hcju/isabelleps huggingface dataset:
elims_3: fixes x :: "atomUni"   and xa :: "real" assumes "\<not> aEvalUni x xa" obtains a :: "real" and b :: "real" and c :: "real" and x :: "real" where "xa__ = EqUni (a__, b__, c__)" and "xa = xb__" and "a__ * xb__\<^sup>2 + b__ * xb__ + c__ \<noteq> 0"   | a :: "real" and b :: "real" and c :: "real" and x :: "real" where "xa__ = LessUni (aa__, ba__, ca__)" and "xa = xc__" and "\<not> aa__ * xc__\<^sup>2 + ba__ * xc__ + ca__ < 0"   | a :: "real" and b :: "real" and c :: "real" and x :: "real" where "xa__ = LeqUni (ab__, bb__, cb__)" and "xa = xd__" and "\<not> ab__ * xd__\<^sup>2 + bb__ * xd__ + cb__ \<le> 0"   | a :: "real" and b :: "real" and c :: "real" and x :: "real" where "xa__ = NeqUni (a, b, c)" and "xa = x" and "\<not> a * x\<^sup>2 + b * x + c \<noteq> 0"
relation_of_Chaos: shows "action.relation_of Chaos = R (false \<turnstile> true)"
quotient_empty: fixes r :: "('a \<times> 'a) set" shows "{} // r = {}"
in2: shows "rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j \<in> gs.span (gs1.gso ` {0..<i})"
prems_1: fixes k :: "nat" assumes "sdrop k w = sconst (w !! k)" shows "thesis"
ras_get_def: fixes s :: "'b array \<times> nat"   and i :: "nat" shows "ras_get s i \<equiv> let (a, n) = s in array_get a (n - (i + 1))"
S1: shows "A = {c_pair x (univ_for_pr x) |x. x \<in> UNIV}"
IH: shows "local.neville_aitken_merge ((xsi, xsj, p_sisj) # rest) = map (\<lambda>i. (x i, x (i + Suc j), local.neville_aitken_main i (i + Suc j))) [Suc l..<Suc l + kk]"
is_glb_def: fixes S :: "'a set"   and x :: "'a" shows "S >>| x = (S >| x \<and> (\<forall>u. S >| u \<longrightarrow> u \<sqsubseteq> x))"
simps_3: fixes i :: "nat" shows "(_Exit_) \<oplus> i = (_Exit_)"
subspace_scale: fixes S :: "('c \<Rightarrow>\<^sub>0 'b) set"   and x :: "'c \<Rightarrow>\<^sub>0 'b"   and c :: "'b" assumes "phull.subspace S"   and "x \<in> S" shows "c \<cdot> x \<in> S"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "inclusion_of_generators f = null"
nat_0_less_mult_iff: fixes m :: "nat"   and n :: "nat" shows "(0 < m * n) = (0 < m \<and> 0 < n)"
cont2contlubE: fixes f :: "'c \<Rightarrow> 'd"   and Y :: "nat \<Rightarrow> 'c" assumes "cont f"   and "chain Y" shows "f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
left_impl_ded: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * a r\<rightarrow> c = a r\<rightarrow> b r\<rightarrow> c"
elims: fixes x :: "D"   and y :: "bool option" assumes "fromC x = y" obtains xa :: "bool" where "x = C xa" and "y = Some xa"   | v :: "int" where "x = A v__" and "y = None"   | v :: "char list" where "x = B v" and "y = None"
iterate_sel_no_map_map_rev_linord_correct_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "(('a \<times> 'b) option \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) option \<Rightarrow> ('a \<times> 'b) option) \<Rightarrow> ('a \<times> 'b) option \<Rightarrow> ('a \<times> 'b) option"   and m :: "'a \<Rightarrow> 'b option"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and k :: "'a"   and v :: "'b" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_rev_linord less_eq it (map_to_set m)"   and "iterate_sel_no_map it P = Some (k, v)" shows "m k = Some v \<and> P (k, v) \<and> (\<forall>k' v'. m k' = Some v' \<and> P (k', v') \<longrightarrow> less_eq k' k)"
strict_trans1: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<preceq>\<^sub>t b"   and "b \<prec>\<^sub>t c" shows "a \<prec>\<^sub>t c"
power2_less_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (power.power one times a 2) zero"
update_conv': fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "map_of (AList.update k v al) = map_of al(k \<mapsto> v)"
