hcju/isabelleps huggingface dataset:
remdups_adj_rev: fixes xs :: "'c list" shows "remdups_adj (rev xs) = rev (remdups_adj xs)"
map_return_pmf: fixes f :: "'i \<Rightarrow> 'h"   and x :: "'i" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
evalOP_AbsN: fixes e :: "db" assumes "val (DBAbsN e)" shows "DBAbsN e \<Down> DBAbsN e"
add_mono_thms_linordered_semiring_4: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i = j \<and> k = l" shows "i + k = j + l"
atoms_lang: fixes w :: "'b list"   and r :: "'b rexp" assumes "w \<in> lang r" shows "set w \<subseteq> atoms r"
transfer: shows "rel_fun (=) (rel_fun cr_uint cr_uint) drop_bit drop_bit"
spr_commonKnowledge_def: fixes envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'e \<times> 'd option"   and cec :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) odlist" shows "spr_commonKnowledge envObs cec \<equiv> \<lambda>as s. let r = \<lambda>a. fromList (concat (map (\<lambda>s'. concat (map (\<lambda>s''. if (s', s'') \<in> SPRViewDet.spr_repRels envObs a then [(s', s'')] else []) (toList cec))) (toList cec))); R = toList (big_union r as) in fromList (memo_list_trancl R s)"
eq_acom_iff_strip_anno: fixes C1 :: "'b acom"   and C2 :: "'b acom" shows "(C1 = C2) = (strip C1 = strip C2 \<and> (\<forall>p<length (annos C1). anno C1 p = anno C2 p))"
fBc: fixes b :: "'a" assumes "b \<in> B" shows "f (generate G {b}) \<in> carrier G"
pi_dom: fixes it :: "('f \<Rightarrow> bool) \<Rightarrow> ('g \<times> 'h \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and it' :: "('i \<Rightarrow> bool) \<Rightarrow> ('g \<times> 'h \<Rightarrow> 'i \<Rightarrow> 'i) \<Rightarrow> 'i \<Rightarrow> 'i" assumes "proper_it it it'" shows "proper_it (map_iterator_dom it) (map_iterator_dom it')"
interleave_append1: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b list" assumes "l \<in> a \<otimes> b" shows "x @ l \<in> x @ a \<otimes> b"
rqrm_lookup_def: fixes rqrm :: "('b, 'a dlist) HashMap.hashmap"   and q :: "'b" shows "rqrm_lookup rqrm q \<equiv> case hm.lookup q rqrm of None \<Rightarrow> ls.empty () | Some s \<Rightarrow> s"
watchCharacterizationCondition_def: fixes w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list"   and clause :: "Literal list" shows "watchCharacterizationCondition w1 w2 M clause = (literalFalse w1 (elements M) \<longrightarrow> (\<exists>l. l el clause \<and> literalTrue l (elements M) \<and> elementLevel l M \<le> elementLevel (opposite w1) M) \<or> (\<forall>l. l el clause \<and> l \<noteq> w1 \<and> l \<noteq> w2 \<longrightarrow> literalFalse l (elements M) \<and> elementLevel (opposite l) M \<le> elementLevel (opposite w1) M))"
XX: shows "\<forall>X\<in>XX. well_ordered_set X (\<sqsubseteq>) \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. \<not> x \<sqsubseteq> y \<longrightarrow> y \<in> X)"
wf_dickson_less_v: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "dickson_grading d" shows "wfP (gd_term.dickson_less_v pair_of_term ord_term_strict d m)"
commute: fixes a :: "'a"   and b :: "'a" shows "gcd a b = gcd b a"
rel_pmf_eqI: fixes x :: "'a pmf" shows "rel_pmf (=) x x"
B_def: shows "B \<equiv> \<Union> (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ` set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' D))"
sred: shows "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"
