sup>\<circ>)\<^sup>\<circ> = x\<^sup>\<circ>"
f0: fixes i :: "nat" assumes "i < n" shows "(nderiv i f {0<..} \<longlongrightarrow> f0 i) (at_right 0)"
z_def: shows "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p"
Abs_quot_inverse: fixes y :: "'d set" assumes "y \<in> quot" shows "Rep_quot (Abs_quot y) = y"
simps_2: fixes x :: "'a \<times> 'b \<times> 'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and v :: "'a"   and e :: "'b"   and v' :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list" shows "replace_by_UnPath (x # xs) (v, e, v') ps = (if x = (v, e, v') then ps @ replace_by_UnPath xs (v, e, v') ps else if x = (v', e, v) then rev_path ps @ replace_by_UnPath xs (v, e, v') ps else x # replace_by_UnPath xs (v, e, v') ps)"
INF_identity_eq: fixes Inf :: "'a set \<Rightarrow> 'a"   and A :: "'a set" shows "Inf ((\<lambda>x. x) ` A) = Inf A"
len_signed: fixes x :: "'c signed itself" shows "len_of x = LENGTH('c)"
set_iterator_genord_axioms: shows "set_iterator_genord it S R"
bits_mod_by_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo a one = zero"
no_loop: fixes x :: "'v \<times> 'v" shows "\<not> edge (web_of_network \<Delta>) x x"
prod_rel_def: fixes R1 :: "('d \<times> 'f) set"   and R2 :: "('e \<times> 'g) set" shows "R1 \<times>\<^sub>r R2 \<equiv> {((a, b), a', b'). (a, a') \<in> R1 \<and> (b, b') \<in> R2}"
ccompare_set_def: shows "ccompare = map_option (\<lambda>c. comp_of_ords (ord.set_less_eq (le_of_comp c)) (ord.set_less (le_of_comp c))) (ID ccompare)"
openz_def: fixes t :: "sterm"   and s :: "sterm"   and p :: "sterm" shows "(t\<^bsup>[s,p]\<^esup>) = {0 \<rightarrow> [s,p]} t"
rtrancl_restrictI_aux: fixes u :: "'b"   and v :: "'b"   and E :: "('b \<times> 'b) set"   and R :: "'b set" assumes "(u, v) \<in> (E - UNIV \<times> R)\<^sup>*"   and "u \<notin> R" shows "(u, v) \<in> (rel_restrict E R)\<^sup>* \<and> v \<notin> R"
0: fixes f :: "'a" assumes "ide f"   and "a \<in> sources f"   and "a' \<in> sources f" shows "a \<cong> a'"
DVDJ_def: fixes i :: "int"   and n :: "int"   and s :: "MIR.num" shows "DVDJ i n s = foldr MIR.disj (map (\<lambda>j. MIR.conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) F"
Cons_in_lists_iff: fixes x :: "'a"   and xs :: "'a list"   and A :: "'a set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
naturality: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "\<chi> (J.cod f) \<cdot> \<chi>.A.map f = local.map f \<cdot> \<chi> (J.dom f)"
CONST_INTF_def: fixes c :: "'b"   and I :: "interface" shows "c ::\<^sub>i I \<equiv> True"
Range_outside_sub_Image_Domain: fixes R :: "('b \<times> 'a) set"   and X :: "'b set" shows "Range (Outside' X R) \<subseteq> R `` (Domain R - X)"
fresh_set: fixes a :: "atom"   and xs :: "'b list" shows "a \<sharp> set xs = a \<sharp> xs"
valid_mmuaux'_mono: fixes args :: "Monitor.args"   and cur :: "nat"   and dt :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and dt' :: "nat" assumes "valid_mmuaux' args cur dt aux auxlist"   and "dt \<le> dt'" shows "valid_mmuaux' args cur dt' aux auxlist"
st_2: shows "\<not> s \<le> Suc (2 * n)"
row_carrier: fixes A :: "'c mat"   and i :: "nat" shows "row A i \<in> carrier_vec (dim_col A)"
lossless_R2: fixes msgs :: "bool \<times> bool"   and \<sigma> :: "bool" shows "lossless_spmf (R2 msgs \<sigma>)"
homotopic_with_sphere_times: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex"   and g :: "'a \<Rightarrow> complex"   and h :: "'a \<Rightarrow> complex" assumes "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) f g"   and "continuous_on S h"   and "\<And>x. x \<in> S \<Longrightarrow> h x \<in> sphere 0 1" shows "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1) (\<lambda>x. f x * h x) (\<lambda>x. g x * h x)"
MOST_Suc_iff: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>n. P (Suc n)) = (\<forall>\<^sub>\<infinity>n. P n)"
carrier_is_submodule: shows "Module.submodule (carrier V) K V"
asms: shows "primitive_extractor (disc, sel) m = (as, ms)"
v_neq_Pinf: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "v cfg \<noteq> \<top>"
SCF_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool" shows "SCF scf A Is Pcond \<equiv> \<forall>P. Pcond A Is P \<longrightarrow> complete A (scf P)"
proper_map_snd: fixes S :: "'a set"   and K :: "'b set"   and T :: "'b set" assumes "compact S"   and "K \<subseteq> T"   and "compact K" shows "compact (S \<times> T \<inter> snd -` K)"
simps_10: fixes op :: "'opinl" shows "norm_instr (Ubx.instr.IOpInl op) = Inca.instr.IOpInl op"
rcut_def: fixes r :: "real" shows "rcut r = {x \<in> \<rat>. x < r}"
gf_G_def: fixes s :: "'s"   and t :: "'s"   and z :: "'a" shows "gf_G s t z = (\<Sum>n. p s t n *\<^sub>R z ^ n)"
ListAskip: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList xs SKIP l s = postList xs l s"
simps_2: fixes x :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "avl (MKT x l r h) = ((height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1) \<and> h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
language_coinduct: fixes R :: "'b list set \<Rightarrow> 'b list set \<Rightarrow> bool"   and K :: "'b list set"   and L :: "'b list set" assumes "R K L"   and "\<And>K L. R K L \<Longrightarrow> ([] \<in> K) = ([] \<in> L)"   and "\<And>K L x. R K L \<Longrightarrow> R (Deriv x K) (Deriv x L)" shows "K = L"
square_eq_1_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times x x = one) = (x = one \<or> x = uminus one)"