hcju/isabelleps huggingface dataset:
obj_ty_def: fixes obj :: "obj" shows "obj_ty obj = (case tag obj of CInst C \<Rightarrow> Class C | Arr T k \<Rightarrow> T.[])"
big_asm: shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
ring_hom_ringI2: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S" shows "ring_hom_ring R S h"
scalar_prod_lincomb_orthogonal: fixes gs :: "'a vec list"   and k :: "nat"   and g :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "orthogonal gs"   and "set gs \<subseteq> carrier_vec n"   and "k \<le> length gs" shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) \<bullet> M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<k]) = (\<Sum>i\<leftarrow>[0..<k]. g i * h i * (gs ! i \<bullet> gs ! i))"
op: fixes inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and delta :: "'opSym" assumes "goodInp inp"   and "goodBinp binp"   and "liftAll phi inp"   and "liftAll phiAbs binp" shows "phi (Op delta inp binp)"
sin_x_le_x: fixes x :: "real" assumes "0 \<le> x" shows "sin x \<le> x"
minus_fun_def: fixes A :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c" shows "A - B = (\<lambda>x. A x - B x)"
Min_set_sorted: fixes xs :: "'a list" assumes "sorted xs"   and "xs \<noteq> []" shows "Min (set xs) = hd xs"
Cons_6: shows "\<forall>s l1 l2. f (f s l1) l2 = f s (l1 @ l2)"
is_irred_minus: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "\<not> punit.is_red F p"   and "\<not> punit.is_red F q" shows "\<not> punit.is_red F (p - q)"
Int_Diff_disjoint: fixes A :: "'h set"   and B :: "'h set" shows "A \<inter> B \<inter> (A - B) = {}"
assms_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
mbisim_simulation2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tl2 :: "'t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r2.redT s2 tl2 s2'"   and "\<not> m\<tau>move2 s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. \<tau>mRed1 s1 s1' \<and> r1.redT s1' tl1 s1'' \<and> \<not> m\<tau>move1 s1' tl1 s1'' \<and> s1'' \<approx>m s2' \<and> tl1 \<sim>T tl2"
psubst_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "goodEnv rho"   and "good X" shows "good X #[rho]"
f2: fixes x :: "atom" assumes "supp T - ba xs = supp S - ba ys"   and "x \<in> ba ys" shows "x \<sharp> f xs T"
holomorphic_on_open: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "open s" shows "(f holomorphic_on s) = (\<forall>x\<in>s. \<exists>f'. (f has_field_derivative f') (at x))"
imod_by_0: fixes a :: "enat" shows "a mod 0 = a"
Liminf_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'b" shows "Liminf F f = (SUP P\<in>{P. eventually P F}. Inf (f ` {x. P x}))"
dirvec_hd_polychain_of: fixes xs :: "'a list"   and p :: "'a" assumes "xs \<noteq> []" shows "dirvec (hd (polychain_of p xs)) = hd xs"
nb\<^sub>1\<^sub>0: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> set (effect_of op)" shows "map_of (effect_of op) v = Some a"
actPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and \<alpha> :: "act" assumes "(P, Q) \<in> Rel" shows "\<alpha>.P \<leadsto><Rel> \<alpha>.Q"
compl_inf_bot: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) x = bot"
prime_multiplicity_other: fixes p :: "'a"   and q :: "'a" assumes "prime p"   and "prime q"   and "p \<noteq> q" shows "multiplicity p q = 0"
order_prop: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.join_semilattice plus less_eq less" shows "less_eq x y = (\<exists>z. plus x z = y)"