hoare_triple_while: shows "\<turnstile>\<^sub>p {adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1} While_P vars2 M0 M1 D {P'}"
l_null: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<zero>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
simps_1: fixes v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat"   and sub :: "('c \<times> nat) \<times> 'c \<times> nat" shows "uses_sub v1 [] v2 sub = (v1 \<noteq> v2 \<and> sub = (v1, v2))"
simps: fixes uu :: "(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)"   and axs :: "Core.term set" shows "axioms (uu, axs) = axs"
start: shows "fst (prod_dfs A B (0, 0)) ! 0 ! 0 = Some 0"
minus_add_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) (plus a b) = b"
inj_on_set: shows "inj_on local.set (Collect ide)"
step_awalk_to_apath: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and q :: "'d list"   and r :: "'d list"   and s :: "'d list" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "wf_digraph.awalk_cyc_decomp G p = (q, r, s)"   and "\<not> distinct (pre_digraph.awalk_verts G u p)" shows "wf_digraph.awalk_to_apath G p = wf_digraph.awalk_to_apath G (q @ s)"
size_gen_2: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "size_dag (Node x21 x22 x23) = size_dag x21 + size_dag x23 + Suc 0"
some_src_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_src (cod \<mu>) = some_src \<mu>"
bigthetaI_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "f \<in> \<Theta>[F](g)"
cases: fixes a1 :: "('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 a2 :: "'t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and a3 :: "('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 "m\<tau>move2 a1 a2 a3" obtains s :: "('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 t :: "'t" and x :: "'x2" and s' :: "('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 x' :: "'x2" and ln' :: "'l \<Rightarrow>f nat" and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" where "a1 = s" and "a2 = (t, ta)" and "a3 = s'" and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "thr s' t = \<lfloor>(x', ln')\<rfloor>" and "\<tau>move2 (x, shr s) ta (x', shr s')"
p0: shows "(\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<noteq> 0"
trans_congruent_rel: fixes b :: "'a" shows "trans (congruent_rel b)"
rec_by_index_is_pr: shows "rec_by_index \<in> PrimRec2"
crdt_ops_distinct_fst: fixes xs :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops xs deps" shows "distinct (map fst xs)"
cases: fixes a1 :: "((nat \<times> nat) \<times> transition) fset"   and a2 :: "nat"   and a3 :: "nat \<Rightarrow>f value option"   and a4 :: "(String.literal \<times> value list) list" assumes "recognises_execution a1 a2 a3 a4" obtains e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" where "a1 = e__" and "a2 = s__" and "a3 = r__" and "a4 = []"   | e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and t :: "(String.literal \<times> value list) list" where "a1 = e" and "a2 = s" and "a3 = r" and "a4 = (l, i) # t" and "\<exists>(s', T)|\<in>|possible_steps e s r l i. recognises_execution e s' (evaluate_updates T i r) t"
diff_left_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and b :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq b a" shows "less_eq (minus c a) (minus c b)"
mset_ran_def: fixes a :: "int \<Rightarrow> 'b"   and r :: "int set" shows "mset_ran a r \<equiv> image_mset a (mset_set r)"
simps_2: fixes x :: "'a"   and ys :: "'a list" shows "sorted (x # ys) = (Ball (set ys) ((\<le>) x) \<and> sorted ys)"
ffUnion_alt_def: fixes x :: "'a"   and A :: "'a fset fset" shows "(x |\<in>| ffUnion A) = fBex A ((|\<in>|) x)"
