hcju/isabelleps huggingface dataset:
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs"   and "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_set_attribute_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs known_ptr get_child_nodes get_child_nodes_locs"
preserves_cod: fixes f :: "'c" assumes "C.arr f" shows "D.cod (G\<epsilon>'_\<eta>G.map f) = G (C.cod f)"
reduce_system_matrix_signs_helper_R: fixes S :: "nat list"   and signs :: "rat list list"   and p :: "real poly"   and subsets :: "(nat list \<times> nat list) list" assumes "\<forall>x. List.member S x \<longrightarrow> x < length signs"   and "p \<noteq> 0" shows "M_mat_R (take_indices signs S) subsets = take_cols_from_matrix (M_mat_R signs subsets) S"
Image_inverse_less_than: fixes k :: "nat" shows "less_than\<inverse> `` {k} = {..<k}"
IH: fixes n :: "Integer" assumes "defined n"   and "\<lbrakk>n\<rbrakk> < 0" shows "xs !! n = \<bottom>"
append_eq_cong: fixes xs :: "'d list"   and ys :: "'d list"   and xs' :: "'d list"   and ys' :: "'d list" assumes "equivclp cancel1 xs ys"   and "equivclp cancel1 xs' ys'" shows "equivclp cancel1 (xs @ xs') (ys @ ys')"
in_rel: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e pmf"   and b :: "'f pmf" shows "rel_pmf R a b = (\<exists>z. z \<in> {x. set_pmf x \<subseteq> {(x, y). R x y}} \<and> map_pmf fst z = a \<and> map_pmf snd z = b)"
maxR_def: fixes d :: "'d"   and x :: "'x"   and y :: "'x" shows "maxR d x y = (if (x, y) \<in> Pd d then y else x)"
rGamma_inverse_Gamma: fixes z :: "'a" shows "rGamma z = inverse (Gamma z)"
precedesTotalOrder: fixes a :: "'a"   and l :: "'a list"   and b :: "'a" assumes "a \<in> set l"   and "b \<in> set l" shows "a = b \<or> precedes a b l \<or> precedes b a l"
l2_inv5D: fixes Rb :: "fid_t"   and gnx :: "msg"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and x :: "l2_state" assumes "hmac \<langle>Number (Suc 0), Exp Gen (NonceF (Rb $ ny)), gnx, Messages.Agent A, Messages.Agent B\<rangle> (hmac Ni (NonceF (Rb $ nr))) \<in> parts (extr (bad x) (ik x) (chan x))"   and "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv5" shows "\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xgnx = Some gnx \<and> sklvl1.guessed_frame Ra xni = Some Ni \<and> sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr)) \<and> sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))"
vec_times_polys_to_mat: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list"   and v :: "'d vec" assumes "Keys (set ps) \<subseteq> set ts"   and "v \<in> carrier_vec (length ps)" shows "row_to_poly ts (v \<^sub>v* polys_to_mat ts ps) = sum_list (map2 (\<cdot>) (list_of_vec v) ps)"
subdegree_eq_0: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "subdegree f = 0"
elims: fixes x :: "'a tm"   and y :: "nat" assumes "time x = y" obtains v :: "'a" and n :: "nat" where "x = TM v n" and "y = n"
le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "le\<cdot>x\<cdot>y = TT"   and "le\<cdot>y\<cdot>z = TT" shows "le\<cdot>x\<cdot>z = TT"
less_diff_iff: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m"   and "k \<le> n" shows "(m - k < n - k) = (m < n)"
cSim: fixes R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>, R, S) \<in> Rel" shows "\<Psi> \<rhd> R \<leadsto>\<^sub>w<Rel'> S"
transp_less: fixes ko :: "'a key_order" shows "transp (ko.lt ko)"
