hcju/isabelleps huggingface dataset:
out_arcs_in_arcs: fixes x :: "'a"   and G :: "('b, 'a) pre_digraph"   and u :: "'b" assumes "x \<in> out_arcs G u" shows "x \<in> arcs G"
premise_free_inf_always_from: fixes \<iota> :: "'f inference"   and N :: "'f set" assumes "\<iota> \<in> Inf_F"   and "prems_of \<iota> = []" shows "\<iota> \<in> no_labels.Inf_from N"
nonneg: fixes xy :: "'a \<times> 'b" shows "0 \<le> (case xy of (x, y) \<Rightarrow> enn2real (h x y))"
ptensor_mat_id: shows "c.ptensor_mat (1\<^sub>m c.d1) (1\<^sub>m c.d2) = 1\<^sub>m c.d0"
meeting_paths_produce_cycle: fixes v :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "path (v # xs)"   and "xs \<noteq> []"   and "path (v # ys)"   and "ys \<noteq> []"   and "last xs = last ys"   and "hd xs \<noteq> hd ys" shows "\<exists>zs. cycle zs \<and> hd zs = v"
ab_diff_conv_add_uminus: fixes a :: "'m"   and b :: "'m" assumes "a \<in> U\<^sub>M"   and "b \<in> U\<^sub>M" shows "a -\<^sub>M b = a +\<^sub>M -\<^sub>M b"
simps: fixes p :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list"   and a :: "'a"   and ts :: "'a rose_tree list" shows "zipper_children (p, Tree (a, ts)) = map (\<lambda>(l, t, r). ((a, l, r) # p, t)) (splits ts)"
Compl_iff: fixes c :: "'d"   and A :: "'d set" shows "(c \<in> - A) = (c \<notin> A)"
vimageI2: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and A :: "'c set" assumes "f a \<in> A" shows "a \<in> f -` A"
nat_pow_distrib: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y) [^]\<^bsub>G\<^esub> n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> y [^]\<^bsub>G\<^esub> n"
disj_1: shows "h\<^sub>0 \<bottom> h\<^sub>1"
test_sup_neg_2: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "(x \<squnion> y) \<sqinter> (! x \<sqinter> ! y) = bot"
add_strict_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b" shows "a + c < b + c"
cprod_def: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "r1 *c r2 = |Field r1 \<times> Field r2|"
continuous_on_closed: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on S f = (\<forall>T. closedin (top_of_set (f ` S)) T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
random_member_singleton: fixes a :: "'a" shows "random_member {|a|} = Some a"
wf_mult1: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (mult1 r)"
subst_cls_def: fixes AA :: "'a literal multiset"   and \<sigma> :: "'s" shows "AA \<cdot> \<sigma> = {#A \<cdot>l \<sigma>. A \<in># AA#}"
eq_snd_iff: fixes b :: "'e"   and p :: "'f \<times> 'e" shows "(b = snd p) = (\<exists>a. p = (a, b))"
get_field_def: fixes s :: "state"   and a :: "loc"   and f :: "fname" shows "get_field s a f \<equiv> the (snd (get_obj s a) f)"
is_preserved_by_base_iso: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b"   and \<phi> :: "'b"   and r' :: "'b" assumes "tabulation V H \<a> \<i> src trg r \<rho> f g"   and "partial_magma.in_hom V \<phi> r r'"   and "category.iso V \<phi>" shows "tabulation V H \<a> \<i> src trg r' (V (H \<phi> f) \<rho>) f g"
\<T>_equals_Yo\<psi>: fixes e :: "'s" assumes "e \<in> Hom.set (a, a')" shows "\<T> e = Y (\<psi> (a, a') e)"
hull_mono: fixes s :: "'b set"   and t :: "'b set"   and S :: "'b set \<Rightarrow> bool" assumes "s \<subseteq> t" shows "S hull s \<subseteq> S hull t"
fd: shows "P \<turnstile> C\<^sub>0 sees F,b:T in D"