prime_sum_upto_eqI': fixes a' :: "nat"   and a :: "real"   and b :: "real"   and b' :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "a' \<le> nat \<lfloor>a\<rfloor>"   and "a \<le> b"   and "nat \<lfloor>b\<rfloor> \<le> b'"   and "\<And>k. k \<in> {a'<..b'} \<Longrightarrow> \<not> prime k" shows "prime_sum_upto f a = prime_sum_upto f b"
solve_equation: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "subgroup H (add_monoid G)"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<oplus> x"
affected_def: shows "affected \<equiv> map_fun (\<langle>$\<rangle>) id (\<lambda>f. {a. f a \<noteq> a})"
word_oa_dist2: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x OR y AND z = (x OR y) AND (x OR z)"
x: fixes s' :: "('i, 'a) state" shows "A \<Turnstile>\<^sub>i \<B>\<I> s' = Q A (<) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<B>\<^sub>i\<^sub>u_update \<I>\<^sub>u \<I>\<^sub>l Leq Geq s'"
isEmpty_def: fixes q :: "('c, 'd) SkewBinomialHeap" shows "isEmpty q \<equiv> Rep_SkewBinomialHeap q = BsSkewBinomialHeapStruc.bs_empty"
inv_inj: shows "inj_on (m_inv G) (carrier G)"
countably_additive_iff_continuous_from_below: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive (Pow A) f"   and "Measure_Space.additive (Pow A) f" shows "countably_additive (Pow A) f = (\<forall>Aa. range Aa \<subseteq> Pow A \<longrightarrow> incseq Aa \<longrightarrow> \<Union> (range Aa) \<in> Pow A \<longrightarrow> (\<lambda>i. f (Aa i)) \<longlonglongrightarrow> f (\<Union> (range Aa)))"
irreducible\<^sub>d_mD_2: fixes m :: "int"   and f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "poly_mod.irreducible\<^sub>d_m m f"   and "degree (poly_mod.Mp m g) < degree (poly_mod.Mp m f)"   and "degree (poly_mod.Mp m h) < degree (poly_mod.Mp m f)" shows "poly_mod.Mp m f \<noteq> poly_mod.Mp m (g * h)"
finite_insert: fixes a :: "'f"   and A :: "'f set" shows "finite (insert a A) = finite A"
add_cancl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "D z x"   and "D z y"   and "z \<oplus> x = z \<oplus> y" shows "x = y"
trm_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and trm :: "'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "elementary_category_with_terminal_object C one trm"   and "partial_magma.ide C a"   and "partial_magma.in_hom C f a one" shows "f = trm a"
cond_spmf_UNIV: fixes p :: "'h spmf" shows "cond_spmf p UNIV = mk_lossless p"
intro: fixes T :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "T ` M \<subseteq> M" shows "ModuleEnd_axioms M T"
scale_minus_right: fixes a :: "'a"   and x :: "'b" shows "a *s - x = - (a *s x)"
onode_comp_def: fixes i :: "nat"   and onp :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and R\<^sub>i :: "nat set" shows "\<langle>i : onp : R\<^sub>i\<rangle>\<^sub>o \<equiv> \<lparr>init = {(\<sigma>, NodeS i s R\<^sub>i) |\<sigma> s. (\<sigma>, s) \<in> init onp}, trans = onode_sos (automaton.trans onp)\<rparr>"
check_poly_strict_mono_discrete_def: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "('d monom \<times> 'c) list"   and v :: "'d" shows "check_poly_strict_mono_discrete gt p v \<equiv> check_poly_gt gt (poly_subst (\<lambda>w. poly_of (if w = v then PSum [PNum (1::'c), PVar v] else PVar w)) p) p"
known_ptr_not_shadow_root_ptr: fixes ptr :: "(_) object_ptr" assumes "local.a_known_ptr ptr"   and "\<not> (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> is_shadow_root_ptr x)" shows "DocumentClass.known_ptr ptr"
dynmethd_declC: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some m"   and "is_class G statC"   and "ws_prog G" shows "(\<exists>d. class G (declclass m) = Some d \<and> table_of (methods d) sig = Some (mthd m)) \<and> G\<turnstile>dynC\<preceq>\<^sub>C declclass m \<and> methd G (declclass m) sig = Some m"
