hcju/isabelleps huggingface dataset:
union_diff2: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) = sum g (A - B) + sum g (B - A) + sum g (A \<inter> B)"
WBT_axioms: shows "WBT 25 10 14 10"
nargs_s_sa_ss: shows "enat (length (sa # ss)) \<le> arity s"
finite_set_avoid: fixes S :: "'a set"   and a :: "'a" assumes "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
qstar_comm: fixes x :: "'b \<times> 'c" shows "times_prod' x (prod.qstar x) = times_prod' (prod.qstar x) x"
le_iff_sup: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
assms_1: shows "inj_on f (set xs \<union> set ys)"
canon_ang_plus_pi1: fixes \<alpha> :: "real" assumes "0 < \<alpha>"   and "\<alpha> \<le> 2 * pi" shows "\<downharpoonright>\<alpha> + pi\<downharpoonleft> = \<alpha> - pi"
choose_dvd: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "fact k * fact (n - k) dvd fact n"
mk_v_concrete: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and sol :: "(real, 'a) vec" shows "mk_v I ODE \<nu> sol = (\<chi>i. (if Inl i \<in> semBV I ODE then sol else fst \<nu>) $ i, \<chi>i. (if Inr i \<in> semBV I ODE then ODE_sem I ODE sol else snd \<nu>) $ i)"
ideal_subset1: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "I \<subseteq> carrier R"
assms: fixes a :: "V"   and b :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "[b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
bounded_err_stepI: fixes n :: "nat"   and ap :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list" assumes "\<forall>p<n. \<forall>s. ap p s \<longrightarrow> (\<forall>(q, s')\<in>set (step p s). q < n)" shows "bounded (err_step n ap step) n"
frange_iff: fixes b :: "V"   and r :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r) = (\<exists>a. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
op_nres_ASSERT_bnd_def: fixes \<Phi> :: "bool"   and m :: "'b nres" shows "op_nres_ASSERT_bnd \<Phi> m \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. m)"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (\<rr> a)"
mm_equiv_sym: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" shows "\<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle> \<approx> \<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle>"
coeffs_1_eq: shows "coeffs 1 = [1::'b]"
step: fixes xs :: "'a llist"   and x :: "'a" assumes "xs \<in> A\<^sup>\<star>"   and "P xs"   and "x \<in> A" shows "P (xs @@ x ## LNil)"
simps_2: fixes x :: "'a"   and l :: "'a tree"   and y :: "'a"   and r :: "'a tree" shows "del_tm x \<langle>l, y, r\<rangle> =1 (case cmp x y of LT \<Rightarrow> del_tm x l \<bind> (\<lambda>l'. return (up_d y r False l')) | EQ \<Rightarrow> if r = \<langle>\<rangle> then return (Some l) else split_min_tm r \<bind> (\<lambda>(a', r'). return (Some \<langle>l, a', r'\<rangle>)) | GT \<Rightarrow> del_tm x r \<bind> (\<lambda>r'. return (up_d y l True r')))"
wt_elim_cases_14: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e0 :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (e0 ? e1 : e2)\<Colon>T" obtains T1 :: "ty" and T2 :: "ty" and T :: "ty" where "Tb__ = Inl T" and "E,dt\<Turnstile>e0\<Colon>-PrimT Boolean" and "E,dt\<Turnstile>e1\<Colon>-T1" and "E,dt\<Turnstile>e2\<Colon>-T2" and "prg E\<turnstile>T1\<preceq>T2 \<and> T = T2 \<or> prg E\<turnstile>T2\<preceq>T1 \<and> T = T1"
obtains_step_append: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r' :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and s'' :: "nat"   and ta :: "transition"   and l :: "String.literal"   and i :: "value list" assumes "obtains s r e s' r' t"   and "(s'', ta) |\<in>| possible_steps e s r l i" shows "obtains s'' (evaluate_updates ta i r) e s' r' (t @ [(l, i)])"
