hcju/isabelleps huggingface dataset:
semisimple: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and U :: "'b set" assumes "FinDimVectorSpace smult V"   and "RModule UNIV smult U \<and> U \<subseteq> V" shows "\<exists>W. (RModule UNIV smult W \<and> W \<subseteq> V) \<and> V = W \<oplus> U"
Sublists_def: fixes A :: "'a list set" shows "Sublists A = {w. \<exists>q\<in>A. sublist w q}"
n_narrow: fixes y :: "'av"   and x :: "'av" assumes "y \<le> x"   and "x \<triangle> y < x" shows "n (x \<triangle> y) < n x"
interval_of_plus: fixes a :: "'b"   and b :: "'b" shows "interval_of (a + b) = interval_of a + interval_of b"
adj_msog_def: fixes xs :: "(accent \<times> 'b) list"   and zs :: "(accent \<times> 'b) list"   and l :: "(accent \<times> 'b) \<times> (accent \<times> 'b) list" shows "adj_msog xs zs l \<equiv> case l of (y, ys) \<Rightarrow> (y, case fst y of Acute \<Rightarrow> ys @ zs | Grave \<Rightarrow> xs @ ys | Macron \<Rightarrow> ys)"
aux'_eq: shows "aux' = (case aux0 of [] \<Rightarrow> [(\<tau> \<sigma> ne, rel2)] | x # aux' \<Rightarrow> if fst x = \<tau> \<sigma> ne then (fst x, snd x \<union> rel2) # aux' else (\<tau> \<sigma> ne, rel2) # x # aux')"
new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr" assumes "new\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a h = (new_character_data_ptr, h')" shows "get ptr h = get ptr h'"
assms_2: assumes "\<Phi>"   and "\<Psi>" shows "m' \<le> c"
evalS_body_def: shows "evalS_body \<equiv> \<Lambda> r e. case undiscr e of const n \<Rightarrow> returnS\<cdot>(Discr n) | add e1 e2 \<Rightarrow> addS\<cdot>(r\<cdot>(Discr e1))\<cdot>(r\<cdot>(Discr e2)) | expr.disj e1 e2 \<Rightarrow> disjS\<cdot>(r\<cdot>(Discr e1))\<cdot>(r\<cdot>(Discr e2)) | fail \<Rightarrow> failS"
pth_9_1: fixes c :: "real"   and x :: "'b"   and z :: "'b"   and d :: "real" shows "c *\<^sub>R x + z + d *\<^sub>R x \<equiv> (c + d) *\<^sub>R x + z"
valid_def: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A \<equiv> \<forall>s t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"
SafeStickyGlobVacantUpto_def: fixes s :: "('c, 'd) configuration stream" shows "SafeStickyGlobVacantUpto s = (\<forall>q t. GlobVacantUpto (shd s) q t \<longrightarrow> alw (holds (\<lambda>c. GlobVacantUpto c q t)) s)"
1: shows "rat_poly_inv (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B) = rat_poly_times A A - rat_poly_times B B"
symp_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "symp (equivclp r)"
ennreal_add_left_cancel: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a + b = a + c) = (a = \<infinity> \<or> b = c)"
add_le_imp_le_left: fixes c :: "'k multiset"   and a :: "'k multiset"   and b :: "'k multiset" assumes "c + a \<subseteq># c + b" shows "a \<subseteq># b"
crel_vs_executeD: fixes P :: "heap \<Rightarrow> bool"   and Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and dp :: "'c \<Rightarrow> 'd"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap"   and heap :: "heap" assumes "dp_heap P Q lookup lookup_st update update_st"   and "dp_heap.crel_vs P Q dp lookup_st R a b"   and "P heap"   and "Q heap"   and "dp_consistency.cmem lookup_st dp heap" obtains x :: "'f" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'" and "dp_consistency.cmem lookup_st dp heap'" and "R a x"
Diff_cover: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "\<exists>C\<subseteq>sets M. finite C \<and> disjoint C \<and> a - b = \<Union> C"
hom_char: fixes a :: "'a"   and b :: "'a" shows "L.hom a b = hom a b \<inter> Collect (left a)"
mat_eye_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>m eye = A"
ptensor_mat_dim_col: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_col (tensor_P m1 m2) = ps2_P.d0"
conc_subset_lists: fixes A :: "'b list set"   and S :: "'b set"   and B :: "'b list set" assumes "A \<subseteq> lists S"   and "B \<subseteq> lists S" shows "A @@ B \<subseteq> lists S"
lead_coeff_smult: fixes c :: "'b"   and p :: "'b poly" shows "lead_coeff (Polynomial.smult c p) = c * lead_coeff p"
neg_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<chi> :: "bool" assumes "neg \<phi> \<pi>l"   and "\<phi> \<pi>l" shows "\<chi>"
memoization_correct_2: fixes dp\<^sub>T :: "'a \<Rightarrow> 'b Heap"   and x :: "'a"   and v :: "'b"   and m :: "heap" assumes "consistentDP dp\<^sub>T"   and "execute (dp\<^sub>T x) empty = Some (v, m)" shows "state_dp_consistency.cmem m"
gauss_sum_conv_gauss_sum_int: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "dcharacter n \<chi>" shows "dcharacter.gauss_sum n \<chi> k = dcharacter.gauss_sum_int n \<chi> (int k)"
ps_after_all_prerecording_events: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "trace init t final"   and "\<forall>i'\<ge>i. \<not> prerecording_event t i'"   and "\<forall>j'<i. \<not> postrecording_event t j'" shows "ps_equal_to_snapshot (S t i) final"
subspace_bounded_eq_trivial: fixes S :: "'a set" assumes "subspace S" shows "bounded S = (S = {0::'a})"
f_1: shows "(f has_contour_integral i) (linepath a c)"
monotone_def: fixes orda :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and ordb :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
coeffs_pCons_eq_cCons: fixes a :: "'b"   and p :: "'b poly" shows "coeffs (pCons a p) = cCons a (coeffs p)"
finite_bounded_functions: fixes S :: "'b set"   and T :: "'b set" assumes "finite S"   and "finite T" shows "finite {f. (\<forall>i\<in>T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
span_induct: fixes x :: "'d"   and S :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> span S"   and "subspace (Collect P)"   and "\<And>x. x \<in> S \<Longrightarrow> P x" shows "P x"
homI: fixes f :: "'a"   and A :: "'o"   and B :: "'o" assumes "f \<in> Ar"   and "Dom f = A"   and "Cod f = B" shows "f \<in> Hom A B"
set_child_nodes_get_disconnected_nodes_is_l_set_child_nodes_get_disconnected_nodes: shows "l_set_child_nodes_get_disconnected_nodes DocumentClass.type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"
rep_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.rep (\<F>C.dom f) = \<^bold>\<parallel>\<F>C.DOM f\<^bold>\<parallel>"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "\<F>C.arr (\<F>C.\<nu> f) = \<F>C.arr f"
cod_simp: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" shows "A_BxA.cod f = (A_B.cod (fst f), A.cod (snd f))"
countable_disjoint_nonempty_interior_subsets: fixes \<F> :: "'b set set" assumes "disjoint \<F>"   and "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}" shows "countable \<F>"
reduce_specs_def: fixes d :: "nat"   and reach_options_o :: "unit"   and X :: "'a list set" shows "reduce_specs d reach_options_o X = SPEC (\<lambda>R. env_len R d \<and> X \<subseteq> R)"
assms_1: shows "\<guillemotleft>rderivs (replicate i as) r\<guillemotright> = \<guillemotleft>r\<guillemotright>"
star_n_export_left: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> !y \<le> !y \<cdot> x" shows "x\<^sup>\<star> \<cdot> !y = !y \<cdot> (x \<cdot> !y)\<^sup>\<star>"
env_delete_esing: fixes x :: "'c"   and n :: "'d" shows "env_delete x (esing x\<cdot>n) = \<bottom>"
assms_5: shows "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 |h \<turnstile> attach_shadow_root element_ptr shadow_root_mode|\<^sub>r"
simps_2: fixes B :: "binding"   and \<Gamma> :: "binding list"   and Y :: "tyvrs"   and T :: "ty" shows "(B # \<Gamma>)[Y \<mapsto> T]\<^sub>e = B[Y \<mapsto> T]\<^sub>b # \<Gamma>[Y \<mapsto> T]\<^sub>e"
Suffixes_singleton: fixes xs :: "'a list" shows "Suffixes {xs} = set (suffixes xs)"
StaticCall: 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\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and ps :: "expr list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and Cs'' :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and Ds :: "char list list"   and vs' :: "val list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs'' "   and "P \<turnstile> C has least M = (Ts, T, pns, body) via Cs'"   and "Ds = (Cs @\<^sub>p Cs'') @\<^sub>p Cs'"   and "length vs = length pns"   and "P \<turnstile> Ts Casts vs to vs' "   and "l\<^sub>2' = [this \<mapsto> Ref (a, Ds), pns [\<mapsto>] vs']"   and "P,E(this \<mapsto> Class (last Ds), pns [\<mapsto>] Ts) \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3)\<rangle>" shows "P,E \<turnstile> \<langle>Call e \<lfloor>C\<rfloor> M ps,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2)\<rangle>"
deleteAround_def: fixes g :: "graph"   and v :: "nat"   and ps :: "(nat \<times> nat) list" shows "deleteAround g v ps \<equiv> let fs = facesAt g v; ws = concat (map (\<lambda>f. if |vertices f| = 4 then [f \<bullet> v, f\<^bsup>2\<^esup> \<bullet> v] else [f \<bullet> v]) fs) in removeKeyList ws ps"
cmp_inj: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "inj_on f A"   and "inj_on g B" shows "inj_on (cmp g f) A"
bind_pure_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> pure (g x) h" shows "pure (f \<bind> g) h"
length_take: fixes n :: "nat"   and xs :: "'k list" shows "length (take n xs) = min (length xs) n"
r_zero: fixes G :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "abelian_monoid G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = x"
nodesp_nodes_eq: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "transition_system_initial.nodesp (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) (\<lambda>p. p = initial A) = (\<lambda>x. x \<in> nodes A)"
monad_def: fixes x :: "'a star" shows "monad x = {y. x \<approx> y}"
proj_stoch_adapted_if_adapted: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure"   and n :: "nat" assumes "Filtration.filtration M F"   and "adapt_stoch_proc F f N" shows "proj_stoch_proc f n \<in> F n \<rightarrow>\<^sub>M stream_space N"
Abs_num1_cases: fixes x :: "1" obtains y :: "unit" where "x = Abs_num1 y" and "y \<in> UNIV"
Roots_ms_def: shows "Roots_ms = {Y. set_mset Y \<subseteq> X \<and> size Y = n ^ m}"
Pre_mem: fixes a :: "'a" assumes "a \<in> carrier D"   and "ExPre D a" shows "Pre D a \<in> carrier D"
cmp_in_hom: fixes V\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>B :: "'d \<Rightarrow> 'd"   and src\<^sub>B :: "'d \<Rightarrow> 'd"   and trg\<^sub>B :: "'d \<Rightarrow> 'd"   and V\<^sub>C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>C :: "'e \<Rightarrow> 'e"   and src\<^sub>C :: "'e \<Rightarrow> 'e"   and trg\<^sub>C :: "'e \<Rightarrow> 'e"   and V\<^sub>D :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and H\<^sub>D :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and \<a>\<^sub>D :: "'f \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and \<i>\<^sub>D :: "'f \<Rightarrow> 'f"   and src\<^sub>D :: "'f \<Rightarrow> 'f"   and trg\<^sub>D :: "'f \<Rightarrow> 'f"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'e \<Rightarrow> 'f"   and \<Phi>\<^sub>G :: "'e \<times> 'e \<Rightarrow> 'f"   and \<mu>\<nu> :: "'d \<times> 'd" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G"   and "partial_magma.arr (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>" shows "partial_magma.in_hom V\<^sub>D (composite_pseudofunctor.cmp V\<^sub>B H\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<mu>\<nu>) (((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D src\<^sub>D trg\<^sub>D (G \<circ> F)) (partial_magma.dom (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>)) ((G \<circ> F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>B (fst \<mu>\<nu>) (snd \<mu>\<nu>))) (partial_magma.cod (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>))"