goal_rm_SOME_m: assumes "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> \<noteq> {}" shows "c_sinvar m (delete_edges (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>) (SOME F. F \<in> c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>))"
pcompose_const: fixes a :: "'c"   and q :: "'c poly" shows "[:a:] \<circ>\<^sub>p q = [:a:]"
intro: fixes G_\<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and G_\<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and G_invar :: "'a \<Rightarrow> bool"   and G_adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and G_empty :: "'a"   and G_add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> 'b option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat" assumes "adt_wgraph G_\<alpha>w G_\<alpha>g G_invar G_adj G_empty G_add_edge"   and "Map M_empty M_update M_delete M_lookup M_invar"   and "PrioMap Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin" shows "Prim_Impl_Adts G_\<alpha>w G_\<alpha>g G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin"
eq_neg_iff_add_eq_0: fixes a :: "'b"   and b :: "'b" shows "(a = - b) = (a + b = (0::'b))"
inited_def: fixes C :: "qtname"   and g :: "oref \<Rightarrow> obj option" shows "inited C g = (g (Inr C) \<noteq> None)"
exP1_Q: shows "adjoint exM1 * Q * exM1 = Q"
kbo_encoding_is_valid_wpo: shows "wpo_with_assms weight_S weight_NS (\<lambda>f g. (pr_strict' f g, pr_weak' f g)) (\<lambda>(f, n). n = 0 \<and> least f) full_status False (\<lambda>f. False)"
"case": fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> ('b, 'c) BsSkewBinomialHeapStruc.BsSkewBinomialTree list \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'c"   and x3 :: "('b, 'c) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "(case BsSkewBinomialHeapStruc.Element x1 x2 x3 of BsSkewBinomialHeapStruc.Element x xa xb \<Rightarrow> f x xa xb) = f x1 x2 x3"
unique_c_uneq: fixes c :: "('a, 'b) MWLsCom"   and c' :: "('a, 'b) MWLsCom" assumes "set (PPc c) \<inter> set (PPc c') = {}" shows "c \<noteq> c'"
R_comm_monoid: fixes m :: "nat" assumes "0 < m" shows "Group.comm_monoid (residue_ring (p ^ m))"
cring_fieldI: fixes R :: "('c, 'd) ring_scheme" assumes "cring R"   and "Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}" shows "field R"
one_im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.one_im_sheaf S one_str f \<equiv> \<lambda>V. one_str (f \<^sup>\<inverse> S V)"
sin_periodic_pi2: fixes x :: "real" shows "sin (pi + x) = - sin x"
convert_obs_initial_eq_empty_TA_2: fixes ta :: "('c, 'd, 'e, 'f, 'g, 'h) thread_action" shows "(\<lbrace>\<rbrace> = convert_obs_initial ta) = (ta = \<lbrace>\<rbrace>)"
gvut1: fixes s :: "('d, 'c) configuration stream"   and q :: "'d"   and t :: "'c" assumes "GlobVacantUpto (shd s) q t"   and "next_performop s" shows "GlobVacantUpto (shd (stl s)) q t"
id_borel_measurable_lebesgue: shows "id \<in> borel_measurable lebesgue"
real_of_card: fixes A :: "'c set" shows "real (card A) = (\<Sum>x\<in>A. 1)"
pred_fmap_id: fixes f :: "'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "pred_fmap id (fmmap f m) = pred_fmap f m"
simps_1: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and u :: "'a"   and v :: "'a" shows "WGraph.path w u [] v = (u = v)"
S_contained_in_set_mult: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" assumes "second_isomorphism_grp H G S" shows "S \<subseteq> H <#>\<^bsub>G\<^esub> S"
class_add_wt_instr: fixes P :: "jvm_prog"   and T :: "ty"   and mxs :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and i :: "instr"   and pc :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P,T,mxs,mpc,xt \<turnstile> i,pc :: \<tau>s"   and "wf_prog wf_md P"   and "\<not> is_class P C" shows "class_add P (C, cdec),T,mxs,mpc,xt \<turnstile> i,pc :: \<tau>s"
