hcju/isabelleps huggingface dataset:
ordLeq_transitive: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r \<le>o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
well_founded_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "well_founded B ((\<sqsubset>) \<restriction> A)"
dg_is_arrD_3: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
bisimResChainPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<sim> Q"   and "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<sim> \<lparr>\<nu>*xvec\<rparr>Q"
not_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
Abs_enat_inject: fixes x :: "nat option"   and y :: "nat option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_enat x = Abs_enat y) = (x = y)"
subgroup_img_is_subgroup: fixes I :: "'a set" assumes "subgroup I G" shows "subgroup (h ` I) H"
x0_hb_bound1: fixes x :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> x"   and "b \<in> set bs" shows "hb / ln x powr (1 + e) < (1 - b) / 2"
independent_image_matrix_vector_mult: fixes B :: "('b, 'c) vec set"   and P :: "(('b, 'c) vec, 'd) vec" assumes "vec.independent B"   and "invertible P" shows "vec.independent ((*v) P ` B)"
cospoly_ub_imp_sinpoly_ub: fixes x :: "real"   and k :: "nat" assumes "0 \<le> x"   and "0 < k"   and "\<And>x. 0 \<le> x \<Longrightarrow> cos x \<le> cospoly (k - 1) x" shows "sin x \<le> sinpoly k x"
fac_body_def: shows "fac_body \<equiv> \<Lambda> r n. If n =\<^sub>B 0 then 1 else n * r\<cdot>(n - 1)"
that: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> \<tau>.P \<Longrightarrow>\<alpha> \<prec> P''"   and "\<Psi> \<rhd> P' \<sim> P''" shows "thesis"
add_0_right: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "monoid_add_ow U plus zero"   and "a \<in> U" shows "plus a zero = a"
po_sw_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" shows "po_sw P E a a' = (E \<turnstile> a \<le>po a' \<or> P,E \<turnstile> a \<le>sw a')"
zero: fixes f :: "'b \<Rightarrow> 'c" assumes "Modules.additive f" shows "f (0::'b) = (0::'c)"
typing_alt_simps_25: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E AnyIteratorCall src k its its_ty body : \<tau>) = (k = ArrowCall \<and> (\<exists>\<tau>' \<rho>. \<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<tau>, \<rho>) \<and> length its \<le> Suc 0 \<and> \<rho> \<le> Boolean[?]))"
fdia_arden_eq: fixes x :: "'a"   and p :: "'a"   and q :: "'a" assumes "\<nabla>x = (zero_class.zero::'a)"   and "d p = d q +  |x\<rangle> p" shows "d p =  |x\<^sup>\<star>\<rangle> q"
support_def: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support P = {x. \<exists>l1 l2 s. (\<forall>y. y \<noteq> x \<longrightarrow> l1 y = l2 y) \<and> P l1 s \<noteq> P l2 s}"
seqI': fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>" shows "C.seq g f"
U0_def: shows "U\<^sub>0 \<equiv> {vs \<odot> 0\<^sup>\<infinity> |vs. vs \<in> UNIV}"
ide_char': fixes a :: "'a" shows "Adj.C.ide a = (Adj.C.arr a \<and> (Adj.C.dom a = a \<or> Adj.C.cod a = a))"
snoc_eq_iff_butlast: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
assoc'_simps_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "cod (assoc' a b c) = local.prod (local.prod a b) c"
enum_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(s.enum i \<le> s.enum j) = (i \<le> j)"
eq_part: fixes x :: "nat" assumes "x < i" shows "gs2.gso x = gs1.gso x"
finite_closed_segment: fixes a :: "'b"   and b :: "'b" shows "finite (closed_segment a b) = (a = b)"
hyps: assumes "j \<le> mod_type_class.from_nat k"   and "column j (Gram_Schmidt_upt_k A k) = column i (Gram_Schmidt_upt_k A k)"   and "k < ncols A" shows "column j (Gram_Schmidt_upt_k A k) = 0"
id: shows "pCons a p \<circ>\<^sub>p q = [:a:] + q * p \<circ>\<^sub>p q"
assms_3: shows "\<rho> \<in> carrier_mat n n"