Q_alt: shows "Q = set (zip A_pos (take (length A_pos) L))"
R_lin_independent_imp_same_scalars: fixes rs :: "'r list"   and ss :: "'r list"   and ms :: "'m list" assumes "length rs = length ss"   and "length rs \<le> length ms"   and "set rs \<subseteq> R"   and "set ss \<subseteq> R"   and "set ms \<subseteq> M"   and "R_lin_independent ms"   and "rs \<bullet>\<cdot> ms = ss \<bullet>\<cdot> ms" shows "rs = ss"
nres_order_simps_3: fixes M :: "'b nres" shows "M \<le> FAIL"
Units_inv_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
ltakeWhile_all: fixes xs :: "'b llist"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>lset xs. P x" shows "ltakeWhile P xs = xs"
antisym_conv3: fixes y :: "'a word"   and x :: "'a word" assumes "\<not> y <s x" shows "(\<not> x <s y) = (x = y)"
less_eq_list_induct: fixes x1 :: "'b list"   and x2 :: "'b list"   and Pa :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "x1 \<preceq> x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>x = y; xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
A: shows "A \<in> sets (Pi\<^sub>F I M)"
mkIde_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>" shows "set_category_given_img.mkIde S img \<AA> A = (if A \<subseteq> Collect (category.terminal S) \<and> |A| <o \<AA> then inv_into (Collect (partial_magma.ide S)) (set_category_data.set S img) A else partial_magma.null S)"
coprime_def: fixes a :: "'a"   and b :: "'a" shows "coprime a b = (\<forall>c. c dvd a \<longrightarrow> c dvd b \<longrightarrow> is_unit c)"
rev_rev_ident: fixes xs :: "'a list" shows "rev (rev xs) = xs"
effect_to_assignments_construction_from_function_graph_1: fixes op :: "'a strips_operator" assumes "set (add_effects_of op) \<inter> set (delete_effects_of op) = {}" shows "effect_to_assignments op = map (\<lambda>v. (v, if ListMem v (add_effects_of op) then True else False)) (add_effects_of op @ delete_effects_of op)"
LocalSecretsComposition_exprChannel_k: fixes x :: "chanID"   and Keys :: "Keys"   and P :: "specID"   and Q :: "specID" assumes "exprChannel x (kE Keys)"   and "\<not> ine P (kE Keys)"   and "\<not> ine Q (kE Keys)"   and "\<not> (x \<notin> ins P \<and> x \<notin> ins Q)" shows "False"
xs3'_def: shows "xs3' = drop (Suc hi) xs'"
I_def: shows "I \<equiv> \<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3 c (wp\<^sub>3 (WHILE b DO c) Q) (ps, n) else Q (ps, n))"
dgrad_sig_setD_rep_list_lt: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "dickson_grading d"   and "p \<in> dgrad_sig_set' j d" shows "d (punit.lt (rep_list p)) \<le> dgrad_max d"
rep_eq: fixes x :: "uint32"   and xa :: "nat" shows "Rep_uint32 (x >>> xa) = signed_drop_bit xa (Rep_uint32 x)"
simps_2: fixes i :: "'a"   and h :: "QDelta ns_constraint"   and t :: "('a \<times> QDelta ns_constraint) list"   and v :: "nat" shows "preprocess' ((i, h) # t) v = (let s' = preprocess' t v; p = poly h; is_monom_h = is_monom p; v' = FirstFreshVariable s'; t' = Tableau s'; a' = Atoms s'; m' = Poly_Mapping s'; u' = UnsatIndices s' in if is_monom_h then IState v' t' ((i, qdelta_constraint_to_atom h v') # a') m' u' else if p = 0 then if zero_satisfies h then s' else IState v' t' a' m' (i # u') else case m' p of None \<Rightarrow> IState (v' + 1) (linear_poly_to_eq p v' # t') ((i, qdelta_constraint_to_atom h v') # a') (m'(p \<mapsto> v')) u' | Some v \<Rightarrow> IState v' t' ((i, qdelta_constraint_to_atom h v) # a') m' u')"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and a :: "'a" shows "depth (Node w t\<^sub>1 t\<^sub>2) a = (if a \<in> alphabet t\<^sub>1 then depth t\<^sub>1 a + 1 else if a \<in> alphabet t\<^sub>2 then depth t\<^sub>2 a + 1 else 0)"
