hcju/isabelleps huggingface dataset:
ni2: fixes i :: "nat" assumes "3 \<le> i"   and "i \<le> k" shows "n i \<noteq> n 2"
n_prod_idealTr: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" shows "(\<forall>k\<le>n. ideal R (J k)) \<longrightarrow> ideal R (i\<Pi>\<^bsub>R,n\<^esub> J)"
entry_rtrancl_is_path: fixes v :: "nat"   and E :: "(nat \<times> nat) set" assumes "(0, v) \<in> E\<^sup>*" obtains p :: "nat list" where "path_entry E p v"
invertible_lf_axioms_def: fixes f :: "'d \<Rightarrow> 'e" shows "invertible_lf_axioms f \<equiv> \<exists>g. g \<circ> f = id \<and> f \<circ> g = id"
first: shows "envObs a (tFirst t) = envObs a (tFirst t')"
meet_def: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "x \<sqinter>\<^bsub>L\<^esub> y = \<Sqinter>\<^bsub>L\<^esub>{x, y}"
antisym: fixes x :: "'b list"   and y :: "'b list" assumes "x \<le>s y"   and "y \<le>s x" shows "x = y"
G_def: fixes A :: "(('d, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "fixed_mat.G A = {(i, j). A $h i $h j \<noteq> (0::'d)}"
IH: fixes P :: "'a simple_packet set" assumes "p \<in> P" shows "simple_fw (rmshadow rs P) p = simple_fw rs p"
interval_integral_endpoints_reverse: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integral lborel a b f = - interval_lebesgue_integral lborel b a f"
fun_u_lt_etc: assumes "is_App u" shows "{#size (fun u), size t, size s#} < {#size u, size t, size s#}"
FiniteEnvironment_axioms: shows "FiniteEnvironment (\<lambda>_. kbp)"
distinct_count_list: fixes xs :: "'b list"   and a :: "'b" assumes "distinct xs" shows "count_list xs a = (if a \<in> set xs then 1 else 0)"
edge'_def: fixes xo :: "'v option"   and yo :: "'v option" shows "edge' xo yo = (case (xo, yo) of (None, None) \<Rightarrow> False | (None, \<langle>y\<rangle>) \<Rightarrow> y \<in> \<^bold>V \<and> y \<notin> A \<Gamma> | (\<langle>x\<rangle>, None) \<Rightarrow> False | (\<langle>x\<rangle>, \<langle>y\<rangle>) \<Rightarrow> edge \<Gamma> x y \<or> edge \<Gamma> y x)"
rassocl\<^sub>w_def: shows "rassocl\<^sub>w = (lsumr, rsuml)"
card_of_cardSuc_finite: fixes A :: "'a set" shows "finite (Field (cardSuc |A|)) = finite A"
h2: shows "\<turnstile>\<^sub>p {adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)} Utrans_P vars1 mat_Ph {adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)}"
PX3d_range: fixes d :: "D2" shows "Field (PX3d d) \<subseteq> {x. X3d x = d}"
preserves_modes_mem_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "gen_refine.preserves_modes_mem \<R> \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x\<^sub>A. mem\<^sub>A x\<^sub>A = mem\<^sub>C (id x\<^sub>A)) \<and> (\<forall>m. id ` mds\<^sub>A m = range id \<inter> mds\<^sub>C m)"
fresh_1: fixes a :: "W.var"   and x1 :: "ty" shows "a \<sharp> Ty x1 = a \<sharp> x1"
proper_ParT_properL: fixes cl :: "('a, 'b, 'c) cmd list" shows "proper (ParT cl) = properL cl"
matches_alist_and_alist_and': fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ls :: "'b negation_type list"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (alist_and' ls) a p = matches \<gamma> (alist_and ls) a p"
lowEquivalence_relevant_nodes_Entry: fixes s :: "('var \<Rightarrow> 'val option) list"   and s' :: "('var \<Rightarrow> 'val option) list"   and S :: "'node SDG_node set" assumes "s \<approx>\<^sub>L s'"   and "(_High_) \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "\<forall>V\<in>rv S (CFG_node (_Entry_)). hd s V = hd s' V"
poly_finite_fourier_poly': fixes m :: "int"   and k :: "nat"   and ws :: "nat \<Rightarrow> complex" assumes "m \<in> {0..<int k}" shows "poly (finite_fourier_poly' ws k) (unity_root k m) = ws (nat m)"
simps_3: fixes m :: "nat"   and n :: "nat" shows "Mn_free (recf.Id m n) = True"
smult_carrier_vec: fixes a :: "'b"   and v :: "'b Matrix.vec"   and n :: "nat" shows "(a \<cdot>\<^sub>v v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