sup_lexord_def: fixes A :: "'b"   and B :: "'b"   and k :: "'b \<Rightarrow> 'c"   and s :: "'b"   and c :: "'b" shows "sup_lexord A B k s c = (if k A = k B then c else if \<not> k A \<le> k B \<and> \<not> k B \<le> k A then s else if k B \<le> k A then A else B)"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'b list" shows "List.enumerate n xs = zip [n..<n + length xs] xs"
hoare_triple_success: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P" shows "success c h"
connected_disjoint_Union_open_subset: fixes A :: "'a set set"   and B :: "'a set set" assumes "disjoint A"   and "disjoint B"   and "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"   and "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"   and "\<Union> A = \<Union> B" shows "A \<subseteq> B"
emeasure_Int_set_pmf: fixes p :: "'b pmf"   and A :: "'b set" shows "emeasure (measure_pmf p) (A \<inter> set_pmf p) = emeasure (measure_pmf p) A"
top_Assertion_def: shows "\<top> = abs_wpt (1::'c)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "local.a_known_ptr ptr = (CharacterDataClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some document_ptr \<Rightarrow> is_document_ptr document_ptr))"
box: shows "box (0, 0) (1, 1) = {0<..<1} \<times> {0<..<1}"
retract_of_refl: fixes S :: "'a set" shows "S retract_of S"
that: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "[M]\<^bsub>v,n\<^esub> = R"   and "\<forall>i\<le>n. \<forall>j\<le>n. M i 0 = \<infinity> \<and> 0 < j \<and> i \<noteq> j \<longrightarrow> M i j = \<infinity> \<and> M j i = \<infinity>"   and "\<forall>i\<le>n. M i i = Le 0"   and "\<forall>i\<le>n. \<forall>j\<le>n. 0 < i \<and> 0 < j \<and> M i 0 \<noteq> \<infinity> \<and> M j 0 \<noteq> \<infinity> \<longrightarrow> (\<exists>d. - int (k (v' j)) \<le> d \<and> d \<le> int (k (v' i)) \<and> M i j = Le (real_of_int d) \<and> M j i = Le (real_of_int (- d)) \<or> - int (k (v' j)) \<le> d - 1 \<and> d \<le> int (k (v' i)) \<and> M i j = Lt (real_of_int d) \<and> M j i = Lt (real_of_int (- d + 1)))"   and "\<forall>i\<le>n. 0 < i \<and> M i 0 \<noteq> \<infinity> \<longrightarrow> (\<exists>d\<le>int (k (v' i)). 0 \<le> d \<and> (M i 0 = Le (real_of_int d) \<and> M 0 i = Le (real_of_int (- d)) \<or> M i 0 = Lt (real_of_int d) \<and> M 0 i = Lt (real_of_int (- d + 1))))"   and "\<forall>i\<le>n. 0 < i \<longrightarrow> (\<exists>d\<ge>- int (k (v' i)). d \<le> 0 \<and> (M 0 i = Le (real_of_int d) \<or> M 0 i = Lt (real_of_int d)))"   and "\<forall>i j. M i j \<noteq> \<infinity> \<longrightarrow> get_const (M i j) \<in> \<int>"   and "\<forall>i\<le>n. \<forall>j\<le>n. M i j \<noteq> \<infinity> \<and> 0 < i \<and> 0 < j \<longrightarrow> (\<exists>d. (M i j = Le (real_of_int d) \<or> M i j = Lt (real_of_int d)) \<and> - int (k (v' j)) \<le> d \<and> d \<le> int (k (v' i)))" shows "thesis"
strict_order_iff: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b <s a) = (a = signed.max a b \<and> a \<noteq> b)"
divide_uexpr_def: fixes u :: "('e, 'f) uexpr"   and v :: "('e, 'f) uexpr" shows "u div v = bop (div) u v"
const_OclIncluding: fixes x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and S :: "'d state \<times> 'd state \<Rightarrow> Set('e)" assumes "const x"   and "const S" shows "const S->including\<^sub>S\<^sub>e\<^sub>t(x)"