P_Suc: fixes n :: "nat" shows "P (Suc n) = st_succ A (w !! n) (g (Suc n)) (P n)"
S4Return: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MemReturn rmCh ires p \<and> $S4 rmhist p \<and> unchanged (e p, c p, r p) \<and> HNext rmhist p \<longrightarrow> S5 rmhist p$"
set_rotate1: fixes xs :: "'b list" shows "set (rotate1 xs) = set xs"
eq_foldr: fixes xs :: "'a list" shows "sum_list xs = foldr (+) xs (0::'a)"
comm_monoid: fixes m :: "int" assumes "residues m" shows "Group.comm_monoid (residue_ring m)"
1: shows "n = l * p ^ k \<and> \<not> p dvd l"
Parallel_empty: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and X :: "'b set" assumes "Ts = []" shows "SEM (Parallel Ts) X = X"
bounded_bilinear_def: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "bounded_bilinear prod \<equiv> ((\<forall>a a' b. prod (a + a') b = prod a b + prod a' b) \<and> (\<forall>a b b'. prod a (b + b') = prod a b + prod a b')) \<and> (\<forall>r a b. prod (r *\<^sub>R a) b = r *\<^sub>R prod a b) \<and> (\<forall>a r b. prod a (r *\<^sub>R b) = r *\<^sub>R prod a b) \<and> (\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K)"
is_Hd_def: fixes tm :: "('c, 'd) tm" shows "is_Hd tm = (\<exists>x1. tm = Hd x1)"
subalgebra_Span_incl: fixes K :: "'a set"   and V :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "subalgebra K V R"   and "set Us \<subseteq> V" shows "Span K Us \<subseteq> V"
assms_2: shows "odd (multiplicity (4 * k + 3) n)"
fps_of_poly_quot_normalize_quot: fixes x :: "'b poly \<times> 'b poly" assumes "snd x \<noteq> 0" shows "fps_of_poly (fst (normalize_quot x)) / fps_of_poly (snd (normalize_quot x)) = fps_of_poly (fst x) / fps_of_poly (snd x)"
ne: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "\<Delta> \<in> \<Delta>s" shows "next \<Delta> \<subseteq> \<Delta>s \<and> unwind_to \<Delta> (next \<Delta>)"
minus_other_side: fixes a :: "'e"   and b :: "'e" assumes "a \<in> carrier W"   and "b \<in> carrier W" shows "(a \<ominus>\<^bsub>W\<^esub> b = \<zero>\<^bsub>W\<^esub>) = (a = b)"
run_one_step_break: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and e :: "e"   and s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" assumes "run_one_step d i (s, vs, ves, e) = (s', vs', RSBreak n res)" shows "e = $Br n \<or> (\<exists>n les es. e = Label n les es)"
per_union_def: fixes R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" shows "per_union R a b \<equiv> R \<union> {(x, y). (x, a) \<in> R \<and> (y, b) \<in> R} \<union> {(y, x). (x, a) \<in> R \<and> (y, b) \<in> R}"
not_stop_eval: fixes c :: "('Var, 'AExp, 'BExp) Stmt" assumes "c \<noteq> Stop"   and "\<not> has_annotated_stop c" shows "\<forall>mds mem. \<exists>c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
R_def: shows "R \<equiv> R3 \<circ> R2 \<circ> R1"
returns_heap_eq: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and h' :: "'ap"   and h'' :: "'ap" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
vbl_cases: fixes x :: "vbl" shows "x = zeroX \<or> (\<exists>y. x = nextX y)"
equal_real_alg: fixes x :: "real_alg"   and y :: "real_alg" shows "equal_class.equal (real_of x) (real_of y) = (x = y)"
arr_cod: fixes f :: "'a1" assumes "C1.arr f" shows "C1.arr (C1.cod f)"
diagvector_mult_left: fixes x :: "'e \<Rightarrow> 'c"   and A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "diagvector x ** A = (\<chi>i j. x i * A $h i $h j)"