Polys_mono: fixes X :: "'b set"   and Y :: "'b set" assumes "X \<subseteq> Y" shows "P[X] \<subseteq> P[Y]"
eigen_vector_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('b, 'c) Finite_Cartesian_Product.vec"   and ev :: "'b" shows "eigen_vector A v ev = (v \<noteq> 0 \<and> A *v v = ev *s v)"
assms_2: shows "pseudonaturally_equivalent V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G H \<Phi>\<^sub>H"
append_insert: fixes xs :: "'a list"   and x :: "'a" shows "set (xs @ [x]) = insert x (set xs)"
length_removeAt: fixes i :: "nat"   and ls :: "'a list" assumes "i < length ls" shows "length (removeAt i ls) = length ls - 1"
ennreal_escape_rate: fixes x :: "'a" shows "ennreal (escape_rate x) = \<integral>\<^sup>+ xa. ennreal (R x xa) \<partial>count_space UNIV"
Root: shows "Q' = (\<Prod>i<n. [:- Root i, 1:])"
NR\<^sub>n_NH\<^sub>N: fixes p :: "real"   and n :: "nat"   and u :: "nat" shows "random_skip_list.NR\<^sub>N p n u 0 = of_nat u * random_skip_list.NH\<^sub>N p n"
assms_1: shows "P \<Longrightarrow>\<tau> \<prec> P'"
timpls_transformable_to_pred_inv_1: fixes A :: "'e set set"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to_pred A (Fun f T) t" shows "is_Fun t"
rep_eq: shows "Linear_Aggregation_Algebras.unit.Rep_unit bot = 0"
cases: fixes T :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b list" assumes "enumerateable T less"   and "enumerateable.sorted less a" obtains "a = []"   | xs :: "'b list" and x :: "'b" where "a = x # xs" and "\<forall>y\<in>set xs. less x y" and "enumerateable.sorted less xs"
lt_dbm_lt: fixes a :: "'c"   and b :: "'c" assumes "Lt a \<preceq> Lt b" shows "a \<le> b"
rec: fixes f :: "'b list \<Rightarrow> 'a"   and x :: "'b list" shows "rec_iarray f (IArray x) = f x"
seqI': fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and a :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and b :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and c :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>S c\<guillemotright>" shows "S.seq g f"
ex1_unique: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "\<exists>!x. P x"   and "P a"   and "P b" shows "a = b"
assms: shows "(t, t') \<in> (\<Union> (relations (mkM T) ` as))\<^sup>*"
db\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('e, 'f) stateful_strand_step list"   and I :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "db\<^sub>s\<^sub>s\<^sub>t S I \<equiv> db'\<^sub>s\<^sub>s\<^sub>t S I []"
assms_2: shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
connected_space_topspace_empty: fixes X :: "'c topology" assumes "topspace X = {}" shows "connected_space X"
llist_all2_lnullD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys" shows "lnull xs = lnull ys"
inv_in_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "local.iso f"   and "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>local.inv f : b \<rightarrow> a\<guillemotright>"
simps_5: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat" shows "exec_step_input P C M pc (Called []) = StepI (instrs_of P C M ! pc)"
sup_measure_def: fixes A :: "'b measure"   and B :: "'b measure" shows "sup A B = sup_lexord A B space (sigma (space A \<union> space B) {}) (sup_lexord A B sets (sigma (space A) (sets A \<union> sets B)) (sup_measure' A B))"
transitionCases: fixes P :: "pi"   and Rs :: "residual"   and F :: "residual \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^Rs"   and "P \<Longrightarrow>\<^sub>l Rs \<Longrightarrow> F Rs"   and "Rs = \<tau> \<prec> P \<Longrightarrow> F (\<tau> \<prec> P)" shows "F Rs"