eq_upto_cong: fixes i :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'b) set"   and C :: "(nat \<Rightarrow> 'b) set" assumes "\<And>j. j < i \<Longrightarrow> f j = g j"   and "B = C" shows "eq_upto B f i = eq_upto C g i"
IH_1: shows "t \<prec> t1 \<longrightarrow> (t, t1) \<in> trm_ord"
semIfSC: fixes e :: "'exp"   and c1 :: "('exp, 'id) MWLfCom"   and c2 :: "('exp, 'id) MWLfCom" assumes "IfSideCondition e c1 c2" shows "\<forall>d. e \<equiv>\<^bsub>d\<^esub> e \<or> [c1] \<approx>\<^bsub>d\<^esub> [c2]"
of_int_of_nat_eq: fixes n :: "nat" shows "ring_1_class.of_int (int n) = semiring_1_class.of_nat n"
squash_def: fixes x :: "'a" shows "squash x = (if (0::'a) < x then 1 else if x < (0::'a) then - 1 else 0)"
sel'_noneD: fixes s :: "'s"   and P :: "'x \<Rightarrow> bool"   and x :: "'x" assumes "invar s"   and "sel' s P = None"   and "x \<in> \<alpha> s" shows "\<not> P x"
simps_1: fixes S :: "'a set" shows "sstart S [] = streams S"
cyclic_group_def: fixes G :: "('b, 'c) cyclic_group_scheme" shows "cyclic_group G \<equiv> group G \<and> cyclic_group_axioms G"
sp_fundef_def: fixes F :: "'c \<Rightarrow> 'd fundef option"   and fd :: "'e fundef"   and xs :: "('f, 'g, 'c, 'op, 'opinl, 'opubx, 'h, 'i) Ubx.instr list" shows "sp_fundef F fd xs \<equiv> Subx.sp F xs (replicate (arity fd) None)"
mk_minsky_map: fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and f :: "nat \<Rightarrow> nat" assumes "mk_minsky P"   and "inj f" shows "mk_minsky (\<lambda>vs vs'. P (vs \<circ> f) (vs' \<circ> f) \<and> (\<forall>x. x \<notin> range f \<longrightarrow> vs x = vs' x))"
possExistenceImpliesNecEx: shows "\<lfloor>\<^bold>\<diamond>(\<lambda>v. \<exists>x. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v) \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v))\<rfloor>"
hoare_star: fixes p :: "'a"   and c :: "'b" assumes "\<lbrace>\<iota> p\<rbrace> c \<lbrace>\<iota> p\<rbrace>" shows "\<lbrace>\<iota> p\<rbrace> c\<^sup>\<star> \<lbrace>\<iota> p\<rbrace>"
L: shows "(\<lambda>x. x $ 1) ` S = cbox a b"
has_sgnx_poly_at_top: fixes p :: "real poly" shows "(poly p has_sgnx sgn_pos_inf p) at_top"
sel_2: fixes x1 :: "'a simple_match"   and x2 :: "simple_action" shows "action_sel (SimpleRule x1 x2) = x2"
hg: fixes v :: "'a" shows "higher_differentiable_on S (\<lambda>x. frechet_derivative g (at x) v) n"
finite_Diff: fixes A :: "'d set"   and B :: "'d set" assumes "finite A" shows "finite (A - B)"
Units_minus_one_closed: shows "\<ominus>\<^bsub>K\<^esub> \<one>\<^bsub>K\<^esub> \<in> Units K"
ord_le_eq_trans: fixes a :: "('x, nat) pp"   and b :: "('x, nat) pp"   and c :: "('x, nat) pp" assumes "ord a b"   and "b = c" shows "ord a c"
agree_plus2: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Plus t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t2)"
CLS_hcomp: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<lbrakk>\<lbrakk>f \<star> g\<rbrakk>\<rbrakk> = Maps.MkArr (src g) (trg f) (Maps.Comp \<lbrakk>f\<rbrakk> \<lbrakk>g\<rbrakk>)"
l2_inv6'I: fixes x :: "l2_state" assumes "\<And>Rb A B gnx. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progress (progress x Rb) xsk; sklvl1.guessed_frame Rb xgnx = Some gnx\<rbrakk> \<Longrightarrow> sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))" shows "x \<in> l2_inv6'"
iso_is_retraction: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.retraction f"