winding_number_join_pos_combined': fixes \<gamma>1 :: "real \<Rightarrow> complex"   and z :: "complex"   and \<gamma>2 :: "real \<Rightarrow> complex" assumes "valid_path \<gamma>1 \<and> z \<notin> path_image \<gamma>1 \<and> 0 < Re (winding_number \<gamma>1 z)"   and "valid_path \<gamma>2 \<and> z \<notin> path_image \<gamma>2 \<and> 0 < Re (winding_number \<gamma>2 z)"   and "pathfinish \<gamma>1 = pathstart \<gamma>2" shows "valid_path (\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image (\<gamma>1 +++ \<gamma>2) \<and> 0 < Re (winding_number (\<gamma>1 +++ \<gamma>2) z)"
preserves_reflects_arr: fixes f :: "'a arrow_of_spans_data" shows "arr (\<ll>.map f) = arr f"
antisym_conv1: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "\<not> x \<sqsubset> y" shows "(x \<sqsubseteq> y) = (x = y)"
e2ennreal_neg: fixes x :: "ereal" assumes "x \<le> 0" shows "e2ennreal x = 0"
inext_cut_less_fix: fixes t :: "nat"   and n :: "nat"   and I :: "nat set" assumes "t \<le> inext n I" shows "inext n (I \<down>< t) = n"
Seq_succ_D: fixes s :: "hf"   and k :: "hf" assumes "Seq s (succ k)" shows "Seq s k"
max2_equals1: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(max2 r a b = a) = ((b, a) \<in> r)"
assms: fixes x :: "'a"   and y1 :: "'a"   and y2 :: "'a" assumes "r x y1"   and "r x y2" shows "\<exists>z. r y1 z \<and> r y2 z"
advantage_multi_def: fixes \<A> :: "'c_a \<Rightarrow> (unit, 'call + 'guess, 'ret + unit) gpv" shows "advantage_multi \<A> = spmf (game_multi \<A>) True"
compl_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<^bold>\<sqinter> y = \<zero>"   and "x \<^bold>\<squnion> y = \<one>" shows "\<sim> x = y"
f: shows "(\<lambda>x. Px (fst x) * Py (snd x)) \<in> borel_measurable (S \<Otimes>\<^sub>M T)"
norm_prod_def: fixes x :: "'a \<times> 'b" shows "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
assms_2: shows "b \<in>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Arr\<rparr>"
simps_9: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" shows "af_letter (\<phi> R\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> and\<^sub>n af_letter \<phi> \<nu> or\<^sub>n (\<phi> R\<^sub>n \<psi>)"
iso_block_in: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and bl :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "bl \<in># \<B>" shows "bij_map ` bl \<in># \<B>'"
diff_numeral_special_9: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "minus one one = zero"
nhdE: fixes U :: "'b set"   and x :: "'b" assumes "U \<in> nhds\<^bsub>T\<^esub> x" obtains m :: "'b set" where "U \<subseteq> carrier\<^bsub>T\<^esub>" and "m open\<^bsub>T\<^esub>" and "x \<in> m" and "m \<subseteq> U"
map_euler_err_fas_nth: fixes X0 :: "'a list"   and d :: "nat"   and h :: "floatarith"   and CX :: "floatarith list" assumes "length X0 = d" shows "map (euler_err_fas_nth X0 h CX) [0..<d] = euler_err_fas X0 h CX"
in_HomD_1: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> Hom a b" shows "F \<noteq> {}"
omega_fdia: fixes x :: "'a"   and y :: "'a" shows "\<Omega> x y = d y \<cdot> ad ( |x\<rangle> y)"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "xs \<down> m \<down> n = xs \<down> min n m"
return_rule: fixes x :: "'b" shows "<emp> return x <\<lambda>r. \<up> (r = x)>"
swap_lassocr\<^sub>w_def: shows "swap_lassocr\<^sub>w \<equiv> rassocl\<^sub>w \<circ>\<^sub>w (swap\<^sub>w |\<^sub>w (id, id) \<circ>\<^sub>w lassocr\<^sub>w)"
trees_of_size_def: fixes n :: "nat" shows "trees_of_size n = {t. count_nodes t = n}"
s_1: shows "s \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)"
le3456_lt__lt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Le Bet Cong C D E F" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B E F"
timpl_closure'_inv': fixes s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" assumes "(s, t) \<in> timpl_closure' TI" shows "(\<exists>x. s = Var x \<and> t = Var x) \<or> (\<exists>f g S T. s = Fun f S \<and> t = Fun g T \<and> length S = length T \<and> (\<forall>i<length T. (S ! i, T ! i) \<in> timpl_closure' TI) \<and> (f \<noteq> g \<longrightarrow> is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> TI\<^sup>+))"