defined_args_valid: fixes d\<^sub>x :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and d\<^sub>y :: "('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'c" assumes "profile_bin_scheme d\<^sub>x d\<^sub>y f g" shows "(\<tau> \<Turnstile> \<delta> f x y) = (\<tau> \<Turnstile> d\<^sub>x x \<and> \<tau> \<Turnstile> d\<^sub>y y)"
HcompNml_Nml: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg (t \<^bold>\<star> u)" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = t \<^bold>\<star> u"
\<phi>\<psi>: shows "internal_equivalence f f\<^sup>~ \<phi> \<psi>"
subsumption_def_alt: fixes t1 :: "transition"   and c :: "nat \<Rightarrow>f value option"   and t2 :: "transition" shows "subsumes t1 c t2 = (Label t2 = Label t1 \<and> Arity t2 = Arity t1 \<and> (\<forall>i. can_take_transition t2 i c \<longrightarrow> can_take_transition t1 i c) \<and> (\<forall>i. can_take_transition t2 i c \<longrightarrow> evaluate_outputs t2 i c = evaluate_outputs t1 i c) \<and> (\<forall>i. can_take_transition t2 i c \<longrightarrow> (\<forall>r' P. P (evaluate_updates t1 i c $ r') \<longrightarrow> evaluate_updates t2 i c $ r' = None \<or> P (evaluate_updates t2 i c $ r'))))"
delta_complexity: fixes d :: "'a"   and def :: "'a" assumes "(0::'a) < d"   and "d \<le> def" shows "mono_matrix_carrier (delta_gt d) def (delta_bound d) delta_mono"
to_nat_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep one = 1"
fmfilter_ran: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'c) fmap" shows "fmran (fmfilter P m) = fmimage m (ffilter P (fmdom m))"
remove_shadow_root_get_tag_name: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr :: "(_) element_ptr" assumes "w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr" shows "r h h'"
carrier_quotient_ring_iff: fixes X :: "('a \<times> 'a) set" shows "(X \<in> qr.carrier_quotient_ring) = qr.valid_frac X"
simps_3: fixes b :: "real"   and c :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalLinearUni b c (LeqUni p) = OrUni (allZero' p) (map_atomUni (linearSubstitutionUni b c) (convertDerivativeUni p))"
narrow2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<triangle> y \<le> x"
wave_bipartiteI: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "separating \<Gamma> (TER f)"   and "current \<Gamma> f" shows "wave \<Gamma> f"
stable_rank_unique: fixes x :: "nat"   and i :: "nat"   and j :: "nat" assumes "stable_rank x i"   and "stable_rank x j" shows "i = j"
geodesic_segment_between_real': fixes x :: "real"   and y :: "real" shows "{x--y} = {min x y..max x y}"
simps_2: fixes xs :: "nat list" shows "eval_wellf S xs \<down>= Suc (xs ! 0)"
cases: fixes r :: "'b dfa" obtains states :: "hf set" and init :: "hf" and final :: "hf set" and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf" where "r = \<lparr>dfa.states = states, init = init, final = final, nxt = nxt\<rparr>"
consistent_def: fixes M :: "('c, 'd) message set" shows "consistent M \<equiv> inj_on I (insert_messages M) \<and> \<Union> (deps ` M) \<subseteq> I ` insert_messages M \<and> wfP (depends_on (insert_messages M)) \<and> (\<exists>a. a_conditions (insert_messages M) a)"
ALG'_det_append: fixes n :: "nat"   and Strat :: "(nat \<times> nat list) list"   and qs :: "'c list"   and a :: "'c list"   and init :: "'c list"   and x :: "'c" assumes "n < length Strat"   and "n < length qs" shows "ALG'_det Strat (qs @ a) init n x = ALG'_det Strat qs init n x"
notStuck_CondFalseD: 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 \<notin> b" shows "\<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
Amin_leTr: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" shows "f \<in> {i. i \<le> n} \<rightarrow> Z\<^sub>\<infinity> \<longrightarrow> (\<forall>j\<in>{i. i \<le> n}. Amin n f \<le> f j)"