iso_iff_section_and_retraction: fixes f :: "'a \<times> 'a" shows "CC.iso f = (CC.section f \<and> CC.retraction f)"
ZFSucZero: shows "Nat2nat (SucNat Empty) = 1"
assms: shows "\<forall>m\<in>#M. k dvd f m"
sup_ge1: fixes x :: "'b"   and y :: "'b" shows "x \<le> x \<sqinter> y"
cnf_to_dimacs_disj: fixes f :: "nat formula" assumes "is_disj f" shows "cnf_to_dimacs f = [disj_to_dimacs f]"
simps: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "local.wpo.wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> weight_NS \<and> wpo.\<sigma> (g, length ts) = [] \<and> (case (g, length ts) of (f, n) \<Rightarrow> n = 0 \<and> least f)) | Fun f ss \<Rightarrow> if (s, t) \<in> weight_S then (True, True) else if (s, t) \<in> weight_NS then if \<exists>i\<in>set (wpo.\<sigma> (f, length ss)). local.wpo.wpo_ns (ss ! i) t then (True, True) else case t of Var x \<Rightarrow> (False, False \<and> False) | Fun g ts \<Rightarrow> case (pr_strict (f, length ss) (g, length ts), pr_weak (f, length ss) (g, length ts)) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (wpo.\<sigma> (g, length ts)). local.wpo.wpo_s s (ts ! j)) then if prs then (True, True) else let ss' = map ((!) ss) (wpo.\<sigma> (f, length ss)); ts' = map ((!) ts) (wpo.\<sigma> (g, length ts)); cf = Lex; cg = Lex in if cf = Lex \<and> cg = Lex then lex_ext local.wpo.wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext local.wpo.wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
iarray_exhaust2: fixes xs :: "'d iarray"   and ys :: "'d iarray" shows "(xs = ys) = (IArray.list_of xs = IArray.list_of ys)"
restrict_def: fixes v :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "restrict v f c \<equiv> f c \<sqinter> lan v"
min_top: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.order_top less_eq less top" shows "ord.min less_eq top x = x"
ctermsl_refl: fixes p :: "('a, 'b, 'c, 'd) seqp" assumes "not_choice p" shows "p \<in> ctermsl p"
is_map_from_is_homomorphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (\<rho> U V) (\<FF> U) (\<FF> V)"
piecewise_C1_differentiable_affine: fixes f :: "real \<Rightarrow> 'b"   and m :: "real"   and c :: "real"   and S :: "real set" assumes "f piecewise_C1_differentiable_on (\<lambda>x. m * x + c) ` S" shows "f \<circ> (\<lambda>x. m *\<^sub>R x + c) piecewise_C1_differentiable_on S"
I3: assumes "Arr v" shows "\<guillemotleft>\<lbrace>v\<rbrace> : \<lbrace>Dom v\<rbrace> \<rightarrow> \<lbrace>Cod v\<rbrace>\<guillemotright>"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and L1 :: "'a Literal"   and L2 :: "'a Literal" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.literal_ordering atom_ordering L1 L2 = ((atom L1, atom L2) \<in> atom_ordering)"
IH: shows "sum f F \<ominus> sum g F \<le> (\<Sum>x\<in>F. f x \<ominus> g x)"
duplicateEdge_is_duplicateEdge_eq: fixes g :: "graph"   and f :: "face"   and a :: "nat"   and b :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "a \<in> \<V> f"   and "b \<in> \<V> f" shows "duplicateEdge g f a b = is_duplicateEdge g f a b"
linear_iff: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" shows "Vector_Spaces.linear s1 s2 f = (vector_space s1 \<and> vector_space s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
assms_2: shows "card {q. prime q \<and> q < p} = n"
bind_cong_All: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and g :: "'a \<Rightarrow> 'b measure" assumes "\<forall>x\<in>space M. f x = g x" shows "M \<bind> f = M \<bind> g"
rbt_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t" shows "rbt (RBT_Set2.delete x t)"
rt_Some_sqn: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "rt ip = Some (dsn, dsk, flag, hops, nhip)" shows "sqn rt ip = dsn"
ground_head: fixes s :: "('b, 'a) tm" assumes "ground s" shows "is_Sym (head s)"
arith_simps_5: fixes m :: "num"   and n :: "num" shows "num.Bit0 m + num.Bit0 n = num.Bit0 (m + n)"
coefficients_isconst: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p n" shows "\<forall>q\<in>set (coefficients p). isconstant q"
distinct_11: fixes ty1' :: "ty"   and ty2' :: "ty" shows "tINT \<noteq> ty1' \<rightarrow> ty2'"
solve_system_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "solve_system A b = (let A' = Gauss_Jordan_PA A in (snd A', fst A' *v b))"
size_gen_1: fixes xa :: "'c \<Rightarrow> nat"   and x :: "'d \<Rightarrow> nat"   and x11 :: "char list option"   and x12 :: "'c" shows "Tagged_Prod_Sum.sum.size_sum xa x (sum.Inl x11 x12) = xa x12 + Suc 0"
flows_state_IFS: shows "flows_state (generate_valid_stateful_policy_IFSACS G M edgesList) = set (filter_compliant_stateful_ACS G M (rev edgesList))"
fixp_def: fixes f :: "'a \<Rightarrow> 'a" shows "ccpo_class.fixp f = Sup (ccpo_class.iterates f)"
substBinp_def2: fixes ys :: "'varSort"   and Y :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and y :: "'var" shows "substBinp ys Y y = lift (substAbs ys Y y)"
chain_subset: fixes ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'f set"   and B :: "'f set" assumes "Complete_Partial_Order.chain ord A"   and "B \<subseteq> A" shows "Complete_Partial_Order.chain ord B"
assms_3: shows "A, m \<turnstile> (q # ps, a) # branch"
rel_mset_Zero_iff_2: fixes rel :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'a multiset" shows "rel_mset rel X {#} = (X = {#})"
n_inj_comp_end: fixes S :: "'v \<Rightarrow> 'v"   and T :: "'v \<Rightarrow> 'v" assumes "VEnd S"   and "VEnd T"   and "\<not> inj_on (S \<circ> T) V" shows "\<not> inj_on S V \<or> \<not> inj_on T V"
Cauchy_higher_deriv_bound: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and n :: "nat"   and y :: "complex"   and B0 :: "real" assumes "f holomorphic_on ball z r"   and "continuous_on (cball z r) f"   and "0 < r"   and "0 < n"   and "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0" shows "cmod ((deriv ^^ n) f z) \<le> fact n * B0 / r ^ n"
security_P1: fixes \<eta> :: "nat"   and msgs :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and D :: "('grp \<times> 'grp) \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> bool spmf" shows "sim_def.adv_P1 \<eta> msgs \<sigma> D \<le> local.ddh.advantage \<eta> (R1_inter_adversary \<eta> D msgs) + local.ddh.advantage \<eta> (inter_S1_adversary \<eta> D msgs)"
update_simps_18: fixes V :: "(nat, 'b) mapping"   and s :: "('a, 'b) state" shows "\<U>\<^sub>c (\<V>_update V s) = \<U>\<^sub>c s"
diagonal_step_PQ_PAQ: fixes P :: "(('a, 'b) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and p :: "'a"   and q :: "'a"   and u :: "'a"   and v :: "'a"   and d :: "'a" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)" shows "P ** A ** Q = mult_column (column_add (column_add (row_add (interchange_rows (row_add A (from_nat k) (from_nat i) p) (from_nat i) (from_nat k)) (from_nat k) (from_nat i) (- v)) (from_nat i) (from_nat k) q) (from_nat k) (from_nat i) u) (from_nat k) (- (1::'a))"
lookup_lift_poly_fun_homogenous: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "f 0 = 0" shows "lookup (lift_poly_fun f p) v = lookup (f (proj_poly (component_of_term v) p)) (pp_of_term v)"
wfE_subst: fixes \<Delta> :: "binding list"   and B :: "binding"   and \<Gamma> :: "binding list"   and U :: "POPLmarkRecord.type" assumes "\<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> U]\<^sub>e @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
fun_eq_on_sym: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "fun_eq_on f g A" shows "fun_eq_on g f A"
inext_less_mono_fin: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "finite I"   and "b \<noteq> Max I" shows "inext a I < inext b I"
coreflexive_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i j. i \<noteq> j \<longrightarrow> f (i, j) = bot" shows "matrix_idempotent_semiring.coreflexive f"
no_state_change_if_no_event: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> p" shows "states c p = states c' p \<and> ps c p = ps c' p"
dvd_prodI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A" shows "dvd.dvd times (f a) (comm_monoid_mult.prod times one f A)"
degree_smult_eq: fixes a :: "'c"   and p :: "'c poly" shows "degree (smult a p) = (if a = (0::'c) then 0 else degree p)"
le_SPEC_UNIV_rule: fixes m :: "'b nres" assumes "m \<le> SPEC (\<lambda>_. True)" shows "m \<le> RES UNIV"
representation_zero: fixes basis :: "'b set" shows "vs1.representation basis (0::'b) = (\<lambda>b. 0::'a)"
abs_eq: fixes xa :: "'b set"   and x :: "'b set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "Filters.filter.Abs_filter xa \<sqinter> Filters.filter.Abs_filter x = Filters.filter.Abs_filter (xa \<inter> x)"
subobj_aux: fixes 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 C :: "char list"   and C' :: "char list"   and Cs :: "char list list"   and C'' :: "char list" assumes "is_subobj P (C, C' # Cs @ [C''])" shows "is_subobj P (C, C' # Cs)"
vrrestriction_mono: fixes A :: "V"   and B :: "V"   and r :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "r \<restriction>\<^sup>r\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> B"
inL1: fixes G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "H, G, \<Gamma> \<Rightarrow> \<Delta>"
not_ev_not: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> ev (\<lambda>xs. \<not> \<phi> xs) xs) = alw \<phi>"
weakly_equivalent_iff_not_distinguished: fixes P :: "'state"   and Q :: "'state" shows "S_transform.weakly_logically_equivalent P Q = (\<nexists>x. S_transform.weak_formula x \<and> S_transform.is_distinguishing_formula x P Q)"
Max_in: fixes A :: "'e set" assumes "finite A"   and "A \<noteq> {}" shows "Max A \<in> A"
branch_inv_Branching_Left: fixes b :: "bool list"   and l :: "tree"   and r :: "tree" shows "branch (Left # b) (Branching l r) = branch b l"
distinct_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 # x22 \<noteq> []"
star_of_simps_6: fixes x :: "'c" shows "star_of (inverse x) = inverse (star_of x)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D I\<^sub>D.map I\<^sub>D.cmp GF.map GF.cmp \<eta>\<^sub>0 \<eta>\<^sub>1"
set_rel_br: fixes a :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "\<langle>br a I\<rangle>set_rel = br ((`) a) (\<lambda>X. Ball X I)"
assms_2: shows "{t..t'} \<subseteq> existence_ivl0 x"
IH: shows "(path \<sigma>, k) \<in> scp \<and> (path \<sigma>', k') \<in> scp"
WTrtLAss: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and e :: "char list exp"   and T' :: "ty" assumes "E V = \<lfloor>T\<rfloor>"   and "P,E,h,sh \<turnstile> e : T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h,sh \<turnstile> V:=e : Void"
Pbij_compose_Pbij: fixes \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set" assumes "\<beta> \<in> Pbij"   and "\<gamma> \<in> Pbij" shows "Pbij_compose \<beta> \<gamma> \<in> Pbij"
select_convs_1: fixes carrier :: "'a set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
cases: fixes a1 :: "nat"   and a2 :: "('c, 'd) bintree option list" assumes "is_binqueue a1 a2" obtains l :: "nat" where "a1 = l__" and "a2 = []"   | l :: "nat" and xs :: "('c, 'd) bintree option list" where "a1 = la__" and "a2 = None # xsa__" and "is_binqueue (Suc la__) xsa__"   | l :: "nat" and xs :: "('c, 'd) bintree option list" and t :: "('c, 'd) bintree" where "a1 = l" and "a2 = Some t # xs" and "is_binqueue (Suc l) xs" and "is_bintree_list l (children t)" and "is_heap_list (priority t) (children t)"
llast_def: fixes xs :: "'c llist" shows "llast xs = (case llength xs of enat (Suc n') \<Rightarrow> lnth xs n')"
countable_def: fixes S :: "'d set" shows "countable S = (\<exists>f. inj_on f S)"
conc_fun_RES: fixes R :: "('b \<times> 'c) set"   and X :: "'c set" shows "\<Down> R (RES X) = RES (R\<inverse> `` X)"
apex_composite: shows "H\<mu>H\<nu>\<pi>.apex = \<mu>.leg0 \<down>\<down> \<nu>\<pi>.leg1"
3: fixes x :: "nat" assumes "x < l" shows "count_list (cruel A (Step A ([0..<l], fst A [0..<l]) (last [0..<l])) n) x \<le> n"
computable_def: shows "computable = (\<lambda>A. A \<in> ce_sets \<and> - A \<in> ce_sets)"
prime_factorization_nat_main: fixes ni :: "nat \<times> nat \<times> nat list"   and n :: "nat"   and i :: "nat"   and "is" :: "nat list"   and jj :: "nat"   and res :: "nat list"   and ps :: "nat list" assumes "ni = (n, i, is)"   and "2 \<le> i"   and "2 \<le> n"   and "\<And>j. \<lbrakk>2 \<le> j; j < i\<rbrakk> \<Longrightarrow> \<not> j dvd n"   and "\<And>j. \<lbrakk>i \<le> j; j < jj; prime j\<rbrakk> \<Longrightarrow> j \<in> set is"   and "i \<le> jj"   and "sorted is"   and "distinct is"   and "candidate_invariant jj"   and "set is \<subseteq> {i..<jj}"   and "res = prime_factorization_nat_main n jj is ps" shows "\<exists>qs. res = qs @ ps \<and> Ball (set qs) prime \<and> n = prod_list qs"
\<psi>_\<phi>: fixes t :: "'t" assumes "t \<in> S'.Univ" shows "INV.\<psi> (\<psi> t) = t"
subformulas\<^sub>\<mu>\<^sub>\<nu>_card: fixes \<phi> :: "'a ltln" shows "card (subformulas\<^sub>\<mu> \<phi> \<union> subformulas\<^sub>\<nu> \<phi>) = card (subformulas\<^sub>\<mu> \<phi>) + card (subformulas\<^sub>\<nu> \<phi>)"
lin_dep_crit: fixes A :: "'c set"   and S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite A"   and "A \<subseteq> S"   and "a \<in> A \<rightarrow> carrier K"   and "v \<in> A"   and "a v \<noteq> \<zero>\<^bsub>K\<^esub>"   and "lincomb a A = \<zero>\<^bsub>V\<^esub>" shows "lin_dep S"
b'_2: shows "k = ln l \<or> k = \<star>"
taylor_zcf: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "zcf (T\<^bsub>a\<^esub> f) = to_fun f a"
iso: shows "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
set_return_spmf: fixes x :: "'b" shows "set_spmf (return_spmf x) = {x}"
nn_integral_restrict_space: fixes \<Omega> :: "'h set"   and M :: "'h measure"   and f :: "'h \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = set_nn_integral M \<Omega> f"
d_transitive_var: fixes x :: "'a" assumes "\<forall>y.  |x\<rangle>  |x\<rangle> y \<le>  |x\<rangle> y" shows "\<forall>y.  |x\<rangle> y =  |x\<rangle>  |x\<^sup>\<star>\<rangle> y"
ab_same: fixes k :: "nat" shows "a k \<lless> \<Union> (range (b k))"