option_bind_commute: fixes A :: "'b option"   and B :: "'c option"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'a option" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
set_eq_iff_mset_eq_distinct: fixes x :: "'k list"   and y :: "'k list" assumes "distinct x"   and "distinct y" shows "(set x = set y) = (mset x = mset y)"
game\<^sub>3_guess: fixes x :: "nat"   and y :: "nat" shows "map_spmf fst (game\<^sub>3 (\<lambda>b m\<^sub>1 m\<^sub>2. spmf_of_set (nlists UNIV len_plain)) x y) = coin_spmf"
WT_conv: fixes \<eta> :: "nat" shows "\<I>_common' \<eta>, \<I>_common \<eta> \<turnstile>\<^sub>C conv \<eta> \<surd>"
distr_I_eq_M: fixes n :: "nat" shows "distr (restrict_space lborel {0<..<1}) borel (\<mu>.I n) = \<mu> n"
h1b: shows "\<forall>i. MPoly_Type.degree (sq ^ (i div 2)) var = 0"
unfold_congs_4: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "r = r'"   and "hare r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "hare_update f r = hare_update f' r'"
adjacent_def: fixes x :: "'b set"   and y :: "'b set" shows "x \<sim> y \<equiv> \<exists>z. z \<lhd> x \<and> z \<lhd> y"
L4_Controller_Zero: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l (Suc t) = Zero" shows "z (Suc t) = [] \<and> l t = Zero \<or> z (Suc t) = [Zero] \<and> l t = One"
restrict_indets_comp_dehomogenize: shows "restrict_indets \<circ> dehomogenize None = restrict_indets"
b_least_aux1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "b_least f x = x \<or> b_least f x < x \<and> f x (b_least f x) \<noteq> 0"
step_in: shows "x t + (h * p) *\<^sub>R f (t, x t) \<in> X"
size_tree: fixes t :: "'b tree" shows "Root_Balanced_Tree.size_tree t = size t"
Group: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a" assumes "BinOpSetGroup G binop e" shows "Group (BinOpSetGroup.Abs_G_perm G binop ` G)"
permutation_orbit_step: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "permutation f" shows "orbit f (f s) = orbit f s"
has_integral_factor_content: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b" shows "(f has_integral i) (cbox a b) = (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of cbox a b \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content (cbox a b)))"
borel_measurable_vimage_closed_interval: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S))"
pHom_mem: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and f :: "'a \<Rightarrow> 'e"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S (n, c)" shows "f (polyn_expr R X n (n, c)) = polyn_expr A Y n (n, cmp f c)"
Mp_ident_iff: fixes f :: "int poly" shows "(Mp f = f) = (\<forall>n. coeff f n \<in> {0..<m})"
sorted_merge: fixes xs :: "'a list"   and ys :: "'a list" assumes "Sorting_Algorithms.sorted cmp xs"   and "Sorting_Algorithms.sorted cmp ys" shows "Sorting_Algorithms.sorted cmp (local.merge xs ys)"
diagonalize_is_functor: shows "functor (\<cdot>) local.comp \<F>C.D"
that_1: fixes xs' :: "'a llist" assumes "xs = LCons x xs'" shows "thesis"
c: fixes x :: "'a \<times> 'b" shows "isCont (\<lambda>(x, y). g x y) x"
complex_norm: fixes x :: "real"   and y :: "real" shows "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
min_algs: fixes i :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "i \<in> Field SucFbd" shows "min_algs s1 s2 i = min_H s1 s2 (min_algs s1 s2) i"
surjective_pairing: fixes t :: "'k \<times> 'l" shows "t = (fst t, snd t)"
l_delete_shadow_root_get_disconnected_nodes_def: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs \<equiv> \<forall>h shadow_root_ptr h' r ptr'. h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h'"
A: shows "((s, c), e, s', c') \<in> ntrp fg"