has_contour_integral_linepath_Reals_iff: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex"   and I :: "complex" assumes "a \<in> \<real>"   and "b \<in> \<real>"   and "Re a < Re b" shows "(f has_contour_integral I) (linepath a b) = ((\<lambda>x. f (complex_of_real x)) has_integral I) {Re a..Re b}"
simps_4: fixes P :: "interface"   and D :: "sdiagram"   and E :: "sdiagram"   and Q :: "interface" shows "com_cell (Choose_sdia P D E Q) = Choose (com_sdia D) (com_sdia E)"
decode_DIMACS_model_def: fixes dimacs_M :: "int list"   and h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "decode_DIMACS_model dimacs_M h prob \<equiv> ast_problem.decode_abs_plan prob (rem_noops (map (strips_op_to_sasp (prob_with_noop (ast_problem.abs_prob prob))) (concat (\<Phi>\<inverse> \<phi> prob_with_noop (ast_problem.abs_prob prob)  dimacs_model_to_abs dimacs_M (\<lambda>_. False) \<circ> cnf_to_dimacs.var_to_dimacs (Suc h) (Suc (length (ast_problem.ast\<delta> prob))) h))))"
simps_2: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "hensel p f a" shows "hensel.newton_seq p f a (Suc n) = hensel.newton_step p f (hensel.newton_seq p f a n)"
index_mult_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and B :: "'b mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = Matrix.row A i \<bullet> col B j"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list" shows "take n (take m xs) = take (min n m) xs"
clm_sing: fixes c :: "cars" shows "\<forall>ts v. \<nexists>va u. v=va--u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> len u ts c = ext u \<and> restrict u (clm ts) c = lan u \<and> |lan u| = 1"
double_in_nonpos_Ints_imp: fixes z :: "'a" assumes "(2::'a) * z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "z \<in> \<int>\<^sub>\<le>\<^sub>0 \<or> z + (1::'a) / (2::'a) \<in> \<int>\<^sub>\<le>\<^sub>0"
IH: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "Fvars (rawpsubst \<phi> txs) \<subseteq> Fvars \<phi> - snd ` set txs \<union> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
Ri_prop: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'b" assumes "mono f" shows "(x \<le> f y) = (\<forall>z. Ri x y z \<le> f z)"
simps_28: shows "cospan_shape.comp cospan_shape.Null cospan_shape.BB = cospan_shape.Null"
HNatInfinite_not_Nats_iff: fixes x :: "nat star" shows "(x \<in> HNatInfinite) = (x \<notin> \<nat>)"
SeqHRP_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm"   and s :: "tm"   and s' :: "tm"   and k :: "tm"   and k' :: "tm" assumes "H \<turnstile> x EQ x'"   and "H \<turnstile> y EQ y'"   and "H \<turnstile> s EQ s'"   and "H \<turnstile> k EQ k'" shows "H \<turnstile> SeqHRP x y s k IFF SeqHRP x' y' s' k'"
UPD_IDX_def: fixes av :: "int \<Rightarrow> int"   and i :: "int"   and pv :: "int" shows "UPD_IDX av i pv \<equiv> av(i := pv)"
preserves_lunit_1: fixes f :: "'a" assumes "C.ide f" shows "F \<l>\<^sub>C[f] = \<l>\<^sub>D[F f] \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))"
leq_seq_def: fixes eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xp :: "'b Predicate.seq"   and Q :: "'b Predicate.pred" shows "leq_seq eq xp Q = leq_pred eq (Predicate.pred_of_seq xp) Q"
ce_rel_lm_4: fixes A :: "nat set" assumes "ce_set_to_rel A \<in> ce_rels" shows "A \<in> ce_sets"
simps_4: fixes a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "\<T>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e) = Terms.Let (map_transform Aeta_expand (snd (a, Aheap \<Gamma> e\<cdot>a)) (map_transform transform_syn (snd (a, Aheap \<Gamma> e\<cdot>a)) \<Gamma>)) (\<T>\<^bsub>fst (a, Aheap \<Gamma> e\<cdot>a)\<^esub> e)"
decr_ev: fixes js :: "'ord stream"   and m :: "'ord" assumes "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"   and "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js"   and "(shd js, m) \<in> Ord" shows "ev (\<lambda>js. (shd js, m) \<in> Ord \<and> shd js \<noteq> m) js"
