hcju/isabelleps huggingface dataset:
Extend_commute_safe: fixes j :: "nat"   and i :: "nat"   and k :: "'k"   and \<AA> :: "'i"   and P :: "'v"   and Q :: "'v" assumes "j \<le> i"   and "LEQ k i (#\<^sub>V \<AA>)" shows "Extend k j (Extend k i \<AA> P) Q = Extend k (Suc i) (Extend k j \<AA> Q) P"
mini_maxi_nths_le: fixes xs :: "'c list"   and n :: "nat"   and key :: "'c \<Rightarrow> 'd" assumes "length xs \<le> Suc (Suc n)" shows "length (nths xs (- {mini xs key, maxi xs key})) \<le> n"
assms_1: shows "degree f + (degree g - J) \<le> k"
Rep_pdata_select: fixes D :: "'b pdata" shows "(PDataValue D, PDataSpace D) \<in> pdata"
dvd_mult_cancel_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times c a) (times c b) = (c = zero \<or> dvd.dvd times a b)"
aux: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "a \<in> set (removeShadowRules2 p)" shows "a \<in> set p"
pr0_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.comp (hfsetcat.pr0 (cod f) (cod g)) (hfsetcat.tuple f g) = g"
ret_le_down_conv: fixes m :: "'b nres"   and c :: "'c"   and R :: "('c \<times> 'b) set" assumes "nofail m" shows "(RETURN c \<le> \<Down> R m) = (\<exists>a. (c, a) \<in> R \<and> RETURN a \<le> m)"
inv_equality: fixes y :: "'a \<times> 'c"   and x :: "'a \<times> 'c" assumes "y \<otimes>\<^bsub>G \<times>\<times> H\<^esub> x = \<one>\<^bsub>G \<times>\<times> H\<^esub>"   and "x \<in> carrier (G \<times>\<times> H)"   and "y \<in> carrier (G \<times>\<times> H)" shows "inv\<^bsub>G \<times>\<times> H\<^esub> x = y"
aff_dim_empty_eq: shows "aff_dim {} = - 1"
rel_eq_onp: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "rel_gpv (eq_onp P1) (eq_onp P2) = eq_onp (pred_gpv P1 P2)"
bij_lens_pbij: fixes x :: "'b \<Longrightarrow> 'c" assumes "bij_lens x" shows "pbij_lens x"
simps_2: fixes n :: "nat"   and x :: "'b" shows "replicate (Suc n) x = x # replicate n x"
\<tau>move1Var: fixes P :: "'a prog"   and h :: "'heap"   and V :: "'b" shows "\<tau>move1 P h (Var V)"
rel_resumption_outputD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and r1 :: "('f, 'h, 'j) Resumption.resumption"   and r2 :: "('g, 'i, 'j) Resumption.resumption" assumes "Resumption.resumption.rel_resumption A B r1 r2"   and "\<not> Resumption.resumption.is_Done r1" shows "B (Resumption.resumption.output r1) (Resumption.resumption.output r2)"
subspace_UNIV: shows "local.subspace U\<^sub>V\<^sub>S"
rel_refl_strong: fixes x :: "'e \<rightharpoonup>\<^sub>f 'd"   and Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z. z \<in> fmran' x \<Longrightarrow> Ra z z" shows "fmrel Ra x x"
obsv_round_refines: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and dec_f :: "process \<Rightarrow> val option"   and Ob :: "process set" shows "{obsv_ref_rel \<inter> UNIV \<times> OV_inv1} sv_round r S v dec_f, obsv_round r S v dec_f Ob {> obsv_ref_rel}"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "local.normalize (v # va) = (if lead_coeff (v # va) \<noteq> \<zero> then v # va else local.normalize (tl (v # va)))"
cblinfun_cinner_eqI: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "\<And>\<psi>. \<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) = \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>)" shows "A = B"
between_triv2: fixes a :: "'b"   and b :: "'b" shows "between (a, b) b"
subseq_drop_many: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "subseq xs ys" shows "subseq xs (zs @ ys)"
simps_49: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "safe_formula (formula.Until \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of formula.Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
reciprocal_involution: fixes z :: "complex_homo" shows "reciprocal (reciprocal z) = z"
simps: fixes F :: "'b formula"   and H :: "'b formula" shows "interpolate F H = (let K = atoms F - atoms H in if K = {} then F else let k = Min K in interpolate (F[\<top>/k] \<^bold>\<or> F[\<bottom>/k]) H)"