has_products_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and I :: "'b set" assumes "category C" shows "category.has_products C I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J C D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. category.has_as_product C J D a))"
sum_list_eq0I: fixes xs :: "'a list" assumes "\<forall>x\<in>set xs. x = (0::'a)" shows "sum_list xs = (0::'a)"
sminus_plus: fixes s :: "'a"   and v :: "'t"   and t :: "'a" assumes "s adds\<^sub>p v"   and "t adds\<^sub>p v \<ominus> s" shows "v \<ominus> (s + t) = v \<ominus> s \<ominus> t"
zero_idem: shows "(0::'a) + (0::'a) = (0::'a)"
partition_between_correct: fixes hi :: "nat"   and xs :: "'c list"   and lo :: "nat"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "hi < length xs"   and "lo \<le> hi"   and "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. R (h x) (h y) \<or> R (h y) (h x)" shows "partition_between R h lo hi xs \<le> SPEC (uncurry (partition_spec R h xs lo hi))"
UINF_as_Sup_collect': fixes f :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Sqinter> P \<bullet> f P) = \<Sqinter>range f"
kronecker_inverse_index: fixes r :: "nat"   and A :: "'b mat"   and s :: "nat"   and v :: "nat"   and B :: "'b mat"   and w :: "nat" assumes "r < dim_row A"   and "s < dim_col A"   and "v < dim_row B"   and "w < dim_col B" shows "kronecker_product A B $$ (dim_row B * r + v, dim_col B * s + w) = A $$ (r, s) * B $$ (v, w)"
inverse_scaleR_distrib: fixes a :: "real"   and x :: "'a" shows "inverse (a *\<^sub>R x) = inverse x /\<^sub>R a"
phiUsesE: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" assumes "v \<in> braun_ssa.phiUses g n" obtains n' :: "'node" and v' :: "'var \<times> 'node \<times> Def" and vs :: "('var \<times> 'node \<times> Def) list" where "n' \<in> set (successors g n)" and "(n, v) \<in> set (zip (predecessors g n') vs)" and "phis' g (n', v') = Some vs"
sets_density: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "sets (density M f) = sets M"
iso_assoc: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.iso (Span.assoc f g h)"
exhaust: fixes y :: "'b prefix_match" obtains x1 :: "'b word" and x2 :: "nat" where "y = PrefixMatch x1 x2"
geodesic_segmentI: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "geodesic_segment G"
2: shows "(\<lambda>(x, y). \<exists>z. P x y z) = (\<lambda>xy. \<exists>z. case xy of (x, y) \<Rightarrow> P x y z)"
Cast_final_eq: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E' :: "char list \<Rightarrow> ty option"   and l' :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h, l)\<rangle>"   and "final e"   and "final e'" shows "P,E' \<turnstile> \<langle>\<lparr>C\<rparr>e,(h, l')\<rangle> \<rightarrow> \<langle>e',(h, l')\<rangle>"
ide_def: fixes a :: "Arr" shows "ide a \<equiv> local.comp a a \<noteq> null \<and> (\<forall>f. (local.comp f a \<noteq> null \<longrightarrow> local.comp f a = f) \<and> (local.comp a f \<noteq> null \<longrightarrow> local.comp a f = f))"
rem_unPath_graph: fixes ps :: "('a \<times> 'b \<times> 'a) list"   and G :: "('a, 'b) graph" shows "rem_unPath (rev_path ps) G = rem_unPath ps G"
inter_subgroup_dvd_card: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G" shows "card (H \<inter> J) dvd card H"
esl_def: shows "Err.esl = (\<lambda>(A, r, f). (A, r, \<lambda>x y. OK (f x y)))"
gt_diff: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t >\<^sub>h\<^sub>d head s"   and "is_Sym (head s)"   and "chkchop (>\<^sub>t) t s" shows "t >\<^sub>t s"
abs_eq: fixes x :: "('a, 'b) fmap" shows "Pm_fmap x = Abs_poly_mapping (lookup0 x)"
alphaAbsGood_def: shows "alphaAbsGood \<equiv> \<lambda>qA qB. qGoodAbs qA \<and> qGoodAbs qB \<and> qA $= qB"