unitl_ex: fixes x :: "'a" shows "\<exists>e\<in>E. D e x \<and> e \<oplus> x = x"
greaterThanLessThan_eq: fixes a :: "'a"   and b :: "'a" shows "{a<..<b} = {a<..} \<inter> {..<b}"
typs: shows "list_all (\<lambda>((idx, ty), t). typ_of t = Some ty) xs"
double_SN_aux: fixes a :: "trm"   and b :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "SN a"   and "SN b"   and "\<And>x z. \<lbrakk>\<And>y. x \<mapsto> y \<Longrightarrow> SN y; \<And>y. x \<mapsto> y \<Longrightarrow> P y z; \<And>u. z \<mapsto> u \<Longrightarrow> SN u; \<And>u. z \<mapsto> u \<Longrightarrow> P x u\<rbrakk> \<Longrightarrow> P x z" shows "P a b"
ideal_sum_mult: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier R"   and "A \<noteq> {}"   and "ideal R B" shows "ideal R (sum_mult R A B)"
degree_minus: fixes p :: "'a poly" shows "degree (- p) = degree p"
hcomp_simps_3: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.hseq \<nu> \<mu>" shows "C.dom (\<nu> \<star>\<^sub>C \<mu>) = C.dom \<nu> \<star>\<^sub>C C.dom \<mu>"
length_negHorbit_list_sublist: fixes fs :: "(('f, 'g) aezfun \<Rightarrow> 'v) list"   and gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" assumes "fs \<in> set (negHorbit_list gs T as)" shows "length fs = length as"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "category B"
zero_less_iff_neq_zero: fixes n :: "'e multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
differentiable_onD: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a" assumes "f differentiable_on S"   and "x \<in> S" shows "f differentiable at x within S"
app_iso_inv: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "pre_digraph.digraph_isomorphism G hom" shows "app_iso (pre_digraph.inv_iso G hom) (app_iso hom G) = G"
open_vimage_Proj_iff: fixes T :: "'b proj_space set" shows "open (Proj -` T) = open T"
not_isVar_Op: fixes s :: "'sort"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "\<not> isVar s (Op delta inp binp)"
vsv_vrange_vsubset_vifunion_app: fixes I :: "V"   and A :: "V \<Rightarrow> V" assumes "\<D>\<^sub>\<circ> f = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I)"
left_tree_union_V: fixes s :: "'a"   and t :: "'a" assumes "s \<rightarrow> t" shows "left_tree s t \<union> left_tree t s = V"
mono: fixes f :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and a :: "'a::type"   and c :: "'a::type"   and b :: "'a::type"   and d :: "'a::type" assumes "semilattice_order (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (c::'a::type)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (b::'a::type) (d::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (b::'a::type)) (f (c::'a::type) (d::'a::type))"
CD_on_range': fixes c :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "c \<in> CD_on ds A" shows "c \<in> A \<inter> (\<Union>d\<in>ds. Field (Pd d))"
disjoint_implies_distinct: fixes x :: "i"   and y :: "i" assumes "PM"   and "\<not> O x y" shows "x \<noteq> y"
euler_trail_def: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" shows "pre_digraph.euler_trail G u p v \<equiv> pre_digraph.trail G u p v \<and> set p = arcs G \<and> set (pre_digraph.awalk_verts G u p) = verts G"
RTranClp_RTranCl_eq: fixes succs :: "'a \<Rightarrow> 'a list" shows "RTranClp succs = (\<lambda>x. in_RTranCl x succs)"
