hcju/isabelleps huggingface dataset:
mkdisjoint_disj: fixes A :: "nat \<Rightarrow> 'b set"   and m :: "nat"   and n :: "nat" assumes "\<And>n. A n \<subseteq> A (Suc n)"   and "m \<noteq> n" shows "mkdisjoint A m \<inter> mkdisjoint A n = {}"
aassoc: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)"
neg_cong: fixes H :: "fmla set"   and A :: "fmla"   and A' :: "fmla" assumes "H \<turnstile> A IFF A'" shows "H \<turnstile> neg A IFF neg A'"
rely_quotient_def: fixes c :: "'a::refinement_lattice"   and i :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) // (i::'a::refinement_lattice) \<equiv> \<Sqinter> {d::'a::refinement_lattice. c \<sqsubseteq> d \<parallel> i}"
full_abstraction_wrt_preorders_impl_indRelSTEQ_to_SRel_and_TRel_4: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "trans TRel"   and "refl SRel"   and "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q" shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> (TP, \<lbrakk>SQ\<rbrakk>) \<in> TRel"
assms_2: shows "rt (\<sigma>' i) = rt (\<sigma> i)"
assms_1: shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
cont_surf: shows "continuous_on (proj ` (S - rel_interior S)) surf"
lex_pm_zero_min: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pm 0 s"
results_gpv_try_gpv: fixes \<I> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" shows "results_gpv \<I> (TRY gpv ELSE gpv') = results_gpv \<I> gpv \<union> (if colossless_gpv \<I> gpv then {} else results_gpv \<I> gpv')"
sec_ex_P1_and: fixes a1 :: "bool"   and a2 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "\<exists>A. \<bar>spmf (and_funct (a1, a2) (b1, b2) \<bind> (\<lambda>(s1, s2). S1_and' (a1, a2) (b1, b2) s1 \<bind> D)) True - spmf (and_R1 (a1, a2) (b1, b2) \<bind> D) True\<bar> = \<bar>spmf (coin_spmf \<bind> (\<lambda>\<sigma>. S1_14_OT (\<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) () \<bind> (\<lambda>view. A view \<sigma>))) True - spmf (coin_spmf \<bind> (\<lambda>\<sigma>. R1_14_OT (\<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) (b1, b2) \<bind> (\<lambda>view. A view \<sigma>))) True\<bar>"
sol: shows "solve_factored_ratfps' p fctrs = (b, zs)"
diff_divide_cancel: fixes z :: "real"   and a :: "real"   and b :: "real" assumes "z \<noteq> 0" shows "(a * z - a * (b * z)) / z = a - a * b"
lower_iso: shows "isotone \<X> \<Y> \<pi>\<^sup>*"
uconnected_trans: fixes E :: "'b uprod set" shows "trans (uconnected E)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.seq (C.map g) (C.map f)"
zero_spaceN_iff: fixes x :: "'b"   and N :: "'b quasinorm" shows "(x \<in> zero_space\<^sub>N N) = (eNorm N x = 0)"
qtable1: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<phi>) rel1"
pderiv_coeffs_code: fixes f :: "'b"   and xs :: "'b list"   and n :: "nat" shows "nth_default (0::'b) (pderiv_coeffs_code f xs) n = (f + of_nat n) * nth_default (0::'b) xs n"
path2_nontriv: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<noteq> m" shows "1 < length ns"
refinesI: fixes R :: "('b \<times> 'c) set"   and Sa :: "('b, 'd) spec"   and Sc :: "('c, 'e) spec"   and pi :: "'e \<Rightarrow> 'd" assumes "PO_refines R Sa Sc"   and "obs_consistent R pi Sa Sc" shows "refines R pi Sa Sc"
a3: shows "s @ t setinterleaves ((ta, u), insert tick (ev ` A))"
lconjby_inj: fixes x :: "'a" shows "inj (lconjby x)"
simps_3: fixes f :: "'b list \<Rightarrow> 'a"   and x :: "'b list" shows "rec_iarray f (IArray x) = f x"
assms: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y" shows "\<not> P x"
X: fixes s :: "'a"   and M :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" shows "{P (f s) |P. P \<in> range M} = {P s |P. P \<in> range (\<lambda>i s. M i (f s))}"
that: fixes L :: "real" assumes "\<forall>\<^sub>F u in at_right 0. \<forall>t'\<in>cball t0 (c * u) \<inter> T. L-lipschitz_on (cball x0 u \<inter> X) (f t')" shows "thesis"