mergeable_envsI: fixes S :: "'c list set"   and n :: "nat" assumes "\<And>v1 v2 v. \<lbrakk>v1 \<in> S; v2 \<in> S; length v = n; \<forall>x<n. v ! x = v1 ! x \<or> v ! x = v2 ! x\<rbrakk> \<Longrightarrow> v \<in> S" shows "mergeable_envs n S"
eqpoll_imp_lepoll: fixes A :: "'b set"   and B :: "'c set" assumes "A \<approx> B" shows "A \<lesssim> B"
tmneg_allpolys_npoly: fixes t :: "tm" assumes "allpolys isnpoly t" shows "allpolys isnpoly (tmneg t)"
p_comp_pp: fixes x :: "'c"   and y :: "'c" shows "- (x \<sqinter> - - y) = - (x \<sqinter> y)"
ShiftP_Eats_Eats: fixes f :: "tm"   and k :: "tm"   and del :: "tm"   and g :: "tm"   and a :: "tm"   and a' :: "tm"   and b :: "tm" shows "{ShiftP f k del g, HaddP del a a', a IN k} \<turnstile> ShiftP (Eats f (HPair a b)) k del (Eats g (HPair a' b))"
crit_pair_cbelow_imp_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m F p q" shows "is_Groebner_basis F"
reduce_root_inorder: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'c btree" assumes "split split" shows "inorder (reduce_root t) = inorder t"
new_element_get_child_nodes_is_l_new_element_get_child_nodes: shows "l_new_element_get_child_nodes ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
map_semantics_2: shows "promela_props_ltl APs = map_props (idx (IArray.list_of APs)) \<circ> promela_props"
simps_1: fixes r :: "('a \<times> bool) language" shows "\<oo> (Times_Plus r) = \<oo> r"
normal_def: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" shows "H \<lhd> G \<equiv> subgroup H G \<and> Group.group G \<and> normal_axioms H G"
IH: shows "\<forall>ys. length ys < length p \<longrightarrow> isPath s ys t \<longrightarrow> (\<exists>p'. isSimplePath s p' t)"
f1: shows "dim_col \<langle>|v\<rangle>| = dim_vec v"
intro: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a x y. scale a (x + y) = scale a x + scale a y"   and "\<And>a b x. scale (a + b) x = scale a x + scale b x"   and "\<And>a b x. scale a (scale b x) = scale (a * b) x"   and "\<And>x. scale (1::'a) x = x" shows "vector_space scale"
exp_pi_i: shows "exp (complex_of_real pi * \<i>) = - 1"
T_on_rand'_append: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and A :: "('b \<Rightarrow> 'e pmf) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) pmf)"   and s :: "('b \<times> 'e) pmf"   and xs :: "'c list"   and ys :: "'c list" shows "On_Off.T_on_rand' step t A s (xs @ ys) = On_Off.T_on_rand' step t A s xs + On_Off.T_on_rand' step t A (On_Off.config'_rand step A s xs) ys"
pseudo_proj_True_measurable: fixes n :: "nat" shows "pseudo_proj_True n \<in> bernoulli_stream p \<rightarrow>\<^sub>M bernoulli_stream p"
del_list_notin_Cons: fixes x :: "'b"   and xs :: "'b list" assumes "Sorted_Less.sorted (x # xs)" shows "del_list x xs = xs"
add_set_0: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "add_set f A 0 = A"
scale_eq_0_iff: fixes a :: "complex"   and x :: "'a" shows "(a *\<^sub>C x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
refinesE: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec" assumes "refines R pi Sa Sc" obtains "PO_refines R Sa Sc" and "obs_consistent R pi Sa Sc"
prime_elem_gauss_cnj_iff: fixes z :: "gauss_int" shows "prime_elem (gauss_cnj z) = prime_elem z"
take_the_conv: fixes k :: "nat"   and xs :: "'d list" shows "xs \<down> k = (if length xs \<le> k then xs else THE r. length r = k \<and> (\<exists>r2. r @ r2 = xs))"
sorted_list_of_set_inject: fixes R :: "'a set"   and S :: "'a set" assumes "finite R"   and "finite S" shows "flatten (flatten (=) R) S = (R = S)"
typeof_opubx_complete: fixes opubx :: "opubx"   and xs :: "(dynamic, nat, bool) unboxed list"   and y :: "(dynamic, nat, bool) unboxed" assumes "eval_ubx opubx xs = Some y" shows "typeof_opubx opubx = (map typeof xs, typeof y)"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
normalize_simps_8: fixes c :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "Language.normalize (DynCom c) = DynCom (\<lambda>s. Language.normalize (c s))"