eq: shows "(THE f'. \<forall>x. ((\<lambda>x. 0::'e) has_derivative f' x) (at x)) = (\<lambda>_ _. 0::'e)"
3: fixes x :: "nat" assumes "x < dim_row C * dim_row D" shows "A $$ (i div dim_row B, x div dim_row D) * B $$ (i mod dim_row B, x mod dim_row D) * (C $$ (x div dim_row D, j div dim_col D) * D $$ (x mod dim_row D, j mod dim_col D)) = row A (i div dim_row B) $ (x div dim_row D) * col C (j div dim_col D) $ (x div dim_row D) * (row B (i mod dim_row B) $ (x mod dim_row D) * col D (j mod dim_col D) $ (x mod dim_row D))"
path_from_toE: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w" shows "H.path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
even_BIT: fixes x :: "int"   and b :: "bool" shows "even (x BIT b) = (\<not> b)"
opp_half_apartmentI: fixes x :: "'a set"   and C :: "'a set" assumes "x \<in> X"   and "C \<in> folding_g.\<C> - f \<turnstile> folding_g.\<C>"   and "order.greater_eq C x" shows "x \<in> folding_f.Y"
assms_2: shows "X \<in> obj\<^bsub>C\<^esub>"
rank_invar_hd_cons: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.rank_invar bq"   and "BinomialHeapStruc.rank t < BinomialHeapStruc.rank (hd bq)" shows "BinomialHeapStruc.rank_invar (t # bq)"
spec_ops_rem_cons: fixes x :: "'c \<times> 'd"   and xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "spec_ops (x # xs) deps" shows "spec_ops xs deps"
simps_5: fixes l1 :: "dag"   and vref1 :: "ref"   and r1 :: "dag"   and vref :: "ref"   and l2 :: "dag"   and vref2 :: "ref"   and r2 :: "dag" shows "bdt (Node (Node l1 vref1 r1) vref (Node l2 vref2 r2)) = (\<lambda>bdtvar. if bdtvar vref = 0 \<or> bdtvar vref = 1 then None else case bdt (Node l1 vref1 r1) bdtvar of None \<Rightarrow> None | Some b1 \<Rightarrow> case bdt (Node l2 vref2 r2) bdtvar of None \<Rightarrow> None | Some b2 \<Rightarrow> Some (Bdt_Node b1 (bdtvar vref) b2))"
zero_minus_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n - v = - v"
zero_left: fixes b :: "'b" shows "0 *\<^sub>v b = (0::'a)"
reducible_polyI: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "p = q * r"   and "0 < degree q"   and "0 < degree r" shows "\<not> irreducible p"
chain_subset: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "Complete_Partial_Order.chain ord A"   and "B \<subseteq> A" shows "Complete_Partial_Order.chain ord B"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat list" assumes "tuple_decode x xa = y" obtains i :: "nat" where "x = 0" and "xa = i__" and "y = [i__]"   | n :: "nat" and i :: "nat" where "x = Suc n" and "xa = i" and "y = pdec1 i # tuple_decode n (pdec2 i)"
map_cong_simp: fixes x :: "'a list"   and ya :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set ya =simp=> f z = g z" shows "map f x = map g ya"
antisym: fixes x :: "'g"   and y :: "'g" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
fps_binomial_ODE_unique: fixes a :: "'a fps"   and c :: "'a" shows "(fps_deriv a = fps_const c * a / (1 + fps_X)) = (a = fps_const (fps_nth a 0) * fps_binomial c)"
smult_minus_left: fixes a :: "'b"   and p :: "'b poly" shows "smult (- a) p = - smult a p"
simps_5: shows "cospan_shape.comp cospan_shape.BT cospan_shape.BB = cospan_shape.BT"
invpst_del: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c" assumes "invpst t" shows "invpst (del x t)"
simps: fixes a1 :: "('d, 'e) Term.term"   and a2 :: "('d, 'e) Term.term" shows "a1 \<unrhd> a2 = ((\<exists>t. a1 = t \<and> a2 = t) \<or> (\<exists>u ss t f. a1 = Fun f ss \<and> a2 = t \<and> u \<in> set ss \<and> u \<unrhd> t))"
