hcju/isabelleps huggingface dataset:
is_ref_map_def: fixes f :: "'b \<Rightarrow> 'c"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa" shows "is_ref_map f B A \<equiv> (\<forall>s\<in>start B. f s \<in> start A) \<and> (\<forall>s t a. reachable B s \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> t \<longrightarrow> (\<exists>e. refines e s a t A f))"
path_component_of_subset_topspace: fixes X :: "'c topology"   and x :: "'c" shows "path_component_of_set X x \<subseteq> topspace X"
reflects_isomorphic: fixes f :: "'a"   and f' :: "'a" assumes "ide f"   and "ide f'"   and "L f \<cong> L f'" shows "f \<cong> f'"
fg_def: shows "fg \<equiv> \<lambda>p. f p + g p"
i_Exec_Stream_expand_aggregate_map_take: fixes k :: "nat"   and n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag \<Down> n = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (input \<Down> n \<odot> k) c)) k ag"
induct: fixes x1 :: "'b ptrm"   and x2 :: "'b ptrm"   and P :: "'b ptrm \<Rightarrow> 'b ptrm \<Rightarrow> bool" assumes "x1 \<approx> x2"   and "P PUnit PUnit"   and "\<And>x. P (PVar x) (PVar x)"   and "\<And>A B C D. \<lbrakk>A \<approx> B; P A B; C \<approx> D; P C D\<rbrakk> \<Longrightarrow> P (PApp A C) (PApp B D)"   and "\<And>A B x T. \<lbrakk>A \<approx> B; P A B\<rbrakk> \<Longrightarrow> P (PFn x T A) (PFn x T B)"   and "\<And>a b A B T. \<lbrakk>a \<noteq> b; A \<approx> [a \<leftrightarrow> b] \<bullet> B; P A ([a \<leftrightarrow> b] \<bullet> B); a \<notin> ptrm_fvs B\<rbrakk> \<Longrightarrow> P (PFn a T A) (PFn b T B)"   and "\<And>A B C D. \<lbrakk>A \<approx> B; P A B; C \<approx> D; P C D\<rbrakk> \<Longrightarrow> P (PPair A C) (PPair B D)"   and "\<And>A B. \<lbrakk>A \<approx> B; P A B\<rbrakk> \<Longrightarrow> P (PFst A) (PFst B)"   and "\<And>A B. \<lbrakk>A \<approx> B; P A B\<rbrakk> \<Longrightarrow> P (PSnd A) (PSnd B)" shows "P x1 x2"
pq: shows "p = smult (inverse (rat_of_int c)) (of_int_poly q)"
under_def: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "under r a \<equiv> {b. (b, a) \<in> r}"
mult_less_cancel_left_disj: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
monotone_spmf: fixes x :: "'a" shows "monotone (ord_spmf (=)) (\<le>) (\<lambda>p. spmf p x)"
le_inf_range: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream" assumes "traces ind"   and "traces.le_inf ind w\<^sub>1 w\<^sub>2" shows "sset w\<^sub>1 \<subseteq> sset w\<^sub>2"
nontriv_def: shows "class.nontriv TYPE('a) \<equiv> 1 < CARD('a)"
assms_3: shows "poly (p * q) b \<noteq> 0"
dom_inv: fixes f :: "'b" assumes "D.iso f" shows "D.dom (D.inv f) = D.cod f"
division_union_intervals_exists: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "cbox a b \<noteq> {}" obtains p :: "'a set set" where "insert (cbox a b) p division_of cbox a b \<union> cbox c d"
enat_in_INF: fixes f :: "'a \<Rightarrow> enat"   and S :: "'a set" assumes "Inf (f ` S) \<noteq> top" obtains x :: "'a" where "x \<in> S" and "Inf (f ` S) = f x"
axioms_2: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g" assumes "graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry" shows "graph_Entry_axioms \<alpha>n invar inEdges' Entry"
diff_fun_def: fixes k :: "enat"   and charts :: "('f, 'g) chart set"   and f :: "'f \<Rightarrow> 'h" shows "diff_fun k charts f \<equiv> diff k charts charts_eucl f"
assms: shows "(\<lambda>x. ereal (real (u x))) \<longlonglongrightarrow> \<infinity>"
wf_bs_insensitive: fixes bs :: "'a list"   and bs' :: "'b list"   and p :: "poly" assumes "length bs = length bs'" shows "wf_bs bs p = wf_bs bs' p"
DisjE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]" shows "[\<phi> in v] \<or> [\<psi> in v]"
remove1_idem: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "remove1 x xs = xs"
that: fixes v' :: "'val" assumes "v' \<in> phiUses g m"   and "v' \<in> set vs" shows "thesis"