fun_relD2: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and Ra :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set"   and x' :: "'c"   and r' :: "'d" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f' x' = r'" shows "\<forall>x. (x, x') \<in> Ra \<longrightarrow> (f x, r') \<in> Rr"
wf_theory_imp_wf_osig: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "wf_theory \<Theta>" shows "wf_osig (osig (sig \<Theta>))"
vec_of_poly_as_vec_of_list: fixes f :: "'b poly" assumes "f \<noteq> 0" shows "vec_of_poly f = vec_of_list (rev (coeffs f))"
polynomial_function_add: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "polynomial_function f"   and "polynomial_function g" shows "polynomial_function (\<lambda>x. f x + g x)"
t_def: shows "t \<equiv> \<langle>s, a, \<langle>r, b, u\<rangle>\<rangle>"
assms_2: fixes y :: "variable \<times> String.literal set" assumes "y \<in> tvsT_Set (snd ` set (x # xs))" shows "y \<notin> fst ` set (x # xs)"
fset_of_list_subset: fixes xs :: "'c list"   and ys :: "'c list" assumes "set xs \<subseteq> set ys" shows "fset_of_list xs |\<subseteq>| fset_of_list ys"
frequently_def: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "frequently P F = (\<not> (\<forall>\<^sub>F x in F. \<not> P x))"
abs_eq: shows "0 = Abs_ell2 (\<lambda>_. 0)"
vimage_image_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "f -` f ` A = {y. \<exists>x\<in>A. f x = f y}"
norm_pres_extensionE: fixes g :: "('c \<times> real) set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "g \<in> norm_pres_extensions E p F f" obtains H :: "'c set" and h :: "'c \<Rightarrow> real" where "g = 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"
setgeI: fixes S :: "'a set"   and x :: "'a" assumes "\<forall>y\<in>S. x \<le> y" shows "x <=* S"
fps_lr_inverse_starting0_1: fixes f :: "'c fps" shows "fps_left_inverse f (0::'c) = 0"
finite_vimage_real_of_nat_greaterThanAtMost: fixes y :: "real"   and x :: "real" shows "finite (real -` {y<..x})"
gc_invar_infinity: fixes Ns :: "('f \<times> 'l) set llist" assumes "\<not> lnull Ns"   and "\<forall>i. enat i < llength Ns \<longrightarrow> gc_invar Ns (enat i)" shows "gc_invar Ns \<infinity>"
inducts: fixes x :: "trm"   and P :: "trm \<Rightarrow> bool" assumes "SNa x"   and "\<And>M. \<lbrakk>\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M'; \<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> P M'\<rbrakk> \<Longrightarrow> P M" shows "P x"
c1_3: shows "C1 \<in> carrier_mat nr2 n1"
seqseq_holds: fixes n :: "nat" shows "P n (seqseq (Suc n))"
absorb1: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" shows "a \<inter># b = a"
n2l_candidate: fixes zs :: "'a nel list" shows "map n2l (candidate (\<lambda>x y. l2n (f (n2l x) (n2l y))) zs) = candidate f (map n2l zs)"
subalgebra_indep_var_def: fixes X :: "'a \<Rightarrow> real"   and N :: "'a measure" shows "subalgebra_indep_var X N = (random_variable borel X \<and> subalgebra M N \<and> indep_set (sigma_sets (space M) {X -` A \<inter> space M |A. A \<in> sets borel}) (sets N))"
safe_sqrt_def: fixes x :: "real" shows "safe_sqrt x = (if 0 \<le> x then sqrt x else 0)"
eq_zero_set_bl: fixes w :: "'b word" shows "(w = 0) = (True \<notin> set (to_bl w))"
lm070: fixes f :: "('c \<times> 'd) set"   and X :: "'c set" assumes "runiq (f || X)" shows "sum (toFunction (f || X)) (X \<inter> Domain f) = sum (toFunction f) (X \<inter> Domain f)"