scalar_prod_append: fixes v1 :: "'a vec"   and n1 :: "nat"   and v2 :: "'a vec"   and n2 :: "nat"   and w1 :: "'a vec"   and w2 :: "'a vec" assumes "v1 \<in> carrier_vec n1"   and "v2 \<in> carrier_vec n2"   and "w1 \<in> carrier_vec n1"   and "w2 \<in> carrier_vec n2" shows "(v1 @\<^sub>v v2) \<bullet> (w1 @\<^sub>v w2) = v1 \<bullet> w1 + v2 \<bullet> w2"
sum_bot: fixes x :: "'a" shows "sum_class.sum x + sum_class.sum bot = sum_class.sum x"
\<tau>_mono: fixes i :: "nat"   and j :: "nat"   and s :: "'b trace" assumes "i \<le> j" shows "\<tau> s i \<le> \<tau> s j"
pochhammer_product: fixes m :: "nat"   and n :: "nat"   and z :: "'b" assumes "m \<le> n" shows "pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
mult_mm_inv_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex"   and C :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0"   and "A *\<^sub>m\<^sub>m B = C" shows "B = mat_inv A *\<^sub>m\<^sub>m C"
iTL_defs_2: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iEx I P \<equiv> \<exists>t\<in>I. P t"
induct: fixes P :: "('c \<Rightarrow> 'd \<Rightarrow> 'e) \<Rightarrow> 'c option \<Rightarrow> 'd option \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and a1 :: "'c option"   and a2 :: "'d option" assumes "\<And>f a b. P f (Some a) (Some b)"   and "\<And>uu_ uw_. P uu_ None uw_"   and "\<And>uu_ uv_. P uu_ uv_ None" shows "P a0 a1 a2"
keysFor_image_Key: fixes E :: "nat set" shows "keysFor (Key ` E) = {}"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a" shows "B.VV.arr (B.VV.cod f) = B.VV.arr f"
equ_2: shows "subtopology (lsphere n) (equator n) = nsphere (n - Suc 0)"
conc_conc: fixes u :: "'b list"   and v :: "'b list"   and w :: "nat \<Rightarrow> 'b" shows "u \<frown> v \<frown> w = (u @ v) \<frown> w"
ide_cod: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" shows "ide (cod f)"
vpra_callstack_prefix: fixes cs :: "'edge list"   and cs' :: "'edge list"   and as :: "'edge list" assumes "valid_path_rev_aux (cs @ cs') as" shows "valid_path_rev_aux cs as"
convergent_powser'_gbinomial: fixes abort :: "bool"   and p :: "real" shows "convergent_powser' (gbinomial_series abort p) (\<lambda>x. (1 + x) powr p)"
double_compl: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<^bold>- (\<^bold>- x) = x"
foldli_ball_aux: fixes l :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and b :: "bool" assumes "distinct (map fst l)" shows "foldli l (\<lambda>x. x) (\<lambda>x _. P x) b = (b \<and> op_map_ball (map_of l) P)"
assms: shows "moebius_pt M z \<in> circline_set (moebius_circline M H)"
intros_15: fixes G :: "('a, 'b) form list"   and G' :: "('a, 'b) form list" assumes "\<turnstile> G"   and "set G = set G'" shows "\<turnstile> G'"
segment_overlapping: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> orbit f a"   and "x \<in> orbit f b"   and "bij f" shows "segment f a x \<subseteq> segment f b x \<or> segment f b x \<subseteq> segment f a x"
enc_preserves_pred_iff_source_target_rel_preserves_pred_1: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_preserves_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_preserves_pred Rel Pred)"
binomial_pmf_Suc: fixes p :: "real"   and n :: "nat" assumes "p \<in> {0..1}" shows "binomial_pmf (Suc n) p = bernoulli_pmf p \<bind> (\<lambda>b. binomial_pmf n p \<bind> (\<lambda>k. return_pmf ((if b then 1 else 0) + k)))"
time_length_tm: fixes xs :: "'a list" shows "time (length_tm xs) = length xs + 1"
natural_isomorphism_axioms: shows "natural_isomorphism hom'.comp hom'.comp FG.map hom'.map \<psi>"
nn_transfer_operator_charact: fixes g :: "'a \<Rightarrow> ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "\<And>A. A \<in> sets M \<Longrightarrow> \<integral>\<^sup>+ x. indicator A x * g x \<partial>M = \<integral>\<^sup>+ x. indicator A (T x) * f x \<partial>M"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. nn_transfer_operator f x = g x"
sin_gt_zero2: fixes x :: "real" assumes "0 < x"   and "x < pi / 2" shows "0 < sin x"