LID\<^sub>GE: fixes l :: "'a"   and s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option" assumes "l \<in> LID\<^sub>G s" obtains r :: "'b" and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" where "s r = Some ls" and "l \<in> LID\<^sub>L ls"
viewLn_empty: fixes t :: "('c, 'd) FingerTreeStruc" shows "(t \<noteq> Empty) = (viewLn t \<noteq> None)"
R_def: shows "R = (\<lambda>i. snd (g i))"
trace_callee_eqI: fixes A :: "'h set"   and callee1 :: "'j \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'j) spmf"   and p :: "'j spmf"   and callee2 :: "'k \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'k) spmf"   and q :: "'k spmf" assumes "\<And>xs x. \<lbrakk>set xs \<subseteq> A \<times> UNIV; x \<in> A\<rbrakk> \<Longrightarrow> trace_callee callee1 p xs x = trace_callee callee2 q xs x" shows "trace_callee_eq callee1 callee2 A p q"
condition2_part3: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and ma :: "'f" assumes "echelon_form_upt_k A k"   and "k < ncols A"   and "a < b"   and "\<not> is_zero_row_upt_k a (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "\<not> is_zero_row_upt_k b (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "\<forall>m>mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1). A $ m $ mod_type_class.from_nat k = (0::'d)"   and "mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<le> ma"   and "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'d)" shows "(LEAST n. A $ a $ n \<noteq> (0::'d)) < (LEAST n. A $ b $ n \<noteq> (0::'d))"
simps_2: fixes x2 :: "'d" shows "None \<noteq> Some x2"
qring_ring: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "Ring (R /\<^sub>r I)"
update_by_pair_tc_def: shows "update_by_pair_tc \<equiv> comparator.update_by_pair comparator_of"
cover_def: shows "cover = (\<lambda>z. ball z (ee z / 3)) ` path_image p"
numneg_nb: fixes t :: "Ferrack.num" assumes "numbound0 t" shows "numbound0 (numneg t)"
mult_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
t_ins_aux_set: fixes X :: "'b t_type" shows "t_ins_aux X \<in> t_ins_set X"
unfold_congs_4: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "Literal list"   and f :: "Literal list \<Rightarrow> Literal list"   and f' :: "Literal list \<Rightarrow> Literal list" assumes "r = r'"   and "getC r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "getC_update f r = getC_update f' r'"
evalRR_one: fixes x :: "int" assumes "x \<in> carrier R" shows "eval R R id x \<one>\<^bsub>UP R\<^esub> = \<one>\<^bsub>R\<^esub>"
concat_map_split_eq_map_split_zip: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" shows "map2 f xs ys = map2 f xs ys"
irreducibleI: fixes p :: "'a" assumes "p \<noteq> (0::'a)"   and "\<not> p dvd (1::'a)"   and "\<And>a b. p = a * b \<Longrightarrow> a dvd (1::'a) \<or> b dvd (1::'a)" shows "irreducible p"
set_sel'_axioms: shows "set_sel' \<alpha> invar sel'"
Frame'_conv: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a \<Rightarrow> bool" shows "Frame P Q F = Frame' (P \<and>* \<box>) \<box> (Q \<and>* \<box>) F"
diff_closed: fixes G :: "'a set"   and g :: "'a"   and h :: "'a" assumes "AbGroup G"   and "g \<in> G"   and "h \<in> G" shows "g - h \<in> G"
ereal_less_eq_3: fixes r :: "real"   and p :: "real" shows "(ereal r \<le> ereal p) = (r \<le> p)"
weak_get_put: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
product_nodes: fixes A :: "('f, 'g, 'd) FSM"   and B :: "('f, 'g, 'e) FSM" shows "nodes (product A B) \<subseteq> nodes A \<times> nodes B"
es't: shows "redT_updTs TS (TA # TAS) t = \<lfloor>(x, w)\<rfloor>"