Idomain_def: fixes R :: "('e, 'f) Ring_scheme" shows "Idomain R \<equiv> Ring R \<and> Idomain_axioms R"
B_trusts_YM4: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list"   and NA :: "nat" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, Crypt K (Nonce NB)\<rbrace> \<in> set evs"   and "Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> \<in> set evs"   and "\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"
induct: fixes P :: "('e, 'f) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('e, 'f) FingerTreeStruc" assumes "P Empty"   and "\<And>a. P (Single a)"   and "\<And>uu_ pr m a b. P (Deep uu_ pr m (Two a b))"   and "\<And>uv_ pr m a b c. P (Deep uv_ pr m (Three a b c))"   and "\<And>uw_ pr m a b c d. P (Deep uw_ pr m (Four a b c d))"   and "\<And>ux_ pr m a. P m \<Longrightarrow> P (Deep ux_ pr m (One a))" shows "P a0"
rtranclp_conversep: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*\<inverse>\<inverse>"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b tree list" shows "map_tree f (Fork x21 x22) = Fork (f x21) (map (map_tree f) x22)"
UNIV_not_singleton: fixes x :: "'b" shows "UNIV \<noteq> {x}"
suffix_simps_3: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "\<sigma> |\<^sub>s m |\<^sub>s n = \<sigma> |\<^sub>s m + n"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>nat. P (TVar nat)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"
fn_not_snd: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm"   and P :: "'b trm" shows "Fn x T A \<noteq> Snd P"
hyps_1: shows "x \<rightarrow>\<^bsub>G\<^esub> y"
create_gba_from_nodes_def: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set" shows "create_gba_from_nodes \<phi> qs \<equiv> \<lparr>g_V = qs, g_E = {(q, q'). q \<in> qs \<and> q' \<in> qs \<and> name q \<in> incoming q'}, g_V0 = {q \<in> qs. expand_init \<in> incoming q}, gbg_F = {{q \<in> qs. \<mu> U\<^sub>r \<eta> \<in> old q \<longrightarrow> \<eta> \<in> old q} |\<mu> \<eta>. \<mu> U\<^sub>r \<eta> \<in> subfrmlsr \<phi>}, gba_L = \<lambda>q l. q \<in> qs \<and> {p. prop\<^sub>r(p) \<in> old q} \<subseteq> l \<and> {p. nprop\<^sub>r(p) \<in> old q} \<inter> l = {}\<rparr>"
pivot_vars': fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "let s' = pivot x\<^sub>i x\<^sub>j s in rvars (\<T> s') = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i} \<and> lvars (\<T> s') = lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j}"
in_cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> \<Theta>[F](h)) = (g \<in> \<Theta>[F](h))"
atLeastatMost_subset_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a..b} \<subseteq> {c..d}) = (\<not> a \<le> b \<or> c \<le> a \<and> b \<le> d)"
add_block_def: fixes b :: "'a set" shows "derived_incidence_sys.add_block b \<equiv> \<B>\<^sup>D + {#b#}"
wqo_on_UNIV: shows "wqo_on (\<lambda>_ _. True) UNIV"
tensor_mat_dim_row: fixes dims :: "nat list"   and vars :: "nat set"   and m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_row (tensor_mat dims vars m1 m2) = state_sig.d dims"
region_cover: fixes v :: "'c \<Rightarrow> real" assumes "\<forall>x\<in>X. 0 \<le> v x" shows "\<exists>R. R \<in> \<R>\<^sub>\<beta> \<and> v \<in> R"
inf_sup_absorb: fixes x :: "'b"   and y :: "'b" shows "inf x (sup x y) = x"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_option R \<lfloor>x2\<rfloor> \<lfloor>y2\<rfloor> = R x2 y2"