rows_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rows A = {row i A |i. i \<in> UNIV}"
ext_inject: fixes p_tag_ctstate :: "ctstate"   and more :: "'a"   and p_tag_ctstate' :: "ctstate"   and more' :: "'a" shows "(\<lparr>p_tag_ctstate = p_tag_ctstate::ctstate, \<dots> = more::'a\<rparr> = \<lparr>p_tag_ctstate = p_tag_ctstate'::ctstate, \<dots> = more'::'a\<rparr>) = (p_tag_ctstate = p_tag_ctstate' \<and> more = more')"
cases: fixes r :: "('b, 'd) sa_rec" obtains g_V :: "'b set" and g_E :: "('b \<times> 'b) set" and g_V0 :: "'b set" and sa_L :: "'b \<Rightarrow> 'd" where "r = \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, sa_L = sa_L\<rparr>"
sel_1: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "occurs_on (Trans x11 x12 x13) = x11"
QChain: shows "\<Psi> \<otimes> (p \<bullet> \<Psi>\<^sub>R) \<rhd> p \<bullet> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"
UP_simps_5: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "cod (UP \<mu>) = UP (B.cod \<mu>)"
int_nat_card: shows "int (nat (int CARD('b))) = int CARD('b)"
DA_in: shows "DA \<in> set (prems_of \<iota>\<^sub>0)"
bounded_any_center: fixes S :: "'c set"   and a :: "'c" shows "bounded S = (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
dual_inf: fixes x :: "'c"   and y :: "'c"   and sup :: "'c" shows "is_inf (dual x) (dual y) (dual sup) = is_sup x y sup"
SD_preorderD_3: fixes q :: "'alt pmf"   and p :: "'alt pmf"   and x :: "'alt" assumes "SD (Pareto R) q p"   and "x \<in> alts" shows "measure_pmf.prob q (preferred_alts (Pareto R) x) \<le> measure_pmf.prob p (preferred_alts (Pareto R) x)"
fix_not_deactivation: fixes s :: "'b"   and a :: "'a" assumes "s \<circ> a = a \<and> \<lambda> (s, a) = \<dd>" shows "a = (0::'a)"
InvariantUniqQAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantUniqQ (getQ state)"   and "getQ state \<noteq> []" shows "let state'' = applyUnitPropagate state in InvariantUniqQ (getQ state'')"
c4: shows "CR3_RED t (\<tau> \<rightarrow> \<sigma>)"
may_lock_lock_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "may_lock (lock_lock l t) t = may_lock l t"
cases: fixes a :: "'a stream"   and A :: "'a set" assumes "a \<in> streams A" obtains aa :: "'a" and s :: "'a stream" where "a = aa ## s" and "aa \<in> A" and "s \<in> streams A"
fls_lr_inverse_X_intpow'_1: fixes i :: "int"   and x :: "'a" shows "fls_left_inverse (fls_X_intpow i) x = fls_const x * fls_X_intpow (- i)"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and f :: "'e \<Rightarrow> 'e"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. f s \<in> Q} Basic f Q,A"
Nats_cases: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.semiring_1 one times plus zero"   and "x \<in> semiring_1.Nats one plus zero" obtains n :: "nat" where "x = semiring_1.of_nat one plus zero n"
carrier_vecD: fixes v :: "'d Matrix.vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "dim_vec v = n"
i2: shows "\<forall>B\<in>H. atom i \<sharp> B"
iso_hom_induced_inclusion_contractible: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set"   and p :: "int" assumes "contractible_space X"   and "contractible_space (subtopology X S)"   and "T \<subseteq> S"   and "topspace X \<inter> S \<noteq> {}" shows "hom_induced p (subtopology X S) T X T id \<in> Group.iso (relative_homology_group p (subtopology X S) T) (relative_homology_group p X T)"
o_eq_dest_lhs: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'c \<Rightarrow> 'd"   and v :: "'c" assumes "a \<circ> b = c" shows "a (b v) = c v"
nEq_neq_conv: fixes x :: "('a, 'b) pp \<times> nat"   and y :: "('a, 'b) pp \<times> nat" shows "(nat_term_compare cmp x y \<noteq> Eq) = (x \<noteq> y)"
simps_11: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Sqrt a) vs = (isDERIV x a vs \<and> 0 < interpret_floatarith a vs)"
Mem_SUCC_I2: fixes H :: "fm set"   and u :: "tm"   and t :: "tm" assumes "H \<turnstile> u EQ t" shows "H \<turnstile> u IN SUCC t"