mkarr_extensionality: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "\<not> monoidal_language.Arr C t" shows "free_monoidal_category.mkarr C t = {}"
conga_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "C B A CongA D E F"
NTUadmit_def: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and \<theta> :: "('c + 'b, 'd) trm"   and U :: "('d + 'd) set" shows "NTUadmit \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
fs_prod_inst: assumes "fs TYPE('a) TYPE('b)"   and "fs TYPE('c) TYPE('b)" shows "fs TYPE('a \<times> 'c) TYPE('b)"
case_2: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x21 :: "nat"   and x22 :: "'e"   and x23 :: "'e"   and x24 :: "'f"   and x25 :: "'f"   and x26 :: "'g" shows "(case Send x21 x22 x23 x24 x25 x26 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25 x26"
assms_2: shows "s \<in> set_pmf (abst (action cfg))"
l2_inv6_trans: shows "{l2_inv6} TS.trans l2 {> l2_inv6}"
simps: fixes n :: "nat"   and xs :: "'a list" shows "bal n xs = (if n = 0 then (\<langle>\<rangle>, xs) else let m = n div 2; (l, ys) = bal m xs; (r, y) = bal (n - 1 - m) (tl ys) in (\<langle>l, hd ys, r\<rangle>, y))"
codomains_def: fixes f :: "('o, 'a) arr" shows "codomains f \<equiv> {b. ide b \<and> COMP b f \<noteq> null}"
zero_eq_add_iff_both_eq_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(zero = plus x y) = (x = zero \<and> y = zero)"
Node_4: shows "Sorted_Less.sorted (Tree2.inorder \<langle>l, (a, m), r\<rangle>)"
get_curr_win2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2"   and "t1 = snd (fst (get_curr_win () s1))"   and "t2 = snd (fst (get_curr_win () s2))" shows "low_equal t1 t2"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim>\<^sup>s Q"   and "Q \<sim>\<^sup>s R" shows "P \<sim>\<^sup>s R"
M_def: shows "M = Tensor.mat_of_cols_list (dim_row A * dim_row C) (mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list C))"
Span_eq_combine_set: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "R.Span K Us = {R.combine Ks Us |Ks. set Ks \<subseteq> K}"
rep_eq: fixes x :: "(real, 'a) Finite_Cartesian_Product.vec" shows "st_vec (to_st_vec x) = (if stoch_vec x \<and> non_neg_vec x then x else \<chi>i. if i = undefined then 1 else 0)"
GC_inference_step: fixes l :: "label"   and M :: "('a literal multiset \<times> label) set"   and N :: "('a literal multiset \<times> label) set"   and C :: "'a literal multiset" assumes "l \<noteq> Old"   and "FL.active_subset M = {}"   and "old_concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (fst ` FL.active_subset N) C) \<subseteq> fst ` M" shows "N \<union> {(C, l)} \<leadsto>GC N \<union> {(C, Old)} \<union> M"
assms_1: shows "\<D> e \<lfloor>set (V # Vs)\<rfloor>"
target_alt_def: fixes r :: "('d \<times> 'c) list"   and p :: "'c" shows "target r p = last (p # states r p)"
poly_prod: fixes p :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and x :: "'a" shows "poly (prod p A) x = (\<Prod>k\<in>A. poly (p k) x)"
doublep_infinite_params: fixes S :: "(nat, 'a) form set" shows "infinite (- \<Union> (params ` psubst ((*) 2) ` S))"
PhiDef: assumes "v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1"   and "v' = (v, n, PhiDef)" shows "thesis"
fst_set_drop_indices: fixes A :: "('e \<times> 'g \<times> 'f) set" shows "fst ` (\<lambda>(f, uu_, d). (f, d)) ` A = fst ` A"
neutr_eq_iff: fixes a :: "'b"   and b :: "'b" shows "(top = inf a b) = (a = top \<and> b = top)"
B: shows "\<forall>x\<in>A. cmod (sum (h x) B) \<le> (\<Sum>y\<in>B. cmod (h x y))"
lift_Postdomination: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and nx :: "'h" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "Postdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit"