hcju/isabelleps huggingface dataset:
to_nat_from_nat_i: shows "mod_type_class.to_nat (mod_type_class.from_nat i) \<le> nrows A - Suc 0"
mono_gamma: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "\<gamma> a \<subseteq> \<gamma> b"
lsbis2_incl: fixes B1 :: "'e set"   and B2 :: "'d set"   and s1 :: "'e \<Rightarrow> ('f, 'e, 'd) F1"   and s2 :: "'d \<Rightarrow> ('g, 'e, 'd) F2" shows "lsbis2 B1 B2 s1 s2 \<subseteq> B2 \<times> B2"
d'_2: fixes x :: "'a" assumes "x \<in> {a--b}" shows "0 < d' x"
c: shows "\<forall>fi\<in>set fs. irreducible\<^sub>d fi \<and> 0 < degree fi"
set_map: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b" shows "S.set (HomC.map (b, a)) = HomC.set (b, a)"
subst_pp_extend_indets_subst: fixes t :: "'b \<Rightarrow>\<^sub>0 nat" shows "subst_pp extend_indets_subst t = monomial (1::'c) (\<Sum>y\<in>keys t. monomial (lookup t y) (Some y))"
odlist_all_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b odlist" shows "odlist_all P xs = (\<forall>x\<in>toSet xs. P x)"
cr_view\<^sub>m_def: shows "cr_view\<^sub>m \<equiv> \<lambda>x y. x = from_view\<^sub>m y"
is_nextElem_nth1: fixes ls :: "'a list"   and x :: "'a"   and y :: "'a" assumes "is_nextElem ls x y" shows "\<exists>i j. i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i mod |ls| = j"
subset_minus_empty: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A - B = {}"
refine: fixes ai :: "'b list"   and a :: "'c set"   and sctnsi :: "real list sctn list"   and sctns :: "'c sctn set" assumes "(ai, a) \<in> appr_rel"   and "(sctnsi, sctns) \<in> sctns_rel" shows "(RETURN (sbelow_sctns_impl ai sctnsi), sbelow_sctns $ a $ sctns) \<in> \<langle>bool_rel\<rangle>nres_rel"
multiplicity_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (power.power one times p (factorial_semiring.multiplicity times one p x)) x"
rep_eq: fixes x :: "'b"   and xa :: "'a"   and xb :: "'b \<Rightarrow> 'b"   and xc :: "('a, 'b) alist" shows "alist.impl_of (update_with x xa xb xc) = update_with_aux x xa xb (alist.impl_of xc)"
string_rel_string_assn: fixes c :: "String.literal"   and a :: "char list" shows "\<up> ((c, a) \<in> string_rel) = string_assn a c"
intro: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "\<And>B N. \<lbrakk>B \<in> Bot; calculus.reduc_saturated Inf Red_I Red_F N; entails N {B}\<rbrakk> \<Longrightarrow> \<exists>B'\<in>Bot. B' \<in> N" shows "reducedly_statically_complete_calculus_axioms Bot Inf entails Red_I Red_F"
inj_finite_single: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'e" assumes "inj f" shows "finite {y. x = f y}"
Kleene_iter_lpfp: fixes f :: "'g \<Rightarrow> 'g"   and p :: "'g"   and k :: "nat" assumes "mono f"   and "f p \<le> p" shows "(f ^^ k) bot \<le> p"
Un: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M" shows "a \<union> b \<in> M"
LLq: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "'b"   and t2 :: "'b" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "Syntax_Arith.LLq var Var FvarsT Fvars subst eql exi pls t1 t2 \<in> fmla"
weakBisimParComm: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<approx> Q \<parallel> P"
circ_simulate_left: fixes x :: "'a"   and z :: "'a"   and y :: "'a"   and w :: "'a" shows "x * z \<le> z * y \<squnion> w \<longrightarrow> x\<^sup>\<circ> * z \<le> (z \<squnion> x\<^sup>\<circ> * w) * y\<^sup>\<circ>"