is_Ref_def2: fixes v :: "'a val" shows "is_Ref v = (v = Null \<or> (\<exists>a. v = Addr a))"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b pmf"   and y :: "'c pmf"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "rel_pmf R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf x; yb \<in> set_pmf y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_pmf Ra x y"
split_min_set: fixes t :: "('a \<times> 'b) tree"   and m :: "'a"   and t' :: "('a \<times> 'b) tree" assumes "split_min t = (m, t')"   and "t \<noteq> \<langle>\<rangle>" shows "m \<in> Tree2.set_tree t \<and> Tree2.set_tree t = {m} \<union> Tree2.set_tree t'"
hom_restrict: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme"   and t :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "h \<in> hom G H"   and "\<And>g. g \<in> carrier G \<Longrightarrow> h g = t g" shows "t \<in> hom G H"
sum_atLeastAtMost_int_last: fixes a :: "int"   and b :: "int"   and f :: "int \<Rightarrow> 'a" assumes "a < b" shows "sum f {a..b} = sum f {a..<b} + f b"
lincomb_index: fixes i :: "nat"   and X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < nc"   and "X \<subseteq> mat_kernel (A * B)" shows "AB.lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
wfp_on_UNIV: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfp_on P UNIV = wfP P"
sup_ty_opt_trans: 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 a :: "ty err"   and b :: "ty err"   and c :: "ty err" assumes "P \<turnstile> a \<le>\<^sub>\<top> b"   and "P \<turnstile> b \<le>\<^sub>\<top> c" shows "P \<turnstile> a \<le>\<^sub>\<top> c"
bl_to_bin_append: fixes bs :: "bool list"   and cs :: "bool list" shows "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)"
valid_strategy_supergame: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and V' :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>"   and "ParityGame.strategy (subgame V') p \<sigma>'"   and "\<And>v. v \<in> V' \<Longrightarrow> \<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w)" shows "strategy p (override_on \<sigma> \<sigma>' V')"
abs_power_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.ring_1_abs_real_line plus zero minus less_eq less times abs uminus one" shows "less_eq (power.power one times (abs a) n) (power.power one times (abs b) n) = (n = 0 \<or> less_eq (abs a) (abs b))"
homeomorphism_st_proj1: shows "homeomorphism (UNIV - {top_sphere}) UNIV st_proj1 st_proj1_inv"
tuple_in_None_None: fixes as :: "'a option list" assumes "Mapping.lookup tuple_in as = None" shows "Mapping.lookup tuple_in' as = None"
that: fixes i :: "nat"   and s'' :: "char list \<Rightarrow> val option"   and j :: "nat" assumes "\<langle>c\<^sub>1,s\<rangle> \<rightarrow>\<^bsup>i\<^esup> \<langle>Skip,s''\<rangle>"   and "\<langle>c\<^sub>2,s''\<rangle> \<rightarrow>\<^bsup>j\<^esup> \<langle>Skip,s'\<rangle>"   and "n = i + j + 1" shows "thesis"
vf\<^sub>1: shows "vertices f\<^sub>1 = rev vs @ u # between (vertices f) u v @ [v]"
left_unique_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
distinct_209: fixes x81 :: "floatarith"   and x82 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "floatarith.Max x81 x82 \<noteq> Power x151 x152"
map_pre_coG\<^sub>M_def: shows "map_pre_coG\<^sub>M \<equiv> \<lambda>f1 f2 f3. BNF_Composition.id_bnf \<circ> map_coG0 f1 f3 \<circ> BNF_Composition.id_bnf"
has_integral_null_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "content {a..b} = 0" shows "(f has_integral (0::'a)) {a..b}"
not_zero_i_suc_k: shows "\<not> is_zero_row_upt_k i (Suc k) B"
kuhn_counting_lemma: fixes F :: "'a set"   and S :: "'b set"   and bnd :: "'a \<Rightarrow> bool"   and face :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and compo :: "'b \<Rightarrow> bool"   and compo' :: "'a \<Rightarrow> bool" assumes "finite F"   and "finite S"   and "\<And>f. \<lbrakk>f \<in> F; bnd f\<rbrakk> \<Longrightarrow> card {s \<in> S. face f s} = 1"   and "\<And>f. \<lbrakk>f \<in> F; \<not> bnd f\<rbrakk> \<Longrightarrow> card {s \<in> S. face f s} = 2"   and "\<And>s. \<lbrakk>s \<in> S; compo s\<rbrakk> \<Longrightarrow> card {f \<in> F. face f s \<and> compo' f} = 1"   and "\<And>s. \<lbrakk>s \<in> S; \<not> compo s\<rbrakk> \<Longrightarrow> card {f \<in> F. face f s \<and> compo' f} = 0 \<or> card {f \<in> F. face f s \<and> compo' f} = 2"   and "odd (card {f \<in> F. compo' f \<and> bnd f})" shows "odd (card {s \<in> S. compo s})"