ivl_disj_int_two_7: fixes l :: "'a"   and m :: "'a"   and u :: "'a" shows "{l..<m} \<inter> {m..u} = {}"
rbt_sorted_entries: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "linorder.sorted less_eq (list.map fst (RBT_Impl.entries t))"
rel_transfer: fixes S1a :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S1c :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S2a :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and S2c :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "((S1a ===> S1c ===> (=)) ===> (S2a ===> S2c ===> (=)) ===> rel_prod S1a S2a ===> rel_prod S1c S2c ===> (=)) rel_prod rel_prod"
intro: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h shadow_root_ptr h' r ptr'. \<lbrakk>h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'; r \<in> get_tag_name_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_delete_shadow_root_get_tag_name get_tag_name_locs"
map_le_implies_dom_le: fixes f :: "'b \<Rightarrow> 'c option"   and g :: "'b \<Rightarrow> 'c option" assumes "f \<subseteq>\<^sub>m g" shows "dom f \<subseteq> dom g"
below_rel_def: fixes A :: "('a \<times> 'c) set" shows "\<langle>A\<rangle>below_rel = \<langle>A\<rangle>sctn_rel O br below_halfspace top"
inj_idom_divide_hom_axioms: shows "inj_idom_divide_hom of_real"
graph_homomorphism_def: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> f `` vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> univalent f \<and> edge_preserving f (edges G\<^sub>1) (edges G\<^sub>2))"
sublens_obs_get: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'e"   and \<rho> :: "'f" assumes "mwb_lens X"   and "Y \<subseteq>\<^sub>L X" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> v) = get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<rho> v)"
intros_9: fixes s :: "'a"   and b :: "'a set"   and x :: "'a set"   and y :: "'a \<Rightarrow> nat"   and c :: "'a com" assumes "s \<notin> b" shows "Sem (While b x y c) (Some s) (Some s)"
fds_nth_fds: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "fds_nth (fds f) n = (if n = 0 then 0::'b else f n)"
lborel_integrable_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'b"   and t :: "real" assumes "c \<noteq> 0" shows "integrable lborel (\<lambda>x. f (t + c * x)) = integrable lborel f"
lang_pnorm: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "project \<Sigma> project"   and "alphabet.wf wf_atom n r" shows "project.lang \<Sigma> project lookup n (pnorm r) = project.lang \<Sigma> project lookup n r"
take_bit_nat_eq_self: fixes m :: "nat"   and n :: "nat" assumes "m < 2 ^ n" shows "take_bit n m = m"
set_update_by_pair: fixes xs :: "('a \<times> 'c) list"   and v :: "'c"   and k :: "'a" assumes "oalist_inv_raw xs"   and "v \<noteq> (0::'c)" shows "set (update_by_pair (k, v) xs) = insert (k, v) (set xs - range (Pair k))"
y1prop: shows "r < y1' \<and> (\<forall>x\<in>{r<..y1'}. aEvalUni (EqUni (a, b, c)) x \<or> aEvalUni (LessUni (a, b, c)) x)"
bind_is_OK_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> ok g x" shows "h \<turnstile> ok (f \<bind> g)"
coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<le> b" shows "c \<le> a + b"
coerce_def: shows "COERCE('a, 'b) \<equiv> PRJ('b) oo EMB('a)"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiconnex A less"   and "irreflexive A less" shows "semiconnex_irreflexive A less"
simps_3: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Accept # rs) = Rule v Accept # process_ret rs"
simps_39: fixes f1 :: "'f \<Rightarrow> bool \<Rightarrow> 'e"   and f2 :: "'f \<Rightarrow> 'g \<Rightarrow> 'f option \<Rightarrow> 'e"   and f3 :: "'f \<Rightarrow> 'g \<Rightarrow> 'h option \<Rightarrow> 'e"   and f4 :: "bool \<Rightarrow> 'e"   and f5 :: "bool \<Rightarrow> 'e"   and f6 :: "gc_phase \<Rightarrow> 'e"   and x31 :: "'f"   and x32 :: "'g"   and x33 :: "'h option" shows "(case mw_Mutate_Payload x31 x32 x33 of mw_Mark x xa \<Rightarrow> f1 x xa | mw_Mutate x xa xb \<Rightarrow> f2 x xa xb | mw_Mutate_Payload x xa xb \<Rightarrow> f3 x xa xb | mw_fA x \<Rightarrow> f4 x | mw_fM x \<Rightarrow> f5 x | mw_Phase x \<Rightarrow> f6 x) = f3 x31 x32 x33"