is_projection_on_idem: fixes \<pi> :: "'a \<Rightarrow> 'a"   and M :: "'a set"   and x :: "'a" assumes "is_projection_on \<pi> M" shows "\<pi> (\<pi> x) = \<pi> x"
Cons: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and u :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<Gamma>\<turnstile> \<langle>cs,css,t\<rangle> \<Rightarrow> u" shows "\<Gamma>\<turnstile> \<langle>c # cs,css,s\<rangle> \<Rightarrow> u"
balanced_def: fixes kdt :: "'b kdt" shows "balanced kdt = (height kdt - min_height kdt \<le> 1)"
induced_map_conservative: fixes A :: "'a set" assumes "A \<in> sets M" shows "conservative (restrict_space M A) (induced_map A)"
iso_iff: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "(h \<in> Group.iso G H) = (h \<in> hom G H \<and> h ` carrier G = carrier H \<and> inj_on h (carrier G))"
closure: fixes env :: "v sem_env"   and n :: "char list"   and v0 :: "v" assumes "vs = [Closure env n exp', v0]"   and "env' = update_v (\<lambda>_. nsBind n v0 (sem_env.v env)) env" shows "thesis"
remdups_fwd_acc_distinct: fixes Acc :: "'b set"   and xs :: "'b list" shows "distinct (remdups_fwd_acc Acc xs)"
SDG_return_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a)" shows "n -p\<rightarrow>\<^bsub>ret\<^esub> n'"
e_def: shows "e = (case e_hs of (e, x) \<Rightarrow> e)"
mult_distrib_inverse: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b / b = a"
nb\<^sub>5: fixes op' :: "('a \<times> 'b) strips_operator"   and v :: "'a"   and a :: "'b" assumes "op' \<in> set ops'"   and "(v, a) \<in> set (strips_operator.precondition_of op')" shows "(\<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s) v = Some a"
par_reflexivity: fixes A :: "'p"   and B :: "'p" assumes "A \<noteq> B" shows "A B Par A B"
orderI: fixes a :: "'a word"   and b :: "'a word" assumes "a = signed.max a b" shows "b \<le>s a"
add_diff_cancel_left': fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
omit_branch_mem_diff_formula: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and q :: "('b, 'c) fm" assumes "all_is_branch p i branch xs"   and "q at i in' set branch"   and "p \<noteq> q" shows "q at i in' set (omit_branch xs branch)"
appliedDecide_def: fixes stateA :: "State"   and stateB :: "State"   and decisionVars :: "nat set" shows "appliedDecide stateA stateB decisionVars \<equiv> \<exists>l. var l \<in> decisionVars \<and> \<not> l el elements (getM stateA) \<and> \<not> opposite l el elements (getM stateA) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA @ [(l, True)] \<and> getConflictFlag stateB = getConflictFlag stateA \<and> getC stateB = getC stateA"
a_subid_aux1': fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad x ; y \<subseteq> y"
assms_1: shows "P 0\<^sub>h 1\<^sub>h \<infinity>\<^sub>h"
Pred_def: shows "Formula.Pred \<equiv> (id ---> Formula.formula.Abs_formula) Formula.Pred\<^sub>\<alpha>"
finite_maxsimp: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimp X x" shows "finite x"
us: shows "set us \<subseteq> U \<and> U = RSpan us \<and> R_lin_independent us"
Saturated_Graph_axioms: shows "Saturated_Graph nodes inPorts outPorts vertices nodeOf edges"
sel_3: fixes x21 :: "'c"   and x22 :: "'c list" shows "tl (x21 # x22) = x22"
order_divides_expdiff: fixes n :: "nat"   and a :: "nat"   and d :: "nat"   and e :: "nat" assumes "coprime n a" shows "[a ^ d = a ^ e] (mod n) = [d = e] (mod Pocklington.ord n a)"