lunit_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "trg (bicategory.lunit V H \<a> \<i> src trg f) = trg f"
holomorphic_on_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and s :: "complex set" assumes "\<And>i. i \<in> I \<Longrightarrow> f i holomorphic_on s" shows "(\<lambda>x. \<Sum>i\<in>I. f i x) holomorphic_on s"
iteration_sim: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z \<cdot> y \<le> x \<cdot> z" shows "z \<cdot> y\<^sup>\<infinity> \<le> x\<^sup>\<infinity> \<cdot> z"
comp_in_homE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VV.in_hom (VV.comp g f) a c" obtains b :: "'a \<times> 'a" where "VV.in_hom f a b" and "VV.in_hom g b c"
add_comm: fixes a :: "int"   and b :: "int"   and p\<^sub>1 :: "int point"   and p\<^sub>2 :: "int point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "local.add a p\<^sub>1 p\<^sub>2 = local.add a p\<^sub>2 p\<^sub>1"
fwi_cyc_free_subs: fixes n :: "nat"   and I :: "nat set"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> I \<union> {k} \<longrightarrow> (0::'a) \<le> len m i i xs"   and "canonical_subs n I m"   and "I \<subseteq> {0..n}"   and "k \<le> n" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> I \<union> {k} \<longrightarrow> (0::'a) \<le> len (fwi m n k n n) i i xs"
drop_i_f_join_eq1: fixes n :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "n < card I" shows "f \<Join>\<^sub> I \<up> n = f \<Join>\<^sub> (I \<down>\<ge> (I \<rightarrow> n))"
quote_all_Mem_imp_All2: fixes p :: "perm"   and Vs :: "name set"   and i :: "name"   and i' :: "name"   and \<alpha> :: "fm"   and Fi :: "name \<Rightarrow> tm"   and j :: "name"   and pi :: "perm" assumes "quote_perm p Vs"   and "insert (QuoteP (Var i) (Var i')) (quote_all p Vs) \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)"   and "supp (All2 i (Var j) \<alpha>) \<subseteq> atom ` Vs"   and "atom j \<sharp> (i, \<alpha>)"   and "atom i \<sharp> p"   and "atom i' \<sharp> (i, p, \<alpha>)"   and "pi = (atom i \<rightleftharpoons> atom i') + p"   and "Fi = make_F (insert i Vs) pi" shows "insert (All2 i (Var j) \<alpha>) (quote_all p Vs) \<turnstile> PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs (make_F Vs p))"
le_comp_left_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and u :: "'b"   and v :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq x y"   and "less_eq u v" shows "less_eq (times x u) (times y v)"
minus_add: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> (x \<oplus>\<^bsub>K [X]\<^esub> y) = \<ominus>\<^bsub>K [X]\<^esub> x \<oplus>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> y"
separation_by_Un_closed: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "S \<inter> T = {}"   and "connected (- S)"   and "connected (- T)" shows "connected (- (S \<union> T))"
le_less_trans: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
\<phi>_simps_4: shows "local.dom t\<^sub>0u\<^sub>1.\<phi> = t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1"
ran_def: fixes m :: "'b \<Rightarrow> 'a option" shows "ran m = {b. \<exists>a. m a = \<lfloor>b\<rfloor>}"
inv_convcomp: fixes c :: "'a \<Rightarrow>\<^sub>U 'b" shows "inv\<^sub>C c \<circ>\<^sub>C c = id\<^sub>C"
self_domain_iff_ide: fixes a :: "'a arrow_of_spans_data" shows "(a \<in> V.domains a) = V.ide a"