rational_cboxes: fixes e :: "real"   and x :: "'b" assumes "0 < e" shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> cbox a b \<and> cbox a b \<subseteq> ball x e"
ordLess_ordIso_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r <o r'"   and "r' =o r''" shows "r <o r''"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_goon_left_moving_B m rs (l, r) = (\<exists>rn. l = Oc \<up> Suc m \<and> r = Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
simps_1: fixes b :: "'b set"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and s' :: "'b" shows "iter 0 b S s s' = (s \<notin> b \<and> s = s')"
f_accessible_subset: fixes q :: "'a"   and \<delta> :: "('a, 'b) ta_rule set"   and Q :: "'a set" assumes "q \<in> f_accessible \<delta> Q" shows "q \<in> Q \<union> \<delta>_states \<delta>"
plus_web_greater: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and e :: "'k \<times> 'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" shows "f e \<le> (f \<frown>\<^bsub>\<Gamma>\<^esub> g) e"
nat_into_Ord: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>" shows "Ord n"
col__refl: fixes P :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col P A B" shows "P P ReflectL A B"
at_set_avoiding2: fixes xs :: "atom set"   and c :: "'b"   and x :: "'c" assumes "finite xs"   and "finite (supp c)"   and "finite (supp x)"   and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
simps_1: fixes ys :: "'a list" shows "list_union [] ys = ys"
IH: shows "\<forall>f. (\<forall>i. i < 2 ^ (n + 1) \<and> odd i \<longrightarrow> f i = 0) \<longrightarrow> sum f {0..<2 ^ (n + 1)} = (\<Sum>k = 0..<2 ^ n. f (2 * k))"
basic4: shows "\<forall>u v w. v \<ominus> u = ~ w \<longrightarrow> u = v \<oplus> w"
v0_v1_Digraph_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c" shows "v0_v1_Digraph G v0 v1 \<equiv> Digraph G \<and> v0_v1_Digraph_axioms G v0 v1"
lemma3_2_2: fixes r :: "('b \<times> 'b) set"   and \<sigma> :: "'b list"   and \<tau> :: "'b list" shows "r|\<sigma> @ \<tau>| = r|\<sigma>| + r|\<tau>| -s r \<down>l \<sigma>"
nontrivialE: fixes g :: "'g"   and p :: "'val" assumes "\<not> trivial g p"   and "phi g p \<noteq> None"   and "p \<in> allVars g" obtains r :: "'val" and s :: "'val" where "phiArg g p r" and "phiArg g p s" and "distinct [p, r, s]"
Step: fixes trn :: "('state, 'act, 'out) trans" assumes "reachNT (srcOf trn)"   and "step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn)"   and "\<not> T trn" shows "reachNT (tgtOf trn)"
map_poly_hom_smult: fixes c :: "'b"   and p :: "'b poly" shows "map_poly to_fract (smult c p) = smult (to_fract c) (map_poly to_fract p)"
CastReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s,b\<rangle> \<rightarrow>* \<langle>Cast C e',s',b'\<rangle>"
scaleR_fun_def: fixes a :: "real"   and f :: "'c \<Rightarrow> 'd" shows "a *\<^sub>R f = (\<lambda>i. a *\<^sub>R f i)"
b: shows "(INF n. Sup (u ` {n + 1..})) = (INF n\<in>{1..}. Sup (u ` {n..}))"
assms_2: fixes r :: "char list"   and s :: "char list" shows "b r @ s = b (r @ s)"
hmsetmset_le: fixes M :: "hmultiset"   and N :: "hmultiset" shows "(hmsetmset M \<le> hmsetmset N) = (M \<le> N)"
typ_of_var: fixes idn :: "variable"   and \<tau> :: "typ" shows "typ_of (Fv idn \<tau>) = Some \<tau>"