vars_subset_0_dim_poly: fixes z :: "linear_poly" shows "vars z \<subseteq> {0..<dim_poly z}"
sorted_list_of_set_remove: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A" shows "linorder.sorted_list_of_set less_eq (A - {x}) = remove1 x (linorder.sorted_list_of_set less_eq A)"
subst_bvs1_subst_bvs1': fixes t :: "Core.term"   and n :: "nat"   and s :: "Core.term list" shows "subst_bvs1 t n s = subst_bvs1' t n (map (incr_bv n 0) s)"
assms: shows "vbrelation (vinsert \<langle>a, b\<rangle> r)"
cqt_orig_1: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<beta> :: "'a" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<beta> in v]"
onorm_triangle_le: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and e :: "real" assumes "bounded_linear f"   and "bounded_linear g"   and "onorm f + onorm g \<le> e" shows "onorm (\<lambda>x. f x + g x) \<le> e"
ncdl: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
invertible_0: shows "monoid.invertible {0::'a} (\<lambda>n m. 0::'a) (0::'a) (0::'a)"
T_on_rand'_nn: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf"   and qs :: "'a list" shows "0 \<le> T\<^sub>p_on_rand' A s qs"
vdomain_atE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>)" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>)" and "\<BB>\<lparr>Cod\<rparr>\<lparr>a\<rparr> = b"
notStuck_CondTrueD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Cond b c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "s \<in> b" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
mono: fixes x :: "('b, 'call, 'ret) Resumption.resumption \<times> 's" shows "spmf.mono_body (\<lambda>execp_resumption. case x of (Resumption.resumption.Done x, s) \<Rightarrow> return_pmf (map_option (\<lambda>a. (a, s)) x) | (Resumption.resumption.Pause out c, s) \<Rightarrow> callee s out \<bind> (\<lambda>(input, s'). curry execp_resumption (c input) s'))"
min_dep_imp_supset_circuit: fixes X :: "'a set"   and C :: "'a set"   and x :: "'a" assumes "indep X"   and "circuit C"   and "C \<subseteq> insert x X" shows "x \<in> C"
confs_hext: fixes P :: "'a prog"   and h :: "heap"   and vs :: "val list"   and Ts :: "ty list"   and h' :: "heap" assumes "P,h \<turnstile> vs [:\<le>] Ts"   and "h \<unlhd> h'" shows "P,h' \<turnstile> vs [:\<le>] Ts"
rstar_rec_isotone: fixes y :: "'a" shows "isotone (Rf y)"
empty_boxes_correct: fixes n :: "nat"   and lm :: "nat list" assumes "n \<le> length lm" shows "{\<lambda>nl. nl = lm} empty_boxes n {\<lambda>nl. nl = 0 \<up> n @ drop n lm}"
g: fixes x :: "'b" assumes "x \<in> topspace X" shows "\<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
greedy_path_LNil: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "greedy_conforming_path p \<sigma> \<sigma>' v0 \<noteq> LNil"
power_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a"   and n :: "nat" assumes "z \<squnion> y \<cdot> x \<le> y" shows "z \<cdot> x ^ n \<le> y"
vsubstAbs_def: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" shows "vsubstAbs ys y1 y2 \<equiv> substAbs ys (Var ys y1) y2"
ereal_mult_less_right: fixes b :: "ereal"   and a :: "ereal"   and c :: "ereal" assumes "b * a < c * a"   and "0 < a"   and "a < \<infinity>" shows "b < c"
norm_map_in_poly_ring_carrier: fixes p :: "'a list"   and f :: "'a \<Rightarrow> 'a list" assumes "p \<in> carrier (poly_ring R)"   and "\<And>a. a \<in> carrier R \<Longrightarrow> f a \<in> carrier (poly_ring R)" shows "UP.normalize (map f p) \<in> carrier (poly_ring (poly_ring R))"
below_lubI: fixes x :: "'e"   and S :: "'e set" assumes "x \<in> S" shows "x \<sqsubseteq> lub S"
lasso_accepted: fixes L :: "'Q lasso" assumes "is_lasso L" shows "is_acc_run (run_of_lasso L)"
portfolio_def: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "portfolio p = finite (support_set p)"
