hcju/isabelleps huggingface dataset:
induct: fixes P :: "'a \<Rightarrow> bool"   and a0 :: "'a" assumes "\<And>v. (\<not> (v = s \<or> dist v = \<infinity> \<or> v \<notin> verts G) \<Longrightarrow> P (tail G (the (parent_edge v)))) \<Longrightarrow> P v" shows "P a0"
hyps: assumes "\<And>i. i < length ts \<Longrightarrow> 0 < f i" shows "sum_list (map weight ts) \<le> sum_list (map weight (concat (map2 (\<lambda>x i. replicate (f i) x) ts [0..<length ts])))"
assms: shows "cont (\<lambda>p. f (fst p) (snd p))"
in_hom_def: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr"   and b :: "('o, 'a) arr" shows "in_hom f a b \<equiv> a \<in> domains f \<and> b \<in> codomains f"
is_PureE: fixes generat :: "('d, 'e, 'f) generat" assumes "is_Pure generat" obtains x :: "'d" where "generat = Pure x"
algebraicE: fixes x :: "'b" assumes "algebraic x" obtains p :: "'b poly" where "\<And>i. coeff p i \<in> \<int>" and "p \<noteq> 0" and "poly p x = (0::'b)"
base: fixes ks :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "P ks [] fs (ks, fs)"
assms_3: shows "ipresIGSwapAbs hA' MOD' MOD''"
restrict_UNIV: fixes f :: "'c \<Rightarrow> 'd" shows "restrict f UNIV = f"
rb_aux_inv_D3_3: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv (bs, ss, ps)"   and "Inl (p, q) \<in> set ps" shows "is_regular_spair p q"
path_cons_simp: fixes v :: "'form \<times> nat list"   and v' :: "'form \<times> nat list"   and e :: "(('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent"   and pth :: "((('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent) list" shows "path v v' (e # pth) = (fst (fst e) = v \<and> e \<in> edges \<and> path (fst (snd e)) v' pth)"
chain1: fixes f :: "'b \<Rightarrow> 'a" assumes "f \<in> Z" shows "Complete_Partial_Order.chain (\<le>) (f ` Y)"
fdia_rel_im2: fixes R :: "('a \<times> 'a) set"   and P :: "('a \<times> 'a) set" shows "s2r (R\<inverse> `` r2s (rdom P)) = relfdia R P"
discrL_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list" assumes "PL_Indis cval indis" shows "PL_Indis.discrL aval tval cval indis cl \<equiv> \<forall>c\<in>set cl. PL_Indis.discr aval tval cval indis c"
b_ag_def: fixes A :: "('c, 'd) aGroup_scheme" shows "b_ag A = \<lparr>carrier = carrier A, Group.top = (\<plusminus>\<^bsub>A\<^esub>), iop = mop A, one = \<zero>\<^bsub>A\<^esub>\<rparr>"
ln_Gamma_series_complex_converges': fixes z :: "complex" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "\<exists>d>0. uniformly_convergent_on (ball z d) (\<lambda>n z. ln_Gamma_series z n)"
eval_preserves_type: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E \<turnstile> e :: T" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
assms_2: shows "c \<turnstile> msg \<mapsto> c'"
distinct_1: fixes x11 :: "char list"   and x12 :: "fterm list"   and x2 :: "char list" shows "Fun x11 x12 \<noteq> Var x2"
convergent_LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'c" shows "convergent X = X \<longlonglongrightarrow> lim X"
rep_eq: fixes x :: "'state"   and xa :: "('b, 'pred, 'act) formula" shows "x \<Turnstile> xa = valid_Tree\<^sub>\<alpha> x (Rep_formula xa)"
divides_prod_gcd: fixes A :: "'a mod_ring set"   and h :: "'a mod_ring poly"   and f :: "'a mod_ring poly" assumes "\<forall>c1 c2. c1 \<in> A \<and> c2 \<in> A \<and> c1 \<noteq> c2 \<longrightarrow> comm_monoid_mult_class.coprime (h - [:c1:]) (h - [:c2:])" shows "(\<Prod>c\<in>A. gcd f (h - [:c:])) dvd f"
abs_neg_floor: fixes b :: "int"   and Bnd :: "'b" assumes "\<bar>of_int b\<bar> \<le> Bnd" shows "- \<lfloor>Bnd\<rfloor> \<le> b"
sinh_diff: fixes x :: "'a"   and y :: "'a" shows "sinh (x - y) = sinh x * cosh y - cosh x * sinh y"
eq: shows "{[b, a]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r} = (\<lambda>x. [x\<lparr>1\<^sub>\<nat>\<rparr>, x\<lparr>0\<^sub>\<nat>\<rparr>]\<^sub>\<circ>) ` elts (fpairs r)"
new_element_def: shows "new_element = Heap_Error_Monad.bind get_heap (\<lambda>h. Heap_Error_Monad.bind (return (new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h)) (\<lambda>(new_ptr, h'). Heap_Error_Monad.bind (return_heap h') (\<lambda>_. return new_ptr)))"
atMost_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> ord.atMost (\<lambda>x y. y \<le> x) k) = (k \<le> i)"
exhaust_sel: fixes "term" :: "('a, 'b) Term.term" obtains "term = Var (the_Var term)"   | "term = Fun (un_Fun1 term) (args term)"
that: fixes C :: "'b set" assumes "C \<subseteq> A"   and "B = f ` C"   and "inj_on f C" shows "thesis"
intro: fixes is_open :: "'d set \<Rightarrow> bool"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<And>U. is_open U \<Longrightarrow> ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)"   and "\<And>U V x. \<lbrakk>is_open U; is_open V; V \<subseteq> U; x \<in> \<FF> U\<rbrakk> \<Longrightarrow> (\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
fresh_prod: fixes a :: "'b"   and x :: "'c"   and y :: "'d" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
wf_trm_subst_range_Var: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range Var)"
free_hull_mono: fixes G :: "'b list set"   and H :: "'b list set" assumes "G \<subseteq> H" shows "\<langle>G\<rangle>\<^sub>F \<subseteq> \<langle>H\<rangle>\<^sub>F"
1: shows "dom\<^bsub>CatDom (F ;;: G)\<^esub> f \<in> obj\<^bsub>CatDom F\<^esub> \<and> cod\<^bsub>CatDom (F ;;: G)\<^esub> f \<in> obj\<^bsub>CatDom F\<^esub>"
simps: fixes v :: "'form \<times> nat list" shows "terminal_vertex v = (v |\<in>| vertices \<and> terminal_node (nodeOf v))"
iso_is_retraction: fixes f :: "'s" assumes "local.iso f" shows "retraction f"
l_def: shows "l = Max {i. i < j \<and> k \<le> i \<and> C \<in> lnth Ns i}"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "LNil \<noteq> LCons x21 x22"
monotone_eqI: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "class.preorder ord (mk_less ord)" shows "monotone (=) ord f"
simps: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "stack (\<Gamma>, e, S) = S"
sse_invar_def: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "sse_invar \<Sigma>i R = {(\<Sigma>, W). W \<subseteq> \<Sigma> \<and> \<Sigma> \<subseteq> R\<^sup>* `` \<Sigma>i \<and> (\<forall>\<sigma>\<in>R\<^sup>* `` \<Sigma>i - \<Sigma>. \<exists>\<sigma>h\<in>W. (\<sigma>h, \<sigma>) \<in> (R - UNIV \<times> \<Sigma>)\<^sup>*)}"
Sem_eval_cases_5: fixes s :: "Var \<Rightarrow> Val"   and b :: "BExpr"   and c1 :: "IMP"   and c2 :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , Iff b c1 c2 \<rightarrow>\<^sub>n  t " obtains na :: "nat" where "n = Suc na__" and "evalB b s" and " s , c1 \<rightarrow>\<^sub>na__  t "   | na :: "nat" where "n = Suc na" and "\<not> evalB b s" and " s , c2 \<rightarrow>\<^sub>na  t "
assms_2: shows "(sctnsi, sctns) \<in> sctns_rel"
G_edge_succ_a'_no_loop: assumes "head G a \<noteq> tail G a" shows "edge_succ M a' = a'"
rep_eq: fixes x :: "complex_homo_coords"   and xa :: "complex_homo_coords" shows "(x \<approx> xa) = (Rep_complex_homo_coords x \<approx>\<^sub>v Rep_complex_homo_coords xa)"
is_prefix_def: fixes ls :: "'a list"   and vs :: "'a list" shows "is_prefix ls vs \<equiv> \<exists>bs. vs = ls @ bs"
axiom_allimp_def: fixes x :: "String.literal"   and p :: "fol fm"   and q :: "fol fm" shows "axiom_allimp x p q \<equiv> Thm (Imp (Forall x (Imp p q)) (Imp (Forall x p) (Forall x q)))"
sup_right_divisibility: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<exists>z. z \<squnion> x = y)"
fun_sum_list_commute: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "f (0::'b) = (0::'a)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
cont_Seq_notFinished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and c1 :: "('c, 'd, 'a) cmd"   and s :: "'b"   and i :: "nat"   and c2 :: "('c, 'd, 'a) cmd" assumes "PL cval"   and "\<not> finished (PL.cont aval tval cval c1 s i)" shows "PL.cont aval tval cval (c1 ;; c2) s i = PL.cont aval tval cval c1 s i ;; c2"
assms_3: shows "vsb \<subseteq> fmdom' fmb"
A: fixes n :: "nat" assumes "N \<le> n" shows "p n * f n < p (Suc n) * f (Suc n)"
Span_mem_iff_length_version: fixes Us :: "'a list"   and a :: "'a" assumes "set Us \<subseteq> carrier R" shows "(a \<in> Span K Us) = (\<exists>Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = combine Ks Us)"
ord0_prec_induct: fixes P :: "ord0 \<Rightarrow> bool"   and a :: "ord0" assumes "\<And>x. \<forall>y. (y, x) \<in> ord0_prec\<^sup>+ \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
homeomorphic_minimal: fixes s :: "'c set"   and t :: "'d set" shows "s homeomorphic t = (\<exists>f g. (\<forall>x\<in>s. f x \<in> t \<and> g (f x) = x) \<and> (\<forall>y\<in>t. g y \<in> s \<and> f (g y) = y) \<and> continuous_on s f \<and> continuous_on t g)"
is_processT7_S: fixes s :: "'c event list"   and t :: "'c event list"   and P :: "'c process" assumes "tickFree s"   and "front_tickFree t"   and "s \<in> D P" shows "s @ t \<in> D P"
tl_lt_stack_hd_discover: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and x :: "'c" assumes "DFS_invar G param s"   and "stack s \<noteq> []"   and "x \<in> set (tl (stack s))" shows "the_discovered s x < the_discovered s (hd (stack s))"
infinite_iff_countable_subset: fixes S :: "'c set" shows "infinite S = (\<exists>f. inj f \<and> range f \<subseteq> S)"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (\<ll> a)"
square_free_map_poly: fixes f :: "rat poly" shows "square_free (map_poly of_rat f) = square_free f"
i_Exec_Stream_Init_nth_Suc_calc: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c (Suc n) = trans_fun (input n) (i_Exec_Comp_Stream_Init trans_fun input c n)"
preserves_arr: fixes f :: "'a" assumes "hom.arr f" shows "hom.arr (hom.map f)"
shext_def: fixes sh :: "sheap"   and sh' :: "sheap" shows "sh \<unlhd>\<^sub>s sh' \<equiv> \<forall>C sfs i. sh C = \<lfloor>(sfs, i)\<rfloor> \<longrightarrow> (\<exists>sfs' i'. sh' C = \<lfloor>(sfs', i')\<rfloor> \<and> i \<le>\<^sub>i i')"
adv_P1_def: fixes m1 :: "'a \<times> 'a"   and m2 :: "bool"   and D :: "('a \<times> 'a) \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> bool spmf" shows "sim_def.adv_P1 m1 m2 D \<equiv> \<bar>spmf (R1 m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> D)) True\<bar>"
weakTrans_def: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<equiv> P \<Longrightarrow>\<alpha> \<prec> P' \<or> \<alpha> = \<tau> \<and> P = P'"
right_interface_def: shows "right_interface \<equiv> corec_converter (\<lambda>conv a. map_gpv (map_prod id Inr) id (right_gpv (run_converter conv a)))"
inner_right_distrib: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner x (y + z) = inner x y + inner x z"
partList1_partList2_equi: fixes s :: "'b set"   and ts :: "'b set list" shows "set (partList1 s ts) - {{}} = set (partList2 s ts) - {{}}"
restrict_top_impl_def: fixes e :: "'ni"   and vr :: "'a"   and vl :: "bool"   and s :: "'s" shows "restrict_top_impl e vr vl s = case_ifexi (\<lambda>s. Some (e, s)) (\<lambda>s. Some (e, s)) (\<lambda>v te ee s. Some (if v = vr then if vl then te else ee else e, s)) e s"
Labels_Seq1: fixes c\<^sub>1 :: "cmd"   and l :: "nat"   and c :: "cmd"   and c\<^sub>2 :: "cmd" assumes "labels c\<^sub>1 l c" shows "labels (c\<^sub>1;; c\<^sub>2) l (c;; c\<^sub>2)"
simps_9: fixes v :: "'d" shows "generalize_instr (IStore v) = IStore v"
infinite_imp_nonempty: fixes S :: "'b set" assumes "infinite S" shows "S \<noteq> {}"
healthy_sound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "sound (t P)"
that: fixes a :: "'a"   and b :: "'a" assumes "x \<in> {a<--<b}"   and "transversal_segment a b" shows "thesis"
distinct_4: fixes x31 :: "'d"   and x32 :: "'b"   and x1 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "tPred x31 x32 \<noteq> tConj x1"
that: fixes XGs :: "(('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set \<times> ('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set) set" assumes "\<And>X G. (X, G) \<in> XGs \<Longrightarrow> flowsto X T CX G"   and "Gs = snd ` XGs"   and "X = \<Union> (fst ` XGs)" shows "thesis"
a_def: shows "a = (\<lambda>p. if prime p then - ln (1 - 1 / real p) - 1 / real p else 0)"
path_component_of_disjoint: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "disjnt (path_component_of_set X x) (path_component_of_set X y) = (\<not> path_component_of X x y)"
differentiable_transform_within: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and d :: "real"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable at x within s"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "g differentiable at x within s"
irrefl_trans_imp_antisym: fixes r :: "('b \<times> 'b) set" assumes "irrefl r"   and "trans r" shows "antisym r"
un: fixes f :: "nat \<Rightarrow> real set" assumes "disjoint_family f"   and "\<And>i. f i \<in> sets borel"   and "\<And>i. P (f i)" shows "P (\<Union> (range f))"
compact_attains_sup: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq t s"
next_state_def: fixes s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "next_state s execs = snd (snd (control s (current s) (execs (current s))))"
sub: fixes a :: "'a" assumes "a \<in> I" shows "a \<in> R"
ne_INF_mono: fixes U\<^sub>2 :: "'c set"   and U\<^sub>3 :: "'d set"   and B :: "'c set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'al"   and g :: "'c \<Rightarrow> 'al" assumes "U\<^sub>2 \<noteq> {}"   and "U\<^sub>3 \<noteq> {}"   and "B \<subseteq> U\<^sub>2"   and "A \<subseteq> U\<^sub>3"   and "\<forall>x\<in>U\<^sub>3. f x \<in> U"   and "\<forall>x\<in>U\<^sub>2. g x \<in> U"   and "\<And>m. \<lbrakk>m \<in> U\<^sub>2; m \<in> B\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. f x \<le>\<^sub>o\<^sub>w g m" shows "\<Sqinter>\<^sub>o\<^sub>w (f ` A) \<le>\<^sub>o\<^sub>w \<Sqinter>\<^sub>o\<^sub>w (g ` B)"
inf_of_singletonI: fixes x :: "'a" assumes "x \<in> carrier L" shows "is_glb L x {x}"
is_of_type_map_ofE: fixes params :: "('a \<times> PDDL_STRIPS_Semantics.type) list"   and x :: "'a"   and T :: "PDDL_STRIPS_Semantics.type" assumes "is_of_type (map_of params) x T" obtains i :: "nat" and xT :: "PDDL_STRIPS_Semantics.type" where "i < length params" and "params ! i = (x, xT)" and "of_type xT T"
ann_com_validity_def: fixes c :: "'b ann_com"   and q :: "'b set" shows "\<Turnstile> c q \<equiv> ann_SEM c (pre c) \<subseteq> q"
psubset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
before_conv_take: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "before x xs = set (take (index xs x) xs)"
count_roots_code: fixes p :: "real poly" shows "count_roots p = (if p = 0 then 0 else let ps = sturm p in sign_changes_neg_inf ps - sign_changes_inf ps)"
inline1_in_sub_gpvs: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('c, 'call, 'ret) rpv"   and gpv :: "('c, 'call, 'ret) gpv"   and s :: "'s"   and x :: "'ret"   and s' :: "'s"   and input :: "'ret'" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "(x, s') \<in> results_gpv \<I>' (callee' input)"   and "input \<in> responses_\<I> \<I>' out"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "rpv' x \<in> sub_gpvs \<I> gpv \<and> I s'"
acyclic_4_def: fixes x :: "'a" shows "acyclic_4 x \<equiv> \<forall>y\<le>x. x \<sqinter> y\<^sup>\<star> \<le> - - y"
h_3: shows "isometry_on {dist x y..dist x z} h"
map_cong0: fixes x :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in># x \<Longrightarrow> f z = g z" shows "image_mset f x = image_mset g x"
true_clss_union: fixes I :: "'b set"   and CC :: "'b literal multiset set"   and DD :: "'b literal multiset set" shows "(I \<TTurnstile>s CC \<union> DD) = (I \<TTurnstile>s CC \<and> I \<TTurnstile>s DD)"
l_def: shows "\<forall>m. \<exists>n>m. kth_res_equals k l (s n)"
factorization_def: fixes Factors :: "'a poly set"   and f :: "'a poly"   and cfs :: "'a \<times> 'a poly multiset" shows "factorization Factors f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> f = Polynomial.smult c (\<Prod>\<^sub># fs) \<and> set_mset fs \<subseteq> Factors"
rsd_winners_subset: fixes agents' :: "'agent list"   and alts' :: "'alt set" assumes "set agents' \<subseteq> agents" shows "rsd_winners R alts' agents' \<subseteq> alts'"
less_Liminf_iff: fixes F :: "'b filter"   and X :: "'b \<Rightarrow> 'a"   and C :: "'a" shows "(Liminf F X < C) = (\<exists>y<C. \<exists>\<^sub>F x in F. X x \<le> y)"
z_signs_R1: fixes signs :: "rat list"   and I :: "nat list" assumes "list_all (\<lambda>i. i < length signs) I"   and "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs" shows "prod_list (map ((!) (map_sgas signs)) I) = 1 \<or> prod_list (map ((!) (map_sgas signs)) I) = 0"
has_limits_of_shape_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C" shows "category.has_limits_of_shape C J \<equiv> \<forall>D. diagram J C D \<longrightarrow> (\<exists>a \<chi>. limit_cone J C D a \<chi>)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "semigroup_add_hom hom"   and "injective hom" shows "inj_semigroup_add_hom hom"
lookup_tensor_from_lookup: fixes "is" :: "nat list"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'a" assumes "is \<lhd> ds" shows "lookup (tensor_from_lookup ds e) is = e is"
atm_of_map_literal: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b literal" shows "atm_of (map_literal f l) = f (atm_of l)"
preserves_arr: fixes f :: "'d" assumes "D.arr f" shows "C.arr (F f)"
abs_fresh_2: fixes b :: "coname"   and a :: "coname"   and x :: "'a" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
P_widen: fixes c :: "'a acom"   and c' :: "'a acom" assumes "P c"   and "P c'" shows "P (c \<nabla>\<^sub>c c')"
parse_text_consumes: fixes ts :: "char list"   and t :: "char list option"   and ts' :: "char list" assumes "ts \<noteq> []"   and "hd ts \<noteq> CHR ''<''"   and "parse_text ts = Inr (t, ts')" shows "length ts' < length ts"
Worder_axioms_def: fixes D :: "('c, 'd) Order_scheme" shows "Worder_axioms D \<equiv> \<forall>X. X \<subseteq> carrier D \<and> X \<noteq> {} \<longrightarrow> (\<exists>x. minimum_elem D X x)"
mono_nat_linear_lb: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat"   and k :: "nat" assumes "\<And>m n. m < n \<Longrightarrow> f m < f n" shows "f m + k \<le> f (m + k)"
bnds_powr_def: fixes prec :: "nat"   and l1 :: "float"   and u1 :: "float"   and l2 :: "float"   and u2 :: "float" shows "bnds_powr prec l1 u1 l2 u2 = (if l1 = 0 \<and> u1 = 0 then Some (0, 0) else if l1 = 0 \<and> 1 \<le> l2 then let uln = the (ub_ln prec u1) in Some (0, ub_exp prec (float_round_up prec (uln * (if 0 \<le> uln then u2 else l2)))) else if l1 \<le> 0 then None else Some (map_bnds lb_exp ub_exp prec (bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2)))"
gfp_eqI: fixes F :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "mono F"   and "F x = x"   and "\<And>z. F z = z \<Longrightarrow> z \<le> x" shows "gfp F = x"
intros: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "\<And>x a. ae x = up\<cdot>a \<Longrightarrow> eq a (\<rho>1 x) (\<rho>2 x)" shows "eq\<rho> ae \<rho>1 \<rho>2"
simps_1: fixes arr :: "(nat \<times> 'a) list" shows "add_spvec arr [] = arr"
run_fail_state: fixes fail :: "'a"   and s :: "'b" shows "run_state (fail_state fail) s = fail"
older_seniors_recursive: fixes x :: "nat"   and n :: "nat" assumes "older_seniors x n \<noteq> {}" shows "older_seniors x n = {Max (older_seniors x n)} \<union> older_seniors (Max (older_seniors x n)) n"
fold_congs_1: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "CPU_register \<Rightarrow> word32"   and f :: "(CPU_register \<Rightarrow> word32) \<Rightarrow> CPU_register \<Rightarrow> word32"   and f' :: "(CPU_register \<Rightarrow> word32) \<Rightarrow> CPU_register \<Rightarrow> word32" assumes "r = r'"   and "cpu_reg r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "cpu_reg_update f r = cpu_reg_update f' r'"
lnth_lappend: fixes xs :: "'c llist"   and ys :: "'c llist"   and n :: "nat" shows "lappend xs ys $ n = (if enat n < llength xs then xs $ n else ys $ n - the_enat (llength xs))"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Dom (t \<^bold>\<otimes> u) = Dom t \<^bold>\<otimes> Dom u"
delete_not_domA: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" assumes "x \<notin> domA \<Gamma>" shows "delete x \<Gamma> = \<Gamma>"
no_loop: fixes x :: "'v vertex" shows "\<not> edge \<Delta>'' x x"
m_comm: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> y = y \<oplus>\<^bsub>F\<^esub> x"
finite_insert: fixes a :: "'e"   and A :: "'e set" shows "finite (insert a A) = finite A"