last_vec_two_block_non_inter_is_last_vec: fixes A :: "'a mat"   and B :: "'a mat"   and v :: "'a vec"   and n :: "nat" assumes "dim_col A + dim_col B = dim_vec v"   and "dim_row B = n" shows "vec_last (two_block_non_interfering A B *\<^sub>v v) n = B *\<^sub>v vec_last v (dim_col B)"
Units_r_inv_ex: fixes x :: "'a mat" assumes "x \<in> Units (ring_mat TYPE('a) n ())" shows "\<exists>y\<in>carrier (ring_mat TYPE('a) n ()). x \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> y = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>"
Sigma_mset_empty2: fixes A :: "'b multiset" shows "A \<times># {#} = {#}"
verts_add_arc_conv: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "verts (pre_digraph.add_arc G a) = verts G \<union> {tail G a, head G a}"
inj_image_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj f" shows "(f ` A = f ` B) = (A = B)"
characteriseSeq: fixes C :: "('a, 'b) sequent" shows "\<exists>A B. C = ( A \<Rightarrow>* B)"
Levellist_unique: fixes hds :: "ref list"   and "next" :: "ref \<Rightarrow> ref"   and ll :: "ref list list"   and ll' :: "ref list list" assumes "Levellist hds next ll"   and "Levellist hds next ll'" shows "ll = ll'"
H: shows "\<Delta> @ \<Gamma> \<turnstile> P <: Q"
sccs_verts_disjoint: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> G.sccs_verts"   and "T \<in> G.sccs_verts"   and "S \<noteq> T" shows "S \<inter> T = {}"
UNIF1_vars_mset_leq: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "vars_mset E' \<subseteq> vars_mset E"
G'_gba_fields: shows "gba_L G' = (\<lambda>q l. if q \<in> f ` V then L (fi q) l else False)"
xyz: shows "\<forall>w. P w (x \<otimes> (y \<otimes> z)) = (P w x \<and> P w (y \<otimes> z))"
frechet_derivative_def: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" shows "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
card_finite: fixes S :: "'a set" assumes "card S = CARD('b)" shows "finite S"
proj_poly_syz_plus: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "proj_poly_syz n (p + q) = proj_poly_syz n p + proj_poly_syz n q"
g_def: fixes i :: "'a" shows "g i = (if i \<in> U then f i else (0::'a))"
card_of_ordIso_subst: fixes A :: "'d set"   and B :: "'d set" assumes "A = B" shows "|A| =o |B|"
swap_events_perm: fixes i :: "nat"   and j :: "nat"   and t :: "'d list" assumes "i < j"   and "j < length t" shows "swap_events i j t <~~> t"
mult_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c * a = c * b) = (c = (0::'a) \<or> a = b)"
ide_is_iso: fixes a :: "'b" assumes "D.ide a" shows "D.iso a"
filter_min_subset: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (General.filter_min rel xs) \<subseteq> set xs"
coboundedI: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "a \<in> A" shows "a \<preceq> ordered_powerprod_lin.Max A"
solves_ODE_def: fixes I :: "interp"   and F :: "real \<Rightarrow> variable \<Rightarrow> real"   and x :: "char"   and \<theta> :: "trm" shows "solves_ODE I F x \<theta> \<equiv> \<forall>\<zeta>. Vagree (F 0) (F \<zeta>) (- {RVar x, DVar x}) \<and> F \<zeta> (DVar x) = deriv (\<lambda>t. F t (RVar x)) \<zeta> \<and> F \<zeta> (DVar x) = term_sem I \<theta> (F \<zeta>)"
prj_simps_9: shows "C.cod fgf.Prj\<^sub>0 = f.apex"
mono2: fixes x :: "'b" shows "monotone ordb (\<le>) (f x)"
scalar_coeff: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "'c list" assumes "ring R"   and "a \<in> carrier R" shows "ring.coeff R (map ((\<otimes>\<^bsub>R\<^esub>) a) p) = (\<lambda>i. a \<otimes>\<^bsub>R\<^esub> ring.coeff R p i)"
less_imp_neq: fixes x :: "'a word"   and y :: "'a word" assumes "x <s y" shows "x \<noteq> y"
Diff: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> sigma_sets UNIV ((\<lambda>(a, i). {x. x \<bullet> i < a}) ` (UNIV \<times> A))"   and "b \<in> sigma_sets UNIV ((\<lambda>(a, i). {x. x \<bullet> i < a}) ` (UNIV \<times> A))" shows "a - b \<in> sigma_sets UNIV ((\<lambda>(a, i). {x. x \<bullet> i < a}) ` (UNIV \<times> A))"