Foundation_lemma: fixes i :: "name"   and z :: "tm" assumes "atom i \<sharp> z" shows "{All2 i z (Neg (Disjoint (Var i) z))} \<turnstile> Neg (Var i IN z) AND Disjoint (Var i) z"
power2_less_imp_less: 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 x :: "'a"   and y :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less (power.power one times x 2) (power.power one times y 2)"   and "less_eq zero y" shows "less x y"
OPT2_B1: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) = real (length qs) / 2"
sees_methods_decl_above: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and M :: "char list"   and m :: "ty list \<times> ty \<times> 'b"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "P \<turnstile> C \<preceq>\<^sup>* D"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> toy_init i | Some s \<Rightarrow> s, snd \<sigma>)"
mono_Sync_sym: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process"   and A :: "'c set" assumes "P \<sqsubseteq> Q" shows "(S \<lbrakk>A\<rbrakk> P) \<sqsubseteq> (S \<lbrakk>A\<rbrakk> Q)"
isoD_1: fixes f :: "'a1 \<times> 'a2" assumes "local.iso f" shows "C1.iso (fst f)"
C_Ord_induct: fixes P :: "C \<Rightarrow> bool"   and n :: "C" assumes "P (C [])"   and "\<And>ns. P (C ns) \<Longrightarrow> P (C (C [] # ns))"   and "\<And>n ns ms. (\<And>i. P (C (funC (C (n # ns)) i @ ms))) \<Longrightarrow> P (C (C (n # ns) # ms))" shows "P n"
prod_primeTr: fixes P :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "prime_ideal R P"   and "ideal R A"   and "\<not> A \<subseteq> P"   and "ideal R B"   and "\<not> B \<subseteq> P" shows "\<exists>x. x \<in> A \<and> x \<in> B \<and> x \<notin> P"
io_def: shows "io = (ARG_MIN length io. io \<in> {io \<in> L M1. io \<notin> L M2})"
hyps_3: assumes "U \<subseteq> X" shows "(\<Sum>i\<in>U. i * f i) \<in> X"
ipath_mkTree_saturated: fixes s :: "'state"   and rs :: "'rule stream"   and steps :: "('state \<times> 'rule) stream"   and r :: "'rule" assumes "s \<in> S"   and "fair rs"   and "ipath (mkTree rs s) steps"   and "r \<in> R" shows "saturated r steps"
trancl_unfold_right: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>+ = r\<^sup>* O r"
cbox_interval: fixes a :: "'e"   and b :: "'e" shows "cbox a b = {a..b}"
inverse_fds_def: fixes f :: "'b fds" shows "inverse f = (\<chi>n. dirichlet_inverse (($) f) (inverse (f $ 1)) n)"
assms_2: shows "v \<preceq>\<^sub>t u"
finite_measure_lebesgue_on: fixes S :: "'b set" assumes "S \<in> lmeasurable" shows "finite_measure (lebesgue_on S)"
set_child_nodes_typess_preserved: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs"   and "w \<in> set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
base: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "f x \<in> orbit f x"
Union_quotient: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "equiv A r" shows "\<Union> (A // r) = A"
left: fixes x :: "'a"   and y :: "'b" assumes "x \<in> carrier \<X>\<^bsub>G\<^esub>"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>"   and "x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y"
return_env_def: fixes return :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "return_env return x = EnvT (\<lambda>_. return x)"
assms_3: shows "ini.safe_reach_direct program_step safe_free_flowing (ts, m, \<S>)"
VSEqualityBlocks: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSEqualityBlocks \<phi>) xs = eval \<phi> xs"
Ord_trans: fixes i :: "V"   and j :: "V"   and k :: "V" assumes "i \<in>\<^sub>\<circ> j"   and "j \<in>\<^sub>\<circ> k"   and "Ord k" shows "i \<in>\<^sub>\<circ> k"
fresh_star_upds_list: fixes as :: "Vars.var set"   and S :: "stack_elem list" assumes "atom ` as \<sharp>* S" shows "atom ` as \<sharp>* upds_list S"
adjoin_subgroup: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "subgroup (adjoin (G\<lparr>carrier := adjoin G H a\<rparr>) H a) (G\<lparr>carrier := adjoin G H a\<rparr>)"
swap_cs_Send_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"