iIN_plus: fixes x :: "nat"   and n :: "nat"   and d :: "nat"   and k :: "nat" assumes "x \<in> [n\<dots>,d]"   and "k \<le> n + d - x" shows "x + k \<in> [n\<dots>,d]"
sqrt_power2_le: fixes n :: "nat" shows "(Discrete.sqrt n)\<^sup>2 \<le> n"
a: shows "x:M \<in> ANDLEFT2 (B AND C) (\<parallel>(C)\<parallel>)"
mset_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "mset (linorder.sort_key less_eq k xs) = mset xs"
sign_simps_2: fixes af :: "'b"   and bf :: "'b" shows "af + bf = bf + af"
mIg_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and I :: "'c set" shows "mIg\<^bsub> K n P I\<^esub> = \<Sigma>\<^sub>e K (\<lambda>k. Zl_mI K P I k \<cdot>\<^sub>r\<^bsub>K\<^esub> mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m_zmax_pdsI K n P I\<^esup>) n"
abs_prts: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add_abs abs plus zero minus uminus less_eq less inf sup" shows "abs a = minus (lattice_ab_group_add.pprt zero sup a) (lattice_ab_group_add.nprt zero inf a)"
unit_disc_fix_f_def: fixes f :: "complex_homo \<Rightarrow> complex_homo" shows "unit_disc_fix_f f = (\<exists>M. unit_disc_fix M \<and> (f = moebius_pt M \<or> f = moebius_pt M \<circ> conjugate))"
serial_strips_equivalent_to_serial_sas_plus_i: fixes ops :: "('c, 'd) sas_plus_operator list"   and \<Pi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list" assumes "ops \<in> set (\<phi>\<^sub>P\<inverse> \<Pi> List_Supplement.embed \<pi>)" obtains op :: "('c \<times> 'd) strips_operator" where "op \<in> set \<pi>" and "ops = [\<phi>\<^sub>O\<inverse> \<Pi> op]"
integral_g_holomorphic: fixes b :: "real"   and c :: "real"   and A :: "complex set" assumes "0 < b" shows "(\<lambda>s. integral {b..c} (g s)) holomorphic_on A"
unclocked_single_determ: fixes env :: "v sem_env"   and s :: "'a state"   and e :: "exp0"   and r3a :: "'a state \<times> (v, v) result"   and r3b :: "'a state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s e r3a"   and "Big_Step_Unclocked_Single.evaluate env s e r3b" shows "r3a = r3b"
lnth_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'b llist" assumes "enat (n + m) < llength xs" shows "lnth (ldropn n xs) m = lnth xs (m + n)"
polynomial_pow_division: fixes p :: "'a list list"   and n :: "nat"   and m :: "nat" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "n \<le> m" shows "p [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> p [^]\<^bsub>poly_ring (poly_ring R)\<^esub> m"
monic_degree_one_root_condition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R" shows "is_root [\<one>, \<ominus> a] b = (a = b)"
list_all2_dropI: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (drop n xs) (drop n ys)"
cmod_plus_Re_le_0_iff: fixes z :: "complex" shows "(cmod z + Re z \<le> 0) = (Re z = - cmod z)"
rot_img_eq: assumes "0 < d" shows "cubeImage circle_cube = cubeImage rot_circle_cube"
DenyAllowDisj: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and p :: "protocol \<times> int" assumes "dom (Cp (AllowPortFromTo a b p)) \<noteq> {}" shows "dom (Cp (DenyAllFromTo a b)) \<inter> dom (Cp (AllowPortFromTo a b p)) \<noteq> {}"
homology_exactness_axiom_2: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "exact_seq ([homology_group (p - 1) X, homology_group (p - 1) (subtopology X S), relative_homology_group p X S], [hom_induced (p - 1) (subtopology X S) {} X {} id, hom_boundary p X S])"
prime_factorization_eqI: fixes P :: "'a multiset"   and n :: "'a" assumes "\<And>p. p \<in># P \<Longrightarrow> prime p"   and "normalize (\<Prod>\<^sub># P) = normalize n" shows "prime_factorization n = P"