hcju/isabelleps huggingface dataset:
fun_relD: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'d \<Rightarrow> 'e"   and A :: "('b \<times> 'd) set"   and B :: "('c \<times> 'e) set"   and x :: "'b"   and x' :: "'d" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
ideal_prod_eq_Inter: fixes I :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "I \<in> {..n} \<rightarrow> {J. ideal J R}"   and "\<And>i j. \<lbrakk>i \<in> {..n}; j \<in> {..n}; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "finprod (ideals_set R) I {..n} = \<Inter> (I ` {..n})"
P2LL: shows "P2 * LL \<in> carrier_mat m n"
ResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"   and "y \<noteq> a"   and "y \<noteq> x" shows "<\<nu>y>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e <\<nu>y>P'"
sup_compl_top: fixes x :: "'a" shows "x \<squnion> - x = top"
arity_generalize_fundef: fixes fd2 :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" shows "arity (Subx.generalize_fundef fd2) = arity fd2"
pr_conv_1_to_3_lm: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "pr_conv_1_to_3 f \<in> PrimRec3"
cone_list_def: fixes Vs :: "'a vec list" shows "n.cone_list Vs = {b. \<exists>c. n.nonneg_lincomb_list c Vs b}"
uverts_edge_ugraph: fixes es :: "nat set set" shows "uverts (ES.edge_ugraph es) = ES.S_verts"
assms_3: fixes x :: "'a" assumes "x \<in> S"   and "s x = (0::'b)" shows "blinfun_apply (Ds x) (f x) \<noteq> (0::'b)"
wt_kil_complete: fixes P :: "jvm_prog"   and Ts :: "ty list"   and p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and b :: "staticb"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table"   and \<tau>s :: "(ty list \<times> ty err list) option list" assumes "start_context P Ts p C"   and "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s" shows "wt_kildall P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt"
filter_conn_fw_lem: fixes s :: "'a word"   and d :: "'a word"   and c :: "parts_connection"   and rs :: "'a simple_rule list" shows "runFw s d c (filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs) = runFw s d c rs"
pairwise_disjoint_connected_components_of: fixes X :: "'b topology" shows "disjoint (connected_components_of X)"
count_mset_set_2: fixes A :: "'k set"   and x :: "'k" assumes "infinite A" shows "count (mset_set A) x = 0"
has_subprob_density_def: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "has_subprob_density M N f \<equiv> has_density M N f \<and> subprob_space M"
divisors_zero: fixes a :: "'a"   and b :: "'a" assumes "a * b = (0::'a)" shows "a = (0::'a) \<or> b = (0::'a)"
assms: shows "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
sample_var_lookup_same: fixes x :: "'v"   and f :: "int \<Rightarrow> int \<Rightarrow> 'm" shows "bind (memo (sample_var X) x) (\<lambda>i. bind (lookup x) (f i)) = bind (memo (sample_var X) x) (\<lambda>i. f i i)"
vars_mset_singleton: fixes p :: "('d, 'c) Term.term \<times> ('e, 'c) Term.term" shows "vars_mset {#p#} = vars_term (fst p) \<union> vars_term (snd p)"
dupe1_2: shows "B = 0 \<or> degree B < degree D1"
bij_betw_inv_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "bij_betw (inv_into A f) B A"
ffd_fbb_galois: fixes f :: "'c \<Rightarrow> 'd set"   and X :: "'d set"   and Y :: "'c set" shows "(fd\<^sub>\<F> f X \<subseteq> Y) = (X \<subseteq> bb\<^sub>\<F> f Y)"
Since: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and \<psi> :: "('msaux, 'muaux) mformula"   and \<psi>' :: "Formula.formula"   and args :: "Monitor.args"   and \<phi>'' :: "Formula.formula"   and I :: "\<I>"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and aux :: "'msaux" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j P V n R \<psi> \<psi>'"   and "if args_pos args then \<phi>'' = \<phi>' else \<phi>'' = formula.Neg \<phi>'"   and "safe_formula \<phi>'' = args_pos args"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>'"   and "args_R args = fv \<psi>'"   and "fv \<phi>' \<subseteq> fv \<psi>'"   and "wf_mbuf2' \<sigma> P V j n R \<phi>' \<psi>' buf"   and "wf_ts \<sigma> P j \<phi>' \<psi>' nts"   and "wf_since_aux \<sigma> V R args \<phi>' \<psi>' aux (Monitor.progress \<sigma> P (formula.Since \<phi>'' I \<psi>') j)" shows "wf_mformula \<sigma> j P V n R (MSince args \<phi> \<psi> buf nts aux) (formula.Since \<phi>'' I \<psi>')"