iterate_Suc2: fixes n :: "nat"   and F :: "'a \<rightarrow> 'a"   and x :: "'a" shows "iterate (Suc n)\<cdot>F\<cdot>x = iterate n\<cdot>F\<cdot>(F\<cdot>x)"
mult_sheaf_spec_extensional: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and a :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and b :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.mult_sheaf_spec R addition multiplication zero U a b \<in> extensional U"
sel_4: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "fun (App x21 x22) = x21"
lincomb_sum_left: fixes rs :: "'r list"   and ss :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ss \<subseteq> R"   and "set ms \<subseteq> M"   and "length rs \<le> length ss" shows "map2 (+) rs ss \<bullet>\<cdot> ms = rs \<bullet>\<cdot> ms + take (length rs) ss \<bullet>\<cdot> ms"
obs_intra_singleton_disj: fixes n :: "'node"   and S :: "'node SDG_node set" assumes "valid_node n" shows "(\<exists>m. obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {m}) \<or> obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"
proj_orig_basis_nth: fixes i :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and n :: "nat" assumes "i < length bs" shows "proj_orig_basis n bs ! i = proj_poly_syz n (bs ! i)"
UN_equiv_class: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b set"   and a :: "'a" assumes "equiv A r"   and "f respects r"   and "a \<in> A" shows "\<Union> (f ` r `` {a}) = f a"
b_trans: fixes s :: "'a"   and b :: "'a list"   and T :: "('a, 'b) TS_scheme"   and t :: "'a" assumes "s # b \<in> beh T"   and "(s, t) \<in> TS.trans T" shows "t # s # b \<in> beh T"
nbt_subset_carrier: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b" assumes "rational_preference carrier relation"   and "x \<in> carrier" shows "no_better_than x carrier relation \<subseteq> carrier"
case_2: fixes f1 :: "'x"   and f2 :: "'y \<Rightarrow> 'x"   and x2 :: "'y" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
Trans_msg: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isTrans ev" shows "msgs c i = msgs c' i"
counit_simps_5: shows "trg \<xi> = trg f"
returnK_def: shows "returnK \<equiv> \<Lambda> x s f. s\<cdot>x\<cdot>f"
f2: assumes "rk {R, a} = 1" shows "rk {P, Q, R, A', A, B} \<le> 3"
Suc_rtos_div_r_1: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "Suc (rtos r s i) < r * s"   and "i mod s = s - Suc 0"   and "i div s < r - Suc 0" shows "Suc (rtos r s i) div r = i mod s"
STCalWB_def: fixes Source :: "('a, 'c) calculusWithBarbs"   and Target :: "('b, 'c) calculusWithBarbs" shows "STCalWB Source Target \<equiv> \<lparr>Calculus = STCal (Calculus Source) (Calculus Target), HasBarb = \<lambda>P a. (\<exists>SP. P = SourceTerm SP \<and> SP\<down><Source>a) \<or> (\<exists>TP. P = TargetTerm TP \<and> TP\<down><Target>a)\<rparr>"
rpr_less_imp_neq: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c" assumes "x \<^bsub>r\<^esub>\<prec> y" shows "x \<noteq> y"
delete_edges_correct: fixes G :: "'b list_graph"   and E :: "('b \<times> 'b) list" shows "FiniteGraph.delete_edges (list_graph_to_graph G) (set E) = list_graph_to_graph (FiniteListGraph.delete_edges G E)"
pmf_eqI: fixes M :: "'d pmf"   and N :: "'d pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
runit_tensor': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<r>\<^sub>1[a \<otimes>\<^sub>1 b] \<cdot> \<a>\<^sub>1\<^sup>-\<^sup>1[a, b, \<I>\<^sub>1] = a \<otimes>\<^sub>1 \<r>\<^sub>1[b]"
msg_zhops_def: fixes m :: "msg" shows "msg_zhops m \<equiv> case m of Rreq hopsc dipc x xa oipc xb sipc xc \<Rightarrow> hopsc = 0 \<longrightarrow> oipc = sipc | Rrep hopsc dipc x xa sipc \<Rightarrow> hopsc = 0 \<longrightarrow> dipc = sipc | _ \<Rightarrow> True"
subcategory_axioms: shows "subcategory (\<cdot>\<^sub>C) Arr"
RES_bind_choose: fixes x :: "'b"   and X :: "'b set"   and m :: "'c nres"   and f :: "'b \<Rightarrow> 'c nres" assumes "x \<in> X"   and "m \<le> f x" shows "m \<le> RES X \<bind> f"
vimage_eq: fixes a :: "real" shows "X -` {..a} \<inter> space M = {x \<in> space M. X x \<le> a}"
skel_swap: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "good X" shows "skel (X #[x1 \<and> x2]_xs) = skel X"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c Predicate.pred \<Rightarrow> 'b"   and f3 :: "'c Predicate.pred \<Rightarrow> 'c Predicate.seq \<Rightarrow> 'b" shows "(case seq.Empty of seq.Empty \<Rightarrow> f1 | seq.Insert x xa \<Rightarrow> f2 x xa | seq.Join x xa \<Rightarrow> f3 x xa) = f1"
ends_in_segment_2: fixes b :: "'a"   and a :: "'a" shows "b \<in> closed_segment a b"
assms_2: fixes k :: "'a"   and x :: "'b" shows "f k x (0::'c) = x"
add_delete_edge: fixes G :: "('c, 'd) graph"   and a :: "'c"   and w :: "'d"   and c :: "'c" assumes "valid_graph G"   and "(a, w, c) \<in> edges G" shows "add_edge a w c (delete_edge a w c G) = G"
prl2: fixes prc :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and prl :: "'a \<Rightarrow> bool"   and g :: "'a"   and f :: "'a" assumes "precedence prc prl"   and "prl g" shows "fst (prc g f) = False"
uint_nat: fixes w :: "'b word" shows "uint w = int (unat w)"
swapLeaves_id_when_notin_alphabet: fixes a :: "'a"   and t :: "'a tree"   and w :: "nat"   and w' :: "nat" assumes "a \<notin> alphabet t" shows "swapLeaves t w a w' a = t"
distinct_1: fixes x1 :: "'f" shows "Node x1 \<noteq> NewEntry"
simple_path_join_loop: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "arc g1"   and "arc g2"   and "pathfinish g1 = pathstart g2"   and "pathfinish g2 = pathstart g1"   and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" shows "simple_path (g1 +++ g2)"
bij_betw_image_Pow: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Pow A) (Pow B)"
indep_card_eq_dim_span: fixes B :: "('b, 'c) vec set" assumes "vec.independent B" shows "finite B \<and> card B = vec.dim (vec.span B)"
restrict_compose_in_diff_fun_space: fixes x :: "'e \<Rightarrow> real" assumes "x \<in> d23.dest.diff_fun_space" shows "restrict0 d12.dest.carrier (x \<circ> g) \<in> d12.dest.diff_fun_space"
expl_cond_exp_diff: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> real"   and Z :: "'c \<Rightarrow> real"   and Y :: "'c \<Rightarrow> 'd" assumes "integrable M X"   and "integrable M Z" shows "\<forall>w\<in>space M. expl_cond_expect M Y (\<lambda>x. X x - Z x) w = expl_cond_expect M Y X w - expl_cond_expect M Y Z w"
sets_lim: shows "P.events = sets (Pi\<^sub>M UNIV (\<lambda>_. borel))"
neg_less_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less a (divide b c) = less b (times a c)"
orb_a_eq: shows "orbit (edge_succ M) a = out_arcs G v"
fmempty_transfer: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "fmrel P fmempty fmempty"
exp_sum: fixes I :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "finite I" shows "exp (sum f I) = (\<Prod>x\<in>I. exp (f x))"
impl_of_inject: fixes x :: "('c, 'd) trie"   and y :: "('c, 'd) trie" shows "(impl_of x = impl_of y) = (x = y)"
frontier_cbox: fixes a :: "'b"   and b :: "'b" shows "frontier (cbox a b) = cbox a b - box a b"
ereal_real': fixes x :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "ereal (real_of_ereal x) = x"
gcs_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs_powerprod.gcs plus minus lcs s t = minus (plus s t) (lcs s t)"
sconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P,E \<turnstile> s \<surd> \<equiv> let (h, l) = s in P \<turnstile> h \<surd> \<and> P,h \<turnstile> l (:\<le>) E"
simps_4: fixes t :: "nat"   and f :: "nat \<Rightarrow> nat"   and e :: "expr" shows "valid_sops_expr t (Unop f e) = valid_sops_expr t e"
NormalAction: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" shows "t \<turnstile> ((Running, x), m) -convert_TA_initial (convert_obs_initial ta)\<rightarrow>i ((Running, x'), m')"
cases: fixes a :: "('a, 'b) obs_event action" assumes "is_write_action a" obtains ad :: "'a" and hT :: "htype" where "a = NormalAction (NewHeapElem ad__ hT)"   | ad :: "'a" and al :: "addr_loc" and v :: "'a val" where "a = NormalAction (WriteMem ad al v)"