lessE: fixes a :: "'a"   and b :: "'a" assumes "a < b" obtains c :: "'a" where "b = a + c" and "c \<noteq> (0::'a)"
fbox_zero: fixes x :: "'a" shows " |(zero_class.zero::'a)] x = (1::'a)"
omega_sum_1_less: fixes ms :: "nat list"   and ns :: "nat list" assumes "(ms, ns) \<in> lenlex less_than" shows "omega_sum (1 # ms) < omega_sum (1 # ns)"
closed_sequential_limits: fixes S :: "'a set" shows "closed S = (\<forall>x l. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l \<longrightarrow> l \<in> S)"
division_ring_inverse_diff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "minus (inverse a) (inverse b) = times (times (inverse a) (minus b a)) (inverse b)"
assms_2: shows "steps0 (1, l, r) B stp = (0, l', r')"
simps_1: fixes x :: "'a" shows "cmp2.lookup_pair [] x = (0::'c)"
If_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (IF b THEN c1 ELSE c2, s) \<Rightarrow> t = (bval b s \<and> \<pi>: (c1, s) \<Rightarrow> t \<or> \<not> bval b s \<and> \<pi>: (c2, s) \<Rightarrow> t)"
spmf_eq_0_set_spmf: fixes p :: "'h spmf"   and x :: "'h" shows "(spmf p x = 0) = (x \<notin> set_spmf p)"
pnop_axioms: shows "pnop A (2\<^sub>\<nat>) f"
proj_in_iff: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c" assumes "equiv A r" shows "(Equiv_Relations.proj r x \<in> A // r) = (x \<in> A)"
append_take_drop_id: fixes n :: "nat"   and xs :: "'g list" shows "take n xs @ drop n xs = xs"
countable_disjoint_open_subsets: fixes \<F> :: "'b set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S"   and "disjoint \<F>" shows "countable \<F>"
coprime_normalize_quot: fixes x :: "'a \<times> 'a" shows "coprime (fst (normalize_quot x)) (snd (normalize_quot x))"
assms_2: shows "s \<in> reachable (closed (pnet np n)) TT"
assms: shows "\<not> (\<Gamma> = \<Empt> \<and> \<Delta> = \<Empt>)"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Deliver x2 = Deliver y2) = (x2 = y2)"
dickson: fixes P1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A1 :: "'a set"   and P2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A2 :: "'c set" assumes "wqo_on P1 A1"   and "wqo_on P2 A2" shows "wqo_on (prod_le P1 P2) (A1 \<times> A2)"
extract_equi0: fixes sel :: "'a simple_match \<Rightarrow> 'a word \<times> nat"   and rs :: "'a simple_rule list" shows "set (map wordinterval_to_set (extract_IPSets_generic0 sel rs)) = (\<lambda>(base, len). ipset_from_cidr base len) ` sel ` match_sel ` set rs"
GrpE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "BNF_Def.Grp A f x y" obtains "f x = y" and "x \<in> A"
scalar_product_trim: fixes ns :: "nat list"   and xs :: "int list" shows "scalar_product ns xs = scalar_product (take (length xs) ns) xs"
right_neutral: fixes a :: "'a" shows "a \<squnion> bot = a"
Units_nat_pow_Units: fixes a :: "'a"   and c :: "nat" assumes "a \<in> Units G" shows "a [^] c \<in> Units G"
right_inverse: fixes a :: "'ag" assumes "a \<in> U" shows "a +\<^sub>o\<^sub>w -\<^sub>o\<^sub>w a = 0\<^sub>o\<^sub>w"
map_of_ls_update: fixes k :: "'a"   and v :: "'b"   and l :: "('a \<times> 'b) list" shows "map_of (fst (ls_update k v l)) = map_of l(k \<mapsto> v)"
h: fixes x :: "'b" assumes "x \<in> A" shows "h (f x) = x"
eqpoll_free_Abelian_group_infinite: fixes A :: "'b set" assumes "infinite A" shows "carrier (free_Abelian_group A) \<approx> A"
aff_dim_nonempty_interior: fixes S :: "'b set" assumes "interior S \<noteq> {}" shows "aff_dim S = int DIM('b)"