case_maybe_case: fixes f :: "'a"   and g :: "'b \<rightarrow> 'c \<rightarrow> 'a"   and h :: "'d \<rightarrow> 'd llist \<rightarrow> 'b \<times> 'c"   and L :: "'d llist" shows "(case case L of lnil \<Rightarrow> Nothing | x :@ xs \<Rightarrow> Just\<cdot>(h\<cdot>x\<cdot>xs) of Nothing \<Rightarrow> f | Just\<cdot>(a, b) \<Rightarrow> g\<cdot>a\<cdot>b) = (case L of lnil \<Rightarrow> f | x :@ xs \<Rightarrow> g\<cdot>(fst (h\<cdot>x\<cdot>xs))\<cdot>(snd (h\<cdot>x\<cdot>xs)))"
sup_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<sqinter> y = x"
epiI: fixes f :: "'s" assumes "arr f"   and "\<And>g g'. seq g f \<and> seq g' f \<and> g \<cdot> f = g' \<cdot> f \<Longrightarrow> g = g'" shows "epi f"
pow_closed: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "a [^] n \<in> carrier Zp"
sel_2: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real" shows "Im1 (Quat x1 x2 x3 x4) = x2"
r_def: fixes t :: "'a" shows "r t = (SOME l. (\<lambda>n. c n t)\<up>l \<and> l \<le> x t)"
subsetFn: fixes Y :: "'a set"   and X :: "'a set" assumes "Y \<noteq> {}"   and "Y \<subseteq> X" shows "subsetFn X Y ` X = Y"
multpw_implies_mult2_alt_ns: fixes X :: "'b multiset"   and Y :: "'b multiset"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "(X, Y) \<in> multpw ns" shows "(X, Y) \<in> mult2_alt_ns ns s"
assms_1: shows "(f \<longlongrightarrow> c) F"
singleton_conv2: fixes a :: "'a" shows "{x. a = x} = \<eta> a"
run_trace_def: shows "run_trace \<equiv> construct_fun_from_rel trace_rel"
B: fixes W :: "'a option set" shows "W \<subseteq> Y \<longrightarrow> (xs, W) \<in> seq_comp_failures P Q"
unfold_llist_raw_def: fixes stop :: "'d \<Rightarrow> bool"   and head :: "'d \<Rightarrow> 'c"   and tail :: "'d \<Rightarrow> 'd"   and s :: "'d" shows "unfold_llist_raw stop head tail s = (if stop s then Done else Yield (head s) (tail s))"
gc4_def: shows "gc4 = gcd pR4 pI4"
cA_def: shows "cA \<equiv> HMA_Connect.map_matrix complex_of_real A"
intro: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "group G composition unit" shows "right_translations_of_group G composition unit"
mult_inc: fixes x :: "num"   and y :: "num" shows "x * Num.inc y = x * y + x"
comp_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "CC.in_hom f a b"   and "CC.in_hom g b c" shows "CC.in_hom (CC.comp g f) a c"
replicate_length_same: fixes xs :: "'a list"   and x :: "'a" assumes "\<forall>y\<in>set xs. y = x" shows "replicate (length xs) x = xs"
tree_def: fixes G :: "('c, 'd) pre_digraph" shows "tree G \<equiv> Digraph_Component.connected G \<and> forest G"
bind_ask: fixes f :: "'b \<Rightarrow> 'm"   and g :: "'a multiset \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
cases: fixes a1 :: "'b"   and a2 :: "'b" assumes "(a1, a2) \<in> hull_rel" obtains p :: "perm" and x :: "'b" where "a1 = p \<bullet> x" and "a2 = x"
dist_triangle_less_add: fixes x1 :: "'b"   and y :: "'b"   and e1 :: "real"   and x2 :: "'b"   and e2 :: "real" assumes "dist x1 y < e1"   and "dist x2 y < e2" shows "dist x1 x2 < e1 + e2"
skew_product_real_iterates: fixes n :: "nat"   and f :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "real" shows "((\<lambda>(x, y). (T x, y + f x)) ^^ n) (x, y) = ((T ^^ n) x, y + birkhoff_sum f n x)"
pmf_map_inj': fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a pmf"   and x :: "'a" assumes "inj f" shows "pmf (map_pmf f M) (f x) = pmf M x"
trace_measure2_id: fixes M0 :: "'a mat"   and n :: "nat"   and M1 :: "'a mat"   and A :: "'a mat" assumes "M0 \<in> carrier_mat n n"   and "M1 \<in> carrier_mat n n"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m n"   and "A \<in> carrier_mat n n" shows "trace (M0 * A * adjoint M0) + trace (M1 * A * adjoint M1) = trace A"
simps_3: fixes dims :: "nat list"   and S1 :: "com"   and S2 :: "com"   and P :: "complex mat" shows "state_sig.wlp dims (S1;; S2) P = state_sig.wlp dims S1 (state_sig.wlp dims S2 P)"