simps_2: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c" shows "f ^^ Suc n = f \<circ> f ^^ n"
vector_component_7: fixes r :: "real"   and x :: "('b, 'c) vec"   and i :: "'c" shows "(r *\<^sub>R x) $ i = r *\<^sub>R x $ i"
QZ: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<turnstile>\<^sub>Z {P} c {Q}"
no_trailing_unfold: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "no_trailing P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (last xs))"
tm_const_def: fixes c :: "float" shows "tm_const c = TaylorModel (c)\<^sub>p 0"
dlp_ax: fixes x :: "'a" shows "x \<le> d x \<cdot> x"
le_Nil: fixes x :: "'b list" shows "(x \<le> []) = (x = [])"
fsubset_fempty: fixes A :: "'a fset" shows "(A |\<subseteq>| {||}) = (A = {||})"
getParts_disjoint_list: fixes rs :: "'b simple_rule list" shows "disjoint_list (map wordinterval_to_set (getParts rs))"
keys_proj_poly: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "keys (term_powerprod.proj_poly term_of_pair k p) = term_powerprod.pp_of_term pair_of_term ` {x \<in> keys p. term_powerprod.component_of_term pair_of_term x = k}"
axioms_1: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "currying A1 A2 B" shows "category A1"
induct: fixes P :: "ty \<Rightarrow> ty \<Rightarrow> Syntax.tvar \<Rightarrow> bool"   and a0 :: "ty"   and a1 :: "ty"   and a2 :: "Syntax.tvar" assumes "\<And>t' x. P One t' x"   and "\<And>t1 t2 t' x. \<lbrakk>P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (Fun t1 t2) t' x"   and "\<And>t1 t2 t' x. \<lbrakk>P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (Syntax.Sum t1 t2) t' x"   and "\<And>t1 t2 t' x. \<lbrakk>P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (Syntax.Prod t1 t2) t' x"   and "\<And>y t' x t. \<lbrakk>atom y \<sharp> (t', x); P t t' x\<rbrakk> \<Longrightarrow> P (Mu y t) t' x"   and "\<And>y t' x. P (Alpha y) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (AuthT t) t' x" shows "P a0 a1 a2"
interval_lemma2: fixes S :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "S \<subseteq> interval r a b"   and "x \<in> S" shows "(x, b) \<in> r"
rel_funD2: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'b" assumes "(A ===> B) f g"   and "A x x" shows "B (f x) (g x)"
fail_def: fixes S :: "'b \<Rightarrow> 'a" shows "fail S = - S \<top>"
succ_pred: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "isSucc r i" shows "succ r (pred r i) = i"
induction_axiomsp_induction_axioms_eq: shows "induction_axiomsp = (\<lambda>x. x \<in> induction_axioms)"
obj_simps_2: fixes a :: "'a" assumes "obj a" shows "src a = a"
mono: fixes a :: "nat"   and c :: "nat"   and b :: "nat"   and d :: "nat" assumes "a dvd c"   and "b dvd d" shows "gcd a b dvd gcd c d"
in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Fop.Bop.in_hom f a b" obtains "Fop.Bop.arr f" and "Fop.Bop.dom f = a" and "Fop.Bop.cod f = b"
calc_ideal_points_cmat_cvec_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "calc_ideal_points_cmat_cvec H = (if is_poincare_line_cmat H then let (A, B, C, D) = H in {calc_ideal_point1_cvec A B, calc_ideal_point2_cvec A B} else {(- 1, 1), (1, 1)})"
bin_lcp_fst_lcp: shows "symcode.\<alpha> \<le>p u\<^sub>0 \<cdot> symcode.\<alpha>"
AR_retract_of_AR: fixes T :: "'a set"   and S :: "'a set" assumes "AR T"   and "S retract_of T" shows "AR S"
Int_Inter_image: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Inter>x\<in>C. A x \<inter> B x) = \<Inter> (A ` C) \<inter> \<Inter> (B ` C)"
mem_ball: fixes y :: "'c"   and x :: "'c"   and e :: "real" shows "(y \<in> ball x e) = (dist x y < e)"
foo5b: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators"   and c :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C b)"   and "\<And>r. r \<in> set c \<Longrightarrow> x \<notin> dom (C r)" shows "applied_rule_rev C x (b # c) = \<lfloor>b\<rfloor>"
compl_sup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (sup x y) = inf (uminus x) (uminus y)"
nat_filtration_sets: fixes n :: "nat" shows "sets (nat_filtration n) = sigma_sets (space (bernoulli_stream p)) {pseudo_proj_True n -` B \<inter> space M |B. B \<in> sets (bernoulli_stream p)}"
assms_2: shows "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"
assms: assumes "vassn_tag (hn_ctxt A x y)"   and "P x" shows "P' x"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
piecewise_differentiable_on_def: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a set" shows "f piecewise_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> (\<forall>x\<in>i - S. f differentiable at x within i))"
ide_cod: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "VVV.ide (VVV.cod f)"
update\<B>\<I>_def: fixes field_update :: "(('b, 'c \<times> 'd) mapping \<Rightarrow> ('b, 'c \<times> 'd) mapping) \<Rightarrow> 'e \<Rightarrow> 'a"   and i :: "'c"   and x :: "'b"   and c :: "'d"   and s :: "'e" shows "update\<B>\<I> field_update i x c s = field_update (Mapping.update x (i, c)) s"
compactin_subtopology_imp_compact: fixes X :: "'b topology"   and S :: "'b set"   and K :: "'b set" assumes "compactin (subtopology X S) K" shows "compactin X K"
span_eq: fixes Sa :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and T :: "(('a \<Rightarrow> real) \<Rightarrow> real) set" assumes "Sa \<subseteq> sub.tangent_space p"   and "T \<subseteq> sub.tangent_space p" shows "(span Sa = span T) = (Sa \<subseteq> span T \<and> T \<subseteq> span Sa)"
wf3: shows "mat (row_length B1) (length B1) B1"
2: shows "0 \<le> x * sqrt (y\<^sup>2 - 1) + y * sqrt (x\<^sup>2 - 1)"
simps_2: fixes C :: "char list"   and e :: "char list exp" shows "fv (Cast C e) = fv e"
Q'''Trans: shows "Q''' \<longmapsto> \<alpha> \<prec> Q''"
pre_iso_test5: shows "\<forall>g\<in>set Pent. pre_iso_test g"
not_less_0: fixes x :: "V" shows "\<not> x < 0"
subsuming_states_def: fixes problem :: "(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"   and partial :: "nat \<Rightarrow> nat option" shows "ast_problem.subsuming_states problem partial \<equiv> {s \<in> ast_problem.valid_states problem. partial \<subseteq>\<^sub>m s}"
obj_simps_4: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "local.dom a = a"
DBM_reset_reset: fixes k :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and d :: "'c" assumes "0 < k"   and "k \<le> n" shows "DBM_reset M n k d (reset M n k d)"
bisim1AAssThrow3: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr expr1"   and i :: "'addr expr1"   and v' :: "'addr val"   and v :: "'addr val" assumes "P,e,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v', v], loc, length (compE2 a) + length (compE2 i) + pc, \<lfloor>ad\<rfloor>)"
assms_2: shows "\<sigma> (Suc k) = \<sigma> k"
l11_14: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and A' :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B A'"   and "A \<noteq> B"   and "A' \<noteq> B"   and "Bet C B C'"   and "B \<noteq> C"   and "B \<noteq> C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B C'"
subst_subst_compose: fixes t :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<tau> :: "'f \<Rightarrow> ('c, 'd) Term.term" shows "t \<cdot> \<sigma> \<circ>\<^sub>s \<tau> = t \<cdot> \<sigma> \<cdot> \<tau>"
simps_9: fixes x :: "name"   and P :: "pi" shows "valid (<\<nu>x>P) = valid P"
matroid_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> insert x X" shows "matroid \<E> (X_insert.indep_in \<E>)"
bidel_arc_same_face_axioms_def: fixes M :: "'c pre_map"   and a :: "'c" shows "bidel_arc_same_face_axioms M a \<equiv> pre_digraph_map.face_cycle_set M (edge_rev M a) = pre_digraph_map.face_cycle_set M a"
HEndPhase2_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4b s q" shows "HInv4b s' q"
iMODb_subset_0_conv: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and r :: "nat"   and m :: "nat" shows "([ r', mod m', c' ] \<subseteq> [ r, mod m, 0 ]) = (r' = r \<and> (m' = 0 \<or> c' = 0))"
semiring_norm_13: fixes m :: "num"   and n :: "num" shows "num.Bit0 m * num.Bit0 n = num.Bit0 (num.Bit0 (m * n))"
cases: fixes a1 :: "'c"   and a2 :: "'d"   and r :: "('d \<times> 'c) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'d" and b :: "'c" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
Cauchy_subseq_Cauchy: fixes "open" :: "'a set \<Rightarrow> bool"   and uniformity :: "('a \<times> 'a) filter"   and X :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> nat" assumes "class.uniform_space open uniformity"   and "uniform_space.Cauchy uniformity X"   and "strict_mono f" shows "uniform_space.Cauchy uniformity (X \<circ> f)"
preserves_isomorphic: fixes f :: "'c"   and g :: "'c" assumes "C.isomorphic f g" shows "D.isomorphic (G f) (G g)"
preserves_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>\<^sub>SC.cod (\<F>C.D f) = \<F>C.D (\<F>C.cod f)"
map_pmf_of_set_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)"
finite_Collect: fixes S :: "'h set"   and f :: "'i \<Rightarrow> 'h" assumes "finite S"   and "inj f" shows "finite {a. f a \<in> S}"
kerI: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "f a = (0::'a)" shows "a \<in> ker f"
0: shows "\<psi> \<mu>' = \<psi>.map \<mu>'"
uniformity_fmap_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
orthogonal_Gram_Schmidt_matrix: fixes A :: "(('d, 'f) vec, 'e) vec" shows "pairwise orthogonal {column i (Gram_Schmidt_matrix A) |i. i \<in> UNIV}"
lex_fun_strict_def: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "lex_fun_strict s t = strict lex_fun s t"
irreducible_not_unit: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p" shows "\<not> dvd.dvd times p one"
1: shows "\<turnstile> \<not> WF(A)_v = (\<diamond>\<box>Enabled \<langle>A\<rangle>_v \<and> \<not> \<box>\<diamond>\<langle>A\<rangle>_v)"
assms: shows "secure (c_process step out s\<^sub>0) I (c_dom D)"
mkAutoSim_simps_3: fixes a :: "'a" shows "pAct (mkAutoSim a) = simAction a"
rel_mono_strong0: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) F1"   and y :: "('e, 'g, 'i) F1"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3a :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "F1rel R1 R2 R3 x y"   and "\<forall>z1\<in>F1set1 x. \<forall>y1\<in>F1set1 y. R1 z1 y1 \<longrightarrow> R1a z1 y1"   and "\<forall>z2\<in>F1set2 x. \<forall>y2\<in>F1set2 y. R2 z2 y2 \<longrightarrow> R2a z2 y2"   and "\<forall>z3\<in>F1set3 x. \<forall>y3\<in>F1set3 y. R3 z3 y3 \<longrightarrow> R3a z3 y3" shows "F1rel R1a R2a R3a x y"
ct_suffix_cons_ct_list: fixes xs :: "ct list"   and ys :: "ct list"   and zs :: "t list" assumes "ct_suffix (xs @ ys) (to_ct_list zs)" shows "\<exists>as bs. zs = as @ bs \<and> ct_list_eq ys (to_ct_list bs) \<and> ct_suffix xs (to_ct_list as)"
BrnFT_incl_brnL: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "length cl = length dl"   and "properL cl" shows "BrnFT cl dl \<subseteq> {..<brnL cl (length cl)}"
simps_2: fixes v :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and uv :: "unit" shows "BsSkewBinomialHeapStruc.bs_meld (Inr v) (Inl uv) = Inr v"
contour_integral_uniform_limit_circlepath_2: fixes f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and F :: "'a filter"   and l :: "complex \<Rightarrow> complex" assumes "\<forall>\<^sub>F n in F. f n contour_integrable_on circlepath z r"   and "uniform_limit (sphere z r) f l F"   and "F \<noteq> bot"   and "0 < r" shows "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"