subst_typ_AList_clearjunk: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" shows "subst_typ insts T = subst_typ (AList.clearjunk insts) T"
match_2: fixes h :: "'a"   and g :: "'a"   and f :: "'a" assumes "C.seq h (g \<cdot> f)"   and "C.seq g f" shows "C.seq h g"
map_pair_subset: fixes f :: "'e \<times> 'g \<Rightarrow> 'e \<times> 'f"   and xs :: "('e \<times> 'g) list" shows "set (map_pair f xs) \<subseteq> f ` set xs"
l0_f1_f1': shows "f1 / (f1' - f1) < real l0"
T_def: fixes x :: "real" shows "T x = 1 / (2 * x) + 1 / (2 * (x + 1))"
SKIP: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "G \<rhd> Skip : Sec (\<lambda>(s, t, \<beta>). s \<equiv>\<^sub>\<beta> t)"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier K" shows "\<ominus>\<^bsub>K\<^esub> x \<in> carrier K"
power_inject_exp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "(power.power one times a m = power.power one times a n) = (m = n)"
I: fixes idn :: "variable"   and S :: "String.literal set" assumes "idn \<in> set fresh_idns" shows "(idn, S) \<notin> fst ` set insts"
insert_insert': fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c option" shows "insert xs e i = insert' xs e i"
no_initial_process_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration" assumes "computation channel trans send recv init final" shows "\<forall>p. \<not> ps init p \<noteq> None"
add_commute: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" shows "a \<union> b = b \<union> a"
pos_els: shows "\<forall>y. y \<in> set xs \<longrightarrow> 0 < y"
om_unfold: fixes x :: "'a" shows "x\<^sup>\<omega> \<le> x \<cdot> x\<^sup>\<omega>"
rep_eq: fixes x :: "'a" shows "Rep_pos (mk_pos x) = (if (0::'a) \<le> x then x else (0::'a))"
delete_edge_sym: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "del_unEdge v e v' g = del_unEdge v' e v g"
not_mem_dm: shows "p \<notin> r \<down>m {#p#}"
strict_trans2: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subset># b"   and "b \<subseteq># c" shows "a \<subset># c"
sdrop_siterate: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "sdrop n (siterate f x) = siterate f ((f ^^ n) x)"
unique_factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "pl.unique_factorization_m f cfs = (pl.Mf ` Collect (pl.factorization_m f) = {pl.Mf cfs})"
rawpsubst_all: fixes \<phi> :: "'fmla"   and y :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "y \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "y \<notin> snd ` set txs"   and "y \<notin> \<Union> (FvarsT ` fst ` set txs)" shows "rawpsubst (all y \<phi>) txs = all y (rawpsubst \<phi> txs)"
typing_pairE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and B :: "'a trm"   and \<theta> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> SimplyTyped.Pair A B : \<theta>" shows "\<exists>\<tau> \<sigma>. \<theta> = TPair \<tau> \<sigma> \<and> \<Gamma> \<turnstile> A : \<tau> \<and> \<Gamma> \<turnstile> B : \<sigma>"
special_axioms_hold: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> special_axioms" shows "eval_fm e A"
dom_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "local.dom f = (if arr f then MkIde (Dom f) else null)"
steps_appendD1: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph_Defs.steps E (xs @ ys)"   and "xs \<noteq> []" shows "Graph_Defs.steps E xs"
T_on_rand_as_sum: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s0 :: "'c list"   and rs :: "'c list" shows "Move_to_Front.T_on_rand' A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs = sum (Move_to_Front.T_on_rand'_n A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs) {..<length rs}"
assert_disch1: fixes P :: "'b \<Rightarrow> bool"   and \<sigma> :: "'b"   and M :: "bool \<Rightarrow> 'b \<Rightarrow> (bool \<times> 'b) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (assert\<^sub>S\<^sub>E P) M) = (\<sigma> \<Turnstile> M True)"