Inf_finite_2_def: fixes A :: "finite_2 set" shows "\<Sqinter> A = (if a\<^sub>1 \<in> A then a\<^sub>1 else a\<^sub>2)"
mult_norm_subgroup: fixes N :: "'a set"   and H :: "'a set" assumes "N \<lhd> G"   and "subgroup H G" shows "subgroup (N <#> H) G"
assms_3: fixes x :: "Core.term" assumes "x \<in> set Hs" shows "typ_of x = Some propT"
prj_simps_10: shows "C.arr \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1"
prepend_goedel: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>" shows "\<exists>i. \<phi> i = [i] \<odot> f"
induct: fixes x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "exp"   and x3 :: "exp"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> exp \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 \<turnstile> x2 \<rightarrow> x3"   and "\<And>CT C Cf es fi ei. \<lbrakk>fields(CT,C) = Cf; lookup2 Cf es (\<lambda>fd. vdName fd = fi) = Some ei\<rbrakk> \<Longrightarrow> P CT (FieldProj (New C es) fi) ei"   and "\<And>CT m C xs e0 ds es e0'. \<lbrakk>mbody(CT,m,C) = xs . e0; substs [xs [\<mapsto>] ds, this \<mapsto> New C es] e0 = e0'\<rbrakk> \<Longrightarrow> P CT (MethodInvk (New C es) m ds) e0'"   and "\<And>CT C D es. CT \<turnstile> C <: D \<Longrightarrow> P CT (Cast D (New C es)) (New C es)"   and "\<And>CT e0 e0' f. \<lbrakk>CT \<turnstile> e0 \<rightarrow> e0'; P CT e0 e0'\<rbrakk> \<Longrightarrow> P CT (FieldProj e0 f) (FieldProj e0' f)"   and "\<And>CT e0 e0' m es. \<lbrakk>CT \<turnstile> e0 \<rightarrow> e0'; P CT e0 e0'\<rbrakk> \<Longrightarrow> P CT (MethodInvk e0 m es) (MethodInvk e0' m es)"   and "\<And>CT ei ei' e0 m el er. \<lbrakk>CT \<turnstile> ei \<rightarrow> ei'; P CT ei ei'\<rbrakk> \<Longrightarrow> P CT (MethodInvk e0 m (el @ ei # er)) (MethodInvk e0 m (el @ ei' # er))"   and "\<And>CT ei ei' C el er. \<lbrakk>CT \<turnstile> ei \<rightarrow> ei'; P CT ei ei'\<rbrakk> \<Longrightarrow> P CT (New C (el @ ei # er)) (New C (el @ ei' # er))"   and "\<And>CT e0 e0' C. \<lbrakk>CT \<turnstile> e0 \<rightarrow> e0'; P CT e0 e0'\<rbrakk> \<Longrightarrow> P CT (Cast C e0) (Cast C e0')" shows "P x1 x2 x3"
fun_when: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and P :: "bool" assumes "f (0::'c) = (0::'b)" shows "f (a when P) = (f a when P)"
type_to_set1_2: fixes m :: "nat"   and n :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int m}"   and "\<exists>Rep Abs. type_definition Rep Abs {0..<int n}"   and "1 < m"   and "1 < n" shows "m = CARD('b)"
hn_refine_guessI: fixes P :: "assn"   and f :: "'c Heap"   and P' :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and f' :: "'d nres"   and f_conc :: "'c Heap" assumes "hn_refine P f P' R f'"   and "f = f_conc" shows "hn_refine P f_conc P' R f'"
bit_slice1_iff: fixes m :: "nat"   and w :: "'c word"   and n :: "nat" shows "bit (slice1 m w) n = (m - LENGTH('c) \<le> n \<and> n < min LENGTH('b) m \<and> bit w (n + (LENGTH('c) - m) - (m - LENGTH('c))))"
new_document_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'ai" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 new_document_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
is_ground_lit_is_ground_on_var: fixes L :: "('a, 'b) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "is_ground_lit (subst_lit L \<sigma>)"   and "v \<in> vars_lit L" shows "is_ground_atm (\<sigma> v)"
graph_ext_def: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons graph_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple nodes edges) more"
homotopic_paths_imp_homotopic_loops: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q"   and "pathfinish p = pathstart p"   and "pathfinish q = pathstart p" shows "homotopic_loops s p q"
absolutely_integrable_absolutely_integrable_ubound: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f integrable_on S"   and "g absolutely_integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "f absolutely_integrable_on S"
Kcknows_2: shows "toSet X \<subseteq> toSet Y"
empty_iff: fixes c :: "'g" shows "(c \<in> {}) = False"
map_gpv_conv_map_gpv': fixes f :: "'h \<Rightarrow> 'k"   and g :: "'i \<Rightarrow> 'l" shows "map_gpv f g = map_gpv' f g id"