adopt_node_def: fixes document :: "(_) document_ptr"   and node :: "(_) node_ptr" shows "adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M document node = Heap_Error_Monad.bind (get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node)) (\<lambda>old_document. Heap_Error_Monad.bind (get_parent node) (\<lambda>parent_opt. Heap_Error_Monad.bind (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent node) (\<lambda>_. if document \<noteq> old_document then Heap_Error_Monad.bind (get_disconnected_nodes old_document) (\<lambda>old_disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes old_document (remove1 node old_disc_nodes)) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document) (\<lambda>disc_nodes. set_disconnected_nodes document (node # disc_nodes)))) else return ())))"
runit_simps_5: fixes f :: "'b" assumes "D.ide f" shows "D.cod \<r>\<^sub>D[f] = f"
fresh_bij: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and a :: "NominalSequents.var"   and x :: "'a" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
assms_4: shows "arr_mset mod_heap l (r - 1) = arr_mset new_heap l (r - 1)"
binop_axioms: shows "binop \<int>\<^sub>\<circ> vint_plus"
assms_1: shows "m\<^sub>1 $$ k = None"
exhaust_sel: fixes prod :: "'e \<times> 'f" obtains "prod = (fst prod, snd prod)"
fdomain_vsingleton: fixes a :: "V"   and b :: "V" shows "\<D>\<^sub>\<bullet> (set {[a, b]\<^sub>\<circ>}) = set {a}"
orthogonal: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j < length fs"   and "i \<noteq> j" shows "gso i \<bullet> gso j = (0::'a)"
cm_all_def: fixes c0 :: "('loc, 't) Propagate.configuration"   and \<Delta> :: "('loc \<times> 't) zmultiset" shows "cm_all c0 \<Delta> = Finite_Set.fold (\<lambda>(loc, t) c. the_cm c loc t (zcount \<Delta> (loc, t))) c0 (set_zmset \<Delta>)"
comp_in_eq_comp_out: shows "\<forall>x\<in>{0..1} - s. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) = vector_derivative \<gamma> (at x within {0..1}) \<bullet> i"
annot_def: fixes t :: "('b, 'a) FingerTreeStruc" shows "FingerTreeStruc.annot t = FingerTreeStruc.gmft t"
fimage_finsert: fixes f :: "'n \<Rightarrow> 'm"   and a :: "'n"   and B :: "'n fset" shows "f |`| finsert a B = finsert (f a) (f |`| B)"
gauss_cnj_eq_minus_self: fixes z :: "gauss_int" assumes "Re (gauss2complex z) = 0" shows "gauss_cnj z = - z"
distrib_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
Body: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> parts H" shows "X \<in> parts H"
path_split_single: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a"   and t :: "'b list"   and e :: "'b"   and sn :: "'a" assumes "s1 (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> sn" shows "\<exists>s'. s1 t\<Longrightarrow>\<^bsub>SES\<^esub> s' \<and> s' e\<longrightarrow>\<^bsub>SES\<^esub> sn"
m_assoc: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
inv_start: shows "inv_locate_b (as, lm) (n, la, ra) ires"
aff: fixes X :: "'a set" assumes "X \<in> \<G>" shows "aff_dim X \<le> d"
simps: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A_ll :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat" shows "mute mf A_ll k l (r, A) = (let p = A $$ (k, l) in if p = (0::'a) then (r, A) else case mf A_ll p of (q', p', g) \<Rightarrow> (r * q', addrow (- p') k l (multrow k q' A)))"
mod_mult_self2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times b c)) b = modulo a b"
sorted_spmat_minus_spmat: fixes A :: "(nat \<times> (nat \<times> 'a) list) list" assumes "sorted_spmat A" shows "sorted_spmat (minus_spmat A)"
nof: fixes x :: "complex" assumes "cmod (\<xi> - x) = r" shows "cmod (f x) \<le> B"
ag_p_inv: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A" shows "-\<^sub>a (x \<plusminus> y) = -\<^sub>a x \<plusminus> -\<^sub>a y"
L: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x * a \<le> b) = (x \<le> a l\<rightarrow> b)"
ennreal_le_epsilon: fixes y :: "ennreal"   and x :: "ennreal" assumes "\<And>e. \<lbrakk>y < \<top>; 0 < e\<rbrakk> \<Longrightarrow> x \<le> y + ennreal e" shows "x \<le> y"
wf_chain_append: fixes \<Gamma> :: "char list \<Rightarrow> 'b rule list option"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "wf_chain \<Gamma> (rs1 @ rs2) = (wf_chain \<Gamma> rs1 \<and> wf_chain \<Gamma> rs2)"
distinct_7: fixes ty1' :: "ty"   and ty2' :: "ty" shows "Top \<noteq> ty1' \<rightarrow> ty2'"
prems: fixes x :: "'a" assumes "x \<in> set (x__ # xs)" shows "0 \<le> f x"
is_poincare_line_0_real_is_x_axis: fixes l :: "circline"   and x :: "complex_homo" assumes "is_poincare_line l"   and "0\<^sub>h \<in> circline_set l"   and "x \<in> circline_set l \<inter> circline_set x_axis"   and "x \<noteq> 0\<^sub>h"   and "x \<noteq> \<infinity>\<^sub>h" shows "l = x_axis"
intros_7: fixes b :: "bexp"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and c :: "com"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and s\<^sub>3 :: "char list \<Rightarrow> int" assumes "bval b s\<^sub>1"   and "(c, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "(WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "(WHILE b DO c, s\<^sub>1) \<Rightarrow> s\<^sub>3"
filterlim_norm_at_top: shows "filterlim norm at_top at_infinity"
continuous_on_componentwise: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on S f = (\<forall>i\<in>Basis. continuous_on S (\<lambda>x. f x \<bullet> i))"
joinE: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set" assumes "(a, b) \<in> A\<^sup>\<down>" obtains c :: "'a" where "(a, c) \<in> A\<^sup>*" and "(b, c) \<in> A\<^sup>*"
ins_size_1: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "RBTi1 TYPE('b) bal_i"   and "RBTi1.ins bal_i n d x t = Bal t'" shows "size t' = size t + 1"
poly_condE: fixes p :: "int poly" assumes "poly_cond p" obtains "irreducible p" and "0 < lead_coeff p" and "root_free p" and "square_free p" and "p \<noteq> 0"
simps_8: fixes va :: "int"   and vb :: "int"   and vc :: "int list" shows "inf\<^sub>- (Atom (NDvd va vb vc)) = Atom (NDvd va vb vc)"
charpoly_def: fixes A :: "(('c, 'd) vec, 'd) vec" shows "Cayley_Hamilton_Compatible.charpoly A = det (mat (monom (1::'c) (Suc 0)) - mat2matofpoly A)"
signed_take_bit_int_greater_eq: fixes k :: "int"   and n :: "nat" assumes "k < - (2 ^ n)" shows "k + 2 ^ Suc n \<le> signed_take_bit n k"
rel_resumption_resultD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and r1 :: "('f, 'h, 'j) Resumption.resumption"   and r2 :: "('g, 'i, 'j) Resumption.resumption" assumes "Resumption.resumption.rel_resumption A B r1 r2"   and "Resumption.resumption.is_Done r1" shows "rel_option A (Resumption.resumption.result r1) (Resumption.resumption.result r2)"
class_leq_refl: fixes c :: "String.literal" assumes "class_ex cs c" shows "class_leq cs c c"
hd: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv"
Inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "Supi A \<le> z"
negligibleI: fixes f :: "nat \<Rightarrow> real" assumes "\<And>c. 0 < c \<Longrightarrow> f \<in> o(\<lambda>x. inverse (real x powr c))" shows "negligible f"
sorted_poly_list_relD: fixes q :: "(char list list \<times> int) list"   and q' :: "(char list multiset \<times> int) multiset"   and R :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" assumes "(q, q') \<in> sorted_poly_list_rel R" shows "q' = {#(mset a, b). (a, b) \<in># mset q#}"
SUP_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 bot :: "'b"   and top :: "'b"   and A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
DOWN_def: fixes \<AA> :: "('a \<times> 'a) set"   and t :: "('b set, 'b \<Rightarrow> 'b) concrete_category.arr" assumes "setcat \<AA>" shows "setcat.DOWN t \<equiv> the_elem (concrete_category.Dom t)"
permute_hf_def: fixes p :: "perm"   and s :: "hf" shows "p \<bullet> s = s"
simps_2: fixes a :: "Arity"   and as :: "Arity list"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "Fstack (a # as) (Alts e1 e2 # S) = (Texp e1\<cdot>a \<oplus>\<oplus> Texp e2\<cdot>a) \<otimes>\<otimes> Fstack as S"
antisymmetric_union: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "antisymmetric A less_eq"   and "antisymmetric B less_eq"   and "\<forall>a\<in>A. \<forall>b\<in>B. less_eq a b \<longrightarrow> less_eq b a \<longrightarrow> a = b" shows "antisymmetric (A \<union> B) less_eq"
set_insort_key: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = insert x (set xs)"
add_0_l: fixes a :: "'a"   and p :: "'a point" shows "local.add a Infinity p = p"
elims_2: fixes x :: "'a"   and xa :: "'a" assumes "sqrtTest x xa" obtains x :: "'a" and r :: "'a" where "xa__ = x" and "xa = r" and "(0::'a) \<le> r \<and> r * r = x"
index_minus_vec_2: fixes v\<^sub>1 :: "'a vec"   and v\<^sub>2 :: "'a vec" shows "dim_vec (v\<^sub>1 - v\<^sub>2) = dim_vec v\<^sub>2"
is_empty_def: fixes A :: "'d set" shows "Set.is_empty A = (A = {})"
RecvMarker_implies_Marker_in_set: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "ev = RecvMarker cid p q" shows "Marker \<in> set (msgs c cid)"
cong_to_1'_nat: fixes a :: "nat"   and n :: "nat" shows "[a = 1] (mod n) = (a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n))"
rank_invar_cons: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" assumes "rank_invar (t # bq)" shows "rank_invar bq"
le1: shows "tensor_mat m1 m2 \<le>\<^sub>L tensor_mat A m2"
induct: fixes P :: "'a common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'a common_primitive match_expr" assumes "\<And>uu_. P (Match (Src (IpAddrNetmask uu_ 0)))"   and "\<And>uv_. P (Match (Dst (IpAddrNetmask uv_ 0)))"   and "\<And>iface. P (Match (IIface iface))"   and "\<And>iface. P (Match (OIface iface))"   and "P (Match (Prot ProtoAny))"   and "\<And>m c. P (Match (L4_Flags (TCP_Flags m c)))"   and "\<And>ctstate. P (Match (CT_State ctstate))"   and "\<And>va. P (Match (Src (IpAddr va)))"   and "\<And>va vc. P (Match (Src (IpAddrNetmask va (Suc vc))))"   and "\<And>va vb. P (Match (Src (IpAddrRange va vb)))"   and "\<And>va. P (Match (Dst (IpAddr va)))"   and "\<And>va vc. P (Match (Dst (IpAddrNetmask va (Suc vc))))"   and "\<And>va vb. P (Match (Dst (IpAddrRange va vb)))"   and "\<And>va. P (Match (Prot (Proto va)))"   and "\<And>v. P (Match (Src_Ports v))"   and "\<And>v. P (Match (Dst_Ports v))"   and "\<And>v. P (Match (MultiportPorts v))"   and "\<And>v. P (Match (Extra v))"   and "\<And>m. P m \<Longrightarrow> P (MatchNot m)"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)"   and "P MatchAny" shows "P a0"
rebalance_middle_tree_last_order: fixes k :: "nat"   and t :: "'a btree"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list" assumes "almost_order k t"   and "\<forall>s\<in>set (subtrees (ls @ (sub, sep) # rs)). order k s"   and "rs = []"   and "length (ls @ (sub, sep) # rs) \<le> 2 * k"   and "height sub = height t" shows "almost_order k (rebalance_middle_tree k ls sub sep rs t)"
payload_Tag: fixes X :: "tag"   and P :: "bool" assumes "Tag X \<in> payload" shows "P"
mono_def: fixes f :: "'e \<Rightarrow> 'f" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
submodule_is_module: fixes N :: "'c set" assumes "LinearCombinations.submodule K N V" shows "module K (vs N)"
Abs_bd_type_F_inverse: fixes y :: "bd_type_F1 + bd_type_F2" assumes "y \<in> UNIV" shows "Rep_bd_type_F (Abs_bd_type_F y) = y"
cl_ivl_Suc: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {m..Suc n} = (if Suc n < m then 0::'b else sum g {m..n} + g (Suc n))"
mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "((a::'b) \<in> Collect (P::'b \<Rightarrow> bool)) = P a"
scaleR_cancel_right: fixes a :: "real"   and x :: "'a"   and b :: "real" shows "(a *\<^sub>R x = b *\<^sub>R x) = (a = b \<or> x = (0::'a))"