map_ran_append: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and \<Gamma> :: "('b \<times> 'c) list"   and \<Delta> :: "('b \<times> 'c) list" shows "map_ran f (\<Gamma> @ \<Delta>) = map_ran f \<Gamma> @ map_ran f \<Delta>"
Scons_inject: fixes M :: "('b, 'c) Old_Datatype.node set"   and N :: "('b, 'c) Old_Datatype.node set"   and M' :: "('b, 'c) Old_Datatype.node set"   and N' :: "('b, 'c) Old_Datatype.node set" assumes "Scons M N = Scons M' N'" obtains "M = M'" and "N = N'"
has_bochner_integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a"   and j :: "'a" assumes "a \<le> c"   and "c \<le> b"   and "has_bochner_integral (lebesgue_on {a..c}) f i"   and "has_bochner_integral (lebesgue_on {c..b}) f j" shows "has_bochner_integral (lebesgue_on {a..b}) f (i + j)"
lehmers_theorem: fixes p :: "nat"   and a :: "nat" assumes "2 \<le> p"   and "\<And>x. x \<in># prime_factorization (p - 1) \<Longrightarrow> [a ^ ((p - 1) div x) \<noteq> 1] (mod p)"   and "[a ^ (p - 1) = 1] (mod p)" shows "prime p"
step: shows "LeftDerives1 (prefix @ [X] @ b2) e r (b1 @ [X] @ b2)"
has_derivative_id: fixes a :: "'a" shows "(id has_derivative id) (at a)"
set_eq_is_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A {.=}\<^bsub>L\<^esub> B) = (A = B)"
preallocatedE: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and C :: "char list"   and P :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<Rightarrow> char list \<Rightarrow> bool" assumes "preallocated h"   and "C \<in> sys_xcpts"   and "\<And>S. h (addr_of_sys_xcpt C) = \<lfloor>(C, S)\<rfloor> \<Longrightarrow> P h C" shows "P h C"
is_full_pmdlI: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>p. component_of_term ` keys p \<subseteq> component_of_term ` Keys B \<Longrightarrow> p \<in> pmdl B" shows "is_full_pmdl B"
f_prop: shows "f \<in> B - B' \<rightarrow>\<^sub>E P"
select_mset_cong: fixes xs :: "'a list"   and ys :: "'a list"   and k :: "nat" assumes "mset xs = mset ys" shows "select k xs = select k ys"
smc_SemiCAT_Comp: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Comp\<rparr> = (\<lambda>\<GG>\<FF>\<in>\<^sub>\<circ>composable_arrs (smc_SemiCAT \<alpha>). \<GG>\<FF>\<lparr>[]\<^sub>\<circ>\<rparr> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>\<FF>\<lparr>1\<^sub>\<nat>\<rparr>)"
p: shows "atom ` (p' \<bullet> insert i V) \<sharp>* insert i V"
croot_0_2: fixes x :: "complex" shows "croot 0 x = 0"
indefinite_integral_uniformly_continuous: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "f integrable_on cbox a b" shows "uniformly_continuous_on (cbox (a, a) (b, b)) (\<lambda>y. integral (cbox (fst y) (snd y)) f)"
expands_to_add: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a"   and g :: "real \<Rightarrow> real"   and G :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "(g expands_to G) basis" shows "((\<lambda>x. f x + g x) expands_to F + G) basis"
interpret_floatarith_FDERIV_floatarith_eucl_of_env: fixes xs :: "nat list"   and fa :: "floatarith"   and vs :: "real list"   and ds :: "nat list" assumes "\<And>i. i < DIM('a) \<Longrightarrow> isDERIV (xs ! i) fa vs"   and "freshs_floatarith fa ds"   and "length xs = DIM('a)"   and "length ds = DIM('a)"   and "\<And>i. i \<in> set xs \<Longrightarrow> i < length vs"   and "distinct xs"   and "\<And>i. i \<in> set ds \<Longrightarrow> i < length vs"   and "distinct ds" shows "((\<lambda>x. interpret_floatarith fa (list_updates xs (list_of_eucl x) vs)) has_derivative (\<lambda>d. interpret_floatarith (FDERIV_floatarith fa xs (map Var ds)) (list_updates ds (list_of_eucl d) vs))) (at (eucl_of_env xs vs))"