wt_sym_ge_\<epsilon>: fixes f :: "'s" shows "\<epsilon> \<le> wt_sym f"
square_free_factorizationD'_3: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list"   and b :: "'a poly"   and i :: "nat" assumes "square_free_factorization p (c, bs)"   and "(b, i) \<in> set bs" shows "degree b \<noteq> 0"
deg_minus_eq1: fixes p :: "'a" assumes "p \<in> carrier R" shows "deg R S X (-\<^sub>a p) = deg R S X p"
sturm_tarski_below: fixes p :: "real poly"   and b :: "real"   and q :: "real poly" assumes "poly p b \<noteq> 0" shows "taq {x. poly p x = 0 \<and> x < b} q = changes_le_smods b p (pderiv p * q)"
lim: shows "(\<lambda>N. \<Sum>n<N. emeasure M (DW (n + 1))) \<longlonglongrightarrow> emeasure M (induced_map A --` W)"
u_eval: fixes U :: "'a set"   and s :: "'b" assumes "U \<in> I"   and "s \<in> \<FF> U" shows "u \<lfloor>U, s\<rfloor> = \<psi> U s"
scaleC_one: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC 1 x = x"
closed_lattice: fixes A :: "(real, 'a) vec set" assumes "\<And>v i. v \<in> A \<Longrightarrow> v $ i \<in> \<int>" shows "closed A"
zmset_of_subset_iff: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(zmset_of M \<subset>#\<^sub>z zmset_of N) = (M \<subset># N)"
simps_4: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_syn_imp (\<phi> R\<^sub>n \<psi>) = (if \<psi> \<turnstile>\<^sub>s \<phi> then rewrite_syn_imp \<psi> else if \<psi> \<turnstile>\<^sub>s not\<^sub>n \<phi> then mk_globally (rewrite_syn_imp \<psi>) else mk_release (rewrite_syn_imp \<phi>) (rewrite_syn_imp \<psi>))"
ACI_le_ACIEQ: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r ~ s" shows "r \<approx> s"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)" shows "Group.group (G\<lparr>carrier := H__, carrier := H\<rparr>)"
D: shows "Max (offs_set_prev ns (x # xs) index key mi ma i) = j"
Quotient_equiv_abs2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "Quotient R Abs Rep T"   and "R x y" shows "T y (Abs x)"
rel_intros_2: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x2 :: "'l"   and y2 :: "'m" assumes "R x2 y2" shows "rel_option R (Some x2) (Some y2)"
antisym: fixes b :: "'a list"   and a :: "'a list" assumes "prefix b a"   and "prefix a b" shows "a = b"
cos_angle_cmat_def: fixes H1 :: "complex \<times> complex \<times> complex \<times> complex"   and H2 :: "complex \<times> complex \<times> complex \<times> complex" shows "cos_angle_cmat H1 H2 = - Re (mat_det_12 H1 H2) / (2 * sqrt (Re (mat_det H1 * mat_det H2)))"
Diagonalize_Diagonalize: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (monoidal_language.Diagonalize C t) = monoidal_language.Diagonalize C t"
inversion_number_def: fixes xs :: "'a list" shows "inversion_number xs = card (inversions xs)"
continuous_at_compose: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "isCont f a"   and "isCont g (f a)" shows "isCont (g \<circ> f) a"
reindex: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and h :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'c" assumes "comm_monoid_set f z"   and "inj_on h A" shows "comm_monoid_set.F f z g (h ` A) = comm_monoid_set.F f z (g \<circ> h) A"
simps_2: fixes M :: "'b nmultiset multiset" shows "|MSet M| = (let X = set_mset (image_mset depth_nmset M) in if X = {} then 0 else Suc (Max X))"
4: shows "ipd (\<pi> j) pd\<rightarrow> ipd (\<pi> k)"
mult: fixes x :: "'a"   and a :: "real" assumes "x \<in> H" shows "h (a \<cdot> x) = a * h x"
image_snd: fixes a :: "'a"   and b :: "'b"   and X :: "('a \<times> 'b) set" assumes "(a, b) \<in> X" shows "b \<in> snd ` X"
prefix_def: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "prefix f n \<equiv> map (\<lambda>x. the (f x)) [0..<Suc n]"
rel_fun_def: fixes A :: "'k \<Rightarrow> 'm \<Rightarrow> bool"   and B :: "'l \<Rightarrow> 'n \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
Prefixes_subset_\<P>: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u" shows "Prefixes p \<subseteq> \<P> k u"
border_len_3: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "\<^bold>|x\<^bold>| < \<^bold>|w\<^bold>|"