nb\<^sub>2: shows "\<forall>op\<in>set ops. \<not> are_operators_interfering a op"
assms_1: shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"
minus_closed: fixes x :: "('c multiset \<Rightarrow> 'a) list"   and y :: "('c multiset \<Rightarrow> 'a) list" assumes "x \<in> carrier (poly_ring L)"   and "y \<in> carrier (poly_ring L)" shows "x \<ominus>\<^bsub>poly_ring L\<^esub> y \<in> carrier (poly_ring L)"
matrix_to_iarray_Gauss_Jordan_upt_k: fixes k :: "nat"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k < ncols A" shows "matrix_to_iarray (Gauss_Jordan_upt_k A k) = Gauss_Jordan_upt_k_iarrays (matrix_to_iarray A) k"
WTrtThrow: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and T' :: "ty" assumes "P,E,h \<turnstile> e : T"   and "P \<turnstile> T \<le> Class Throwable" shows "P,E,h \<turnstile> throw e : T'"
fls_one_nth: fixes n :: "int" shows "fls_nth 1 n = (if n = 0 then 1::'a else (0::'a))"
i_shrink_defs_1: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat" shows "f \<div> k \<equiv> i_aggregate f k last_message"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct G v0 v1 paths P_new sep_size"   and "ProofStepInductOptimalPaths_axioms G v0 v1 P_new sep_size" shows "ProofStepInductOptimalPaths G v0 v1 paths P_new sep_size"
vrange_vsubset_VUnion2: fixes r :: "V" shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r)"
zero_le_divide_1_iff: fixes a :: "'d" shows "((0::'d) \<le> (1::'d) / a) = ((0::'d) \<le> a)"
assms: shows "run_one_step d i (s, vs, ves, $C x23) = (s', vs', res)"
rGreaterF: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r1 \<le> r2" shows "[Greater \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> False"
cSUP_least: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and M :: "'b multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "Sup (f ` A) \<subseteq># M"
measure_try_spmf: fixes p :: "'h spmf"   and q :: "'h spmf"   and A :: "'h set" shows "Sigma_Algebra.measure (measure_spmf (TRY p ELSE q)) A = Sigma_Algebra.measure (measure_spmf p) A + pmf p None * Sigma_Algebra.measure (measure_spmf q) A"
ipd: shows "\<forall>i\<le>n. \<pi> i \<noteq> ipd k"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "i < nr"   and "j < nc" shows "Matrix.mat nr nc f $$ (i, j) = f (i, j)"
grey_fun_upd: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and fld :: "'f process_name"   and val :: "('g, 'f, 'h, 'e) Model.local_state" assumes "(r \<in> (\<Union>p. WL p (s(fld := val)))) = (r \<in> (\<Union>p. WL p s))" shows "grey r (s(fld := val)) = grey r s"
lfp_omega: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "mono f" shows "omega_fun f \<circ> g = lfp (Omega_fun f g)"
the_adj_chamber_facet: fixes X :: "'b set set"   and C :: "'b set"   and z :: "'b set" assumes "ThinChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "z \<lhd> (THE D. D \<in> X - {C} \<and> z \<lhd> D)"
HAStates_Chi_NoCycles: fixes s :: "'a"   and a :: "('a, 'b, 'c) hierauto"   and t :: "'a" assumes "s \<in> HAStates a"   and "t \<in> HAStates a"   and "s \<in> Chi a t" shows "t \<notin> Chi a s"
that: shows "\<forall>x\<in>set_spmf p. trace_eq_simcl S (f x) q"
exists_k_mod: fixes a :: "'a"   and c :: "'a" shows "\<exists>k. a mod c = a + k * c"
distinct_71: fixes x8 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Floor x8 \<noteq> CF x91 x92 x93"