simps_41: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x5 :: "('c, 'd) fm" shows "(case \<^bold>\<diamond> x5 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f5 x5"
infinite_iff_card_of_countable: fixes B :: "'b set"   and A :: "'c set" assumes "countable B"   and "infinite B" shows "infinite A = (|B| \<le>o |A|)"
snd_Inf: fixes A :: "('d \<times> 'c) set" shows "snd (Inf A) = Inf (snd ` A)"
pgwt_is_empty_synth: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" shows "public_ground_wf_term t = ({} \<turnstile>\<^sub>c t)"
generalized_sfw_def: fixes l :: "('b simple_match \<times> 'c) list"   and p :: "('b, 'd) simple_packet_scheme" shows "generalized_sfw l p \<equiv> find (\<lambda>(m, a). simple_matches m p) l"
get_component_impl: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name" shows "get_component = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component get_root_node to_tree_order"
fimeq: shows "f ` (topspace (nsphere n) \<inter> equator n) \<subseteq> topspace (nsphere n) \<inter> equator n"
equiv_into_ext_equiv: fixes x :: "'a itrm"   and y :: "'a itrm"   and extra_rule :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool" assumes "x \<simeq> y" shows "itrm_cong (special_idiom.idiom_ext_rule extra_rule) x y"
map_color_of: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "color_of (RBT_Impl.map f t) = color_of t"
min_dnf_finite: fixes \<phi> :: "'b ltln" shows "finite (min_dnf \<phi>)"
enum_0_eq_Inf: fixes N :: "nat set" assumes "infinite N" shows "enum N 0 = \<Sqinter> N"
closed_def: fixes F :: "'b program"   and T :: "'b set"   and B :: "'b set"   and L :: "'b set set" shows "closed F T B L \<equiv> \<forall>M. \<forall>act\<in>Acts F. B \<subseteq> M \<and> T \<inter> M \<in> L \<longrightarrow> T \<inter> (B \<union> wp act M) \<in> L"
npMulDistr: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r x^\<^bsup>R m\<^esup> = x^\<^bsup>R (n + m)\<^esup>"
existence_ivl_zero: fixes x0 :: "'a" assumes "x0 \<in> X" shows "0 \<in> existence_ivl0 x0"
of_int_min: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "int"   and y :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus (min x y) = ord.min less_eq (ring_1.of_int one plus zero minus x) (ring_1.of_int one plus zero minus y)"
oseq_msg_ok: fixes i :: "nat"   and U :: "state \<Rightarrow> state \<Rightarrow> bool" shows "optoy i \<Turnstile>\<^sub>A (act TT, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). anycast (msg_ok \<sigma>) a)"
sumCases': fixes P :: "pi"   and Q :: "pi"   and Rs :: "residual" assumes "P \<oplus> Q \<longmapsto> Rs" obtains P :: "pi" and Q :: "pi" where "Pa__ = Pb__ \<and> Qa__ = Qb__" and "Pb__ \<longmapsto> Rs"   | Q :: "pi" and P :: "pi" where "Pa__ = P \<and> Qa__ = Q" and "Q \<longmapsto> Rs"
commutative: fixes x :: "('a set \<times> 'b) set"   and y :: "('a set \<times> 'b) set" assumes "x \<in> carrier_direct_lim"   and "y \<in> carrier_direct_lim" shows "add_rel x y = add_rel y x"
less_sets_fst_grab: fixes A :: "nat set"   and N :: "nat set"   and n :: "nat" assumes "A \<lless> N" shows "A \<lless> fst (grab N n)"
blues_seen_def: fixes w :: "'person \<Rightarrow> 'color"   and p :: "'person" shows "local.blues_seen w p = {p'. w p' = blue} - {p}"
wt_arg_le: fixes s :: "('s, 'v) tm" shows "wt (arg s) \<le> wt s"
norm_unsigned_length: fixes w :: "bit list" shows "length (norm_unsigned w) \<le> length w"
SBRMWReadOnly: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" assumes "\<not> cond (\<theta>(t \<mapsto> m a))" shows "(RMW a t (D, f) cond ret A L R W # is, \<theta>, [], m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>(t \<mapsto> m a), [], m, \<D>, \<O>, \<R>, \<S>)"
WT_gpv_OutD: fixes \<Gamma> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and out :: "'h"   and c :: "('j, 'h, 'i) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
getRel_dom_1: fixes G :: "('c, 'd) labeled_graph"   and a :: "'d"   and b :: "'d"   and l :: "'c" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "a \<in> vertices G"
\<II>\<nn>\<ll>\<OO>\<pp>_correct: fixes xs :: "'dyn list"   and opinl :: "'opinl" assumes "length xs = \<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl)" shows "\<II>\<nn>\<ll>\<OO>\<pp> opinl xs = \<OO>\<pp> (\<DD>\<ee>\<II>\<nn>\<ll> opinl) xs"
ptr'_owner_document_c: shows "h \<turnstile> get_dom_component (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 ptr'_owner_document) \<rightarrow>\<^sub>r ptr'_owner_document_c"
coprime_dvd_mult_left_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "dvd.dvd times a (times b c) = dvd.dvd times a b"
a3: shows "\<not> (\<forall>i. (\<tau> ^^ i) {} \<subset> (\<tau> ^^ (i + 1)) {})"
scale_shift_ms_aux: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and C :: "'a"   and e :: "real" assumes "basis_wf (b # basis)"   and "is_expansion_aux F f (b # basis)"   and "is_expansion C basis" shows "is_expansion_aux (scale_shift_ms_aux (C, e) F) (\<lambda>x. eval C x * b x powr e * f x) (b # basis)"
hyps: assumes "t \<in> set q" shows "tree_to_multiset t \<subseteq># queue_to_multiset q"
assms_2: shows "f holomorphic_on ball 0 k - {0}"
fork_pseudodeterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])" shows "substitution.revision_step subst r s s' = (\<exists>r'. r' \<notin> RID\<^sub>G (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [Rfork e]))) \<and> s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e)))"
nextX_zeroX: fixes a :: "vbl" shows "nextX a \<noteq> zeroX"
All2_E: fixes i :: "name"   and t :: "tm"   and H :: "fm set"   and x :: "tm"   and A :: "fm"   and B :: "fm" assumes "atom i \<sharp> t"   and "H \<turnstile> x IN t"   and "insert (A(i::=x)) H \<turnstile> B" shows "insert (All2 i t A) H \<turnstile> B"
map_add_Some_iff: fixes m :: "'c \<Rightarrow> 'b option"   and n :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and x :: "'b" shows "((m ++ n) k = \<lfloor>x\<rfloor>) = (n k = \<lfloor>x\<rfloor> \<or> n k = None \<and> m k = \<lfloor>x\<rfloor>)"
run_state_def: fixes stateT :: "('d, 'e) stateT" shows "run_state stateT = (case stateT of StateT x \<Rightarrow> x)"
one_smult_vec: fixes v :: "'a Matrix.vec" shows "(1::'a) \<cdot>\<^sub>v v = v"
smult_dvd_cancel: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" assumes "smult a p dvd q" shows "p dvd q"
var_not_fst: fixes x :: "'b"   and P :: "'b trm" shows "Var x \<noteq> Fst P"
inv_is_inverse: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.inverse_arrows f (VV.inv f)"
assms: fixes i :: "'a"   and v1 :: "'b"   and v2 :: "'c" assumes "inp1 i = Some v1"   and "inp2 i = Some v2" shows "phi v1 v2 = phi' v1 v2"
iffb_sup_dual: fixes \<phi> :: "'d set \<Rightarrow> 'c set"   and \<psi> :: "'d set \<Rightarrow> 'c set" shows "fb\<^sup>-\<^sub>\<F> (\<phi> \<squnion> \<psi>) = fb\<^sup>-\<^sub>\<F> \<phi> \<sqinter> fb\<^sup>-\<^sub>\<F> \<psi>"
select_convs_1: fixes ok :: "bool"   and wait :: "bool"   and tr :: "'c list"   and ref :: "'c set"   and more :: "'d" shows "wait \<lparr>ok = ok, wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr> = wait"
abs_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys_pp (PP x) = keys x"
poly_y_x_id: fixes p :: "'c poly poly" shows "poly_y_x (poly_y_x p) = p"
rep_eq: fixes x :: "uint16" shows "Rep_uint16 (- x) = - Rep_uint16 x"
list_rel_mono: fixes R :: "('a \<times> 'b) set"   and R' :: "('a \<times> 'b) set" assumes "R \<subseteq> R'" shows "\<langle>R\<rangle>list_rel \<subseteq> \<langle>R'\<rangle>list_rel"
bij_betw_respects_domain_and_range_permutation: fixes A :: "'a set"   and B :: "'b set" shows "(\<lambda>f. bij_betw f A B) respects domain_and_range_permutation A B"
sel_9: fixes x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "the_elem_term \<langle>x61: x62 \<in> x63\<rangle> = x62"
winding_number: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and e :: "real" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "0 < e" shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
pow_mono_one: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) \<le> a" shows "(1::'a) \<le> a ^ n"
maximal_def: fixes S :: "('a, 'b) fm set" shows "maximal S \<equiv> \<forall>p. p \<notin> S \<longrightarrow> \<not> consistent ({p} \<union> S)"
finfun_to_list_def: fixes f :: "('a, 'b) finfun" shows "finfun_to_list f = (THE xs. set xs = {x. finfun_apply (finfun_dom f) x} \<and> sorted xs \<and> distinct xs)"
\<a>'_def: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" shows "B.\<a>' \<mu> \<nu> \<tau> \<equiv> B.\<alpha>' (\<mu>, \<nu>, \<tau>)"
deriv_zeta_eq': fixes s :: "complex" assumes "0 < Re s"   and "s \<noteq> 1" shows "deriv zeta s = deriv (\<lambda>z. pre_zeta 1 z * (z - 1)) s / (s - 1) - (pre_zeta 1 s * (s - 1) + 1) / (s - 1)\<^sup>2"
some_src_def: fixes \<mu> :: "'a" shows "some_src \<mu> \<equiv> if arr \<mu> then rep (SOME a. a \<in> sources \<mu>) else null"
T_bal_tree_def: fixes n :: "nat"   and xs :: "'b tree" shows "T_bal_tree n xs = time (bal_tree_tm n xs)"
lunit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>MC.lunit a : MC.tensor MC.unity a \<rightarrow> a\<guillemotright>"
sorted_spmat_add_spmat: fixes A :: "(nat \<times> (nat \<times> 'a) list) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list" assumes "sorted_spmat A"   and "sorted_spmat B" shows "sorted_spmat (add_spmat A B)"
D\<^sub>i\<^sub>i: fixes x :: "i" shows "((\<^bold>\<forall>x. (dom x) \<cdot> x \<cong> x \<^bold>\<leftarrow> E ((dom x) \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> (dom x) \<cong> x \<^bold>\<leftarrow> E (x \<cdot> (dom x)))) \<^bold>\<and> x \<cdot> (dom x) \<cong> x \<^bold>\<leftarrow> E x"
map_bset_binsert: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e"   and X :: "'e set['d]" shows "map_bset f (binsert x X) = binsert (f x) (map_bset f X)"