vors_4: shows "vs1 = between (vertices f) ram1 ram2"
prems_2: shows "Suc e \<le> unat (- 1)"
pole_rep_non_zero: fixes l :: "proj2_line" shows "M *v proj2_line_rep l \<noteq> 0"
simps_1: fixes r :: "real"   and rs :: "real list" shows "is_Less (Less r rs) = True"
is_natural_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c"   and G :: "'a \<Rightarrow> 'c"   and \<tau> :: "'a \<Rightarrow> 'c"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
closest_pair_rec_c0_c1: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length xs"   and "distinct xs"   and "(ys, c\<^sub>0, c\<^sub>1) = closest_pair_rec xs" shows "c\<^sub>0 \<in> set xs \<and> c\<^sub>1 \<in> set xs \<and> c\<^sub>0 \<noteq> c\<^sub>1"
Collect_mem_eq: fixes A :: "'c set" shows "_Collect x (x \<in> A) = A"
rep_eq: fixes x :: "'a ccsubspace" shows "(*\<^sub>V) (Proj x) = projection (space_as_set x)"
hyps_2: shows "P' \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P''"
cbelow_on_def: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "cbelow_on A ord z rel a b \<equiv> a = b \<and> b \<in> A \<and> ord b z \<or> cbelow_on_1 A ord z rel a b"
r_both_2: fixes i :: "nat"   and f :: "recf"   and j :: "nat"   and g :: "recf"   and t :: "nat"   and x :: "nat" assumes "\<And>x. eval r_phi [i, x] = eval f [x]"   and "\<And>x. eval r_phi [j, x] = eval g [x]"   and "eval r_result1 [t, i, x] \<down>= 0"   and "eval r_result1 [t, j, x] \<down>= 0" shows "eval r_both [t, i, j, x] \<down>= prod_encode (2, 0)"
cycle_free_diag_dest: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "cycle_free M n" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
rem_effectless_act_set_def: fixes A :: "('d \<times> ('e, 'f) fmap) set" shows "rem_effectless_act_set A \<equiv> {a \<in> A. fmdom' (snd a) \<noteq> {}}"
Val_semilattice_axioms: shows "Val_semilattice \<gamma>_parity num_parity plus_parity"
assms_1: shows "mat (rat_poly.row_length m1) (rat_poly.row_length m2) m1"
w0_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "vmc_path_no_deadend.w0 P \<equiv> lhd (ltl P)"
simps: fixes a :: "'a itrm" shows "(a \<in> CF) = ((\<exists>x. a = Pure x) \<or> (\<exists>f x. a = f \<diamondop> Opaque x \<and> f \<in> CF))"
fds_nth_mult_Suc_0: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_nth (f * g) (Suc 0) = fds_nth f (Suc 0) * fds_nth g (Suc 0)"
map_tpOfV_getVars: fixes \<sigma>l :: "'tp list" shows "map TE.tpOfV (TE.getVars \<sigma>l) = \<sigma>l"
regular_cover_3: shows "locally_finite_on carrier N (domain \<circ> \<psi>)"
tdghm_NTDom: shows "\<NN>\<lparr>NTDom\<rparr> = \<FF>"
normalize_non_0_smult: fixes p :: "'a poly" shows "\<exists>a. a \<noteq> (0::'a) \<and> smult a (primitive_part p) = p"
kruskal3_ref_spec': shows "(uncurry0 kruskal3, uncurry0 (SPEC (\<lambda>r. MSF (\<alpha> ` set r)))) \<in> unit_rel \<rightarrow>\<^sub>f \<langle>Id\<rangle>nres_rel"
results_gpv_\<I>_full: shows "results_gpv \<I>_full = results'_gpv"
Koszul_syz_sigs_is_syz_sig: fixes v :: "'t" assumes "dickson_grading dgrad"   and "distinct fs"   and "0 \<notin> set fs"   and "v \<in> set (Koszul_syz_sigs fs)" shows "is_syz_sig dgrad v"
parent_child_rel_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel" shows "parent_child_rel = (\<lambda>h. {(parent, child). parent |\<in>| object_ptr_kinds h \<and> child \<in> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` set |h \<turnstile> get_child_nodes parent|\<^sub>r})"
ln_float_bounds_2: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 \<le> real_of_float x"   and "real_of_float x < 1" shows "ln (real_of_float x + 1) \<le> real_of_float (x * ub_ln_horner prec (get_odd n) 1 x)"