index_locale_axioms: shows "index_locale (\<le>) (<) local.idx local.size"
weakBisimulationCoinduct: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto>\<^sup>^<(X \<union> weakBisimulation)> S"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> (S, R) \<in> X" shows "P \<approx> Q"
mset_ran_def: fixes a :: "int \<Rightarrow> 'a"   and r :: "int set" shows "mset_ran a r \<equiv> image_mset a (mset_set r)"
66204: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> - (y \<squnion> - y \<sqinter> x) = bot"
f_Exec_eq_f_Exec_Stream_last_if: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun xs c = (if xs = [] then c else last (f_Exec_Comp_Stream trans_fun xs c))"
density_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
add_0_right: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> mbot = a"
push_down_rank_older_seniors: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and i :: "nat" assumes "rank x n = rank y n"   and "rank x n = Some i" shows "older_seniors x n = older_seniors y n"
in_measures_2: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> nat"   and fs :: "('a \<Rightarrow> nat) list" shows "((x, y) \<in> measures (f # fs)) = (f x < f y \<or> f x = f y \<and> (x, y) \<in> measures fs)"
maxc: fixes A :: "'a set" assumes "P A" shows "finite A \<and> card A \<le> N"
impl_of_inverse: fixes x :: "('a, 'b) hashmap" shows "HashMap (impl_of x) = x"
sets_null_measure: fixes M :: "'b measure" shows "sets (null_measure M) = sets M"
directional_derivative_in_tangent_space: fixes k :: "enat"   and a :: "'c"   and x :: "'c" assumes "k \<noteq> 0" shows "directional_derivative k a x \<in> manifold_eucl.tangent_space k a"
a1: shows "low_equal s1 s2 \<and> t1 = in_reg_mod w curr_win rd s1 \<and> t2 = in_reg_mod w curr_win rd s2"
singleton_Un_iff: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" shows "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
IH_2: fixes ass :: "'a \<Rightarrow> bool" assumes "ifex_no_twice e"   and "ifex_sat e = Some ass" shows "val_ifex e ass = True"
return_optionT_def: shows "Monad_Overloading.return = return_option Monad_Overloading.return"
no_initial_channel_snapshot: shows "\<forall>i. cs init i = ([], NotStarted)"
F'_def: fixes m :: "nat" shows "F' m = \<Inter> (E ` {m..})"
wfP_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP r = wf {(x, y). r x y}"
seqp_sendTE: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}send(s\<^sub>m\<^sub>s\<^sub>g) . p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = send (s\<^sub>m\<^sub>s\<^sub>g \<xi>)" and "\<xi>' = \<xi>" and "q = p"
antipar_2: shows "src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)\<^sup>* = trg (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)"
assms_1: shows "carrier G \<subseteq> A <#> B"
DirProd_list_def: fixes Gs :: "('c, 'd) monoid_scheme list" shows "DirProd_list Gs = foldr (\<lambda>G H. image_group (\<lambda>(x, xs). x # xs) (G \<times>\<times> H)) Gs nil_monoid"
flatten_app: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com" shows "flatten (sequence Seq (flatten c1 @ flatten c2)) = flatten c1 @ flatten c2"
primitive_root_unityD_3: fixes k :: "nat"   and x :: "'a"   and k' :: "nat" assumes "primitive_root_unity k x"   and "k' \<noteq> 0"   and "x ^ k' = (1::'a)" shows "k \<le> k'"
Pause_neq_DONE: fixes out :: "'g"   and c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption"   and a :: "'f" shows "Resumption.resumption.Pause out c \<noteq> DONE a"
left_tree_disjoint: fixes s :: "'c"   and t :: "'c" assumes "s \<rightarrow>\<^bsub>T\<^esub> t" shows "T.left_tree s t \<inter> T.left_tree t s = {}"
approximating_bigstep_undecided_to_undecided_in_doubt_allow_approx: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "(\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided \<or> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
Int_Diff_disjoint: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B \<inter> (A - B) = {}"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "I.SAT \<Phi> \<equiv> \<forall>\<xi>. I.wtE \<xi> \<longrightarrow> I.satPB \<xi> \<Phi>"