subset_insertI: fixes B :: "'b set"   and a :: "'b" shows "(B::'b set) \<subseteq> (a::'b) \<triangleright> B"
assms: assumes "vassn_tag A" shows "A \<Longrightarrow>\<^sub>A B"
outstanding_non_volatile_refs_owned_or_read_only: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_owned_or_read_only False \<S> \<O> sb"
CARAT_DERIVD: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "\<forall>z. f z - f x = g z * (z - x)"   and "isNSCont g x" shows "NSDERIV f x :> g x"
pseudo_mod_zero_1: fixes f :: "'b poly" shows "pseudo_mod 0 f = 0"
decI': fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>" shows "Dec G u \<in> lists G\<^sub>+"
per_pref: fixes x :: "'b list"   and r :: "'b list" shows "x \<le>p r\<^sup>\<omega> = ((\<exists>k. x \<le>p r \<^sup>@ k) \<and> r \<noteq> \<epsilon>)"
assms_3: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "r = a \<otimes> b" shows "a \<in> Units R \<or> b \<in> Units R"
1: fixes y :: "'b" assumes "y \<in> indets p" shows "g y \<in> X"
residual_network_sel_4: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "sink (residual_network f) = sink \<Delta>"
ringE_3: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
homeomorphism_compact: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and t :: "'c set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "\<exists>g. homeomorphism s t f g"
sup_V_0_left: fixes a :: "V" shows "0 \<union>\<^sub>\<circ> a = a"
card_le_dim_spanning: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "B \<subseteq> V"   and "V \<subseteq> vec.span B"   and "finite B"   and "card B \<le> vec.dim V" shows "vec.independent B"
const_apply: fixes x :: "'a"   and i :: "'b" shows "pure x i = x"
can_def_alt: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.can C f = (partial_magma.arr (free_monoidal_category.comp C) f \<and> (\<forall>t. t \<in> f \<longrightarrow> monoidal_language.Ide C (monoidal_language.Diagonalize C t)))"
hom_sum: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "c * sum f X = (\<Sum>x\<in>X. c * f x)"
compseriesTr5: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and i :: "nat" assumes "Group G"   and "0 < n"   and "compseries G n f"   and "i \<le> n - Suc 0" shows "f (Suc i) \<subseteq> f i"
mtf2_forward_effect3': fixes q :: "'a"   and xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q - n \<le> index xs x"   and "index xs x < index xs q" shows "index (mtf2 n q xs) (xs ! index xs x) = Suc (index xs (xs ! index xs x)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! index xs x) \<and> index (mtf2 n q xs) (xs ! index xs x) \<le> index xs q"
test_mult_closed: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "test (p \<cdot> q)"
ipd_is_ipd: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" shows "ipd x ipd\<rightarrow> x"
lists_succ_Nil_pf: shows "lists_succ [] = id"
subst_apply_pairs_def: fixes F :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) list"   and \<theta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> \<equiv> map (\<lambda>f. f \<cdot>\<^sub>p \<theta>) F"
rel_G_pos_distr_cond_eq: fixes tytok :: "('r \<times> 's \<times> 't \<times> 'u \<times> 'v \<times> 'w \<times> 'x) itself" shows "rel_G_pos_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
finite_span: fixes S :: "'a vec set" assumes "finite S"   and "S \<subseteq> carrier_vec n" shows "local.span S = {uu_. \<exists>a. uu_ = lincomb a S}"
Rep_inverse: fixes x :: "'a word" shows "word_of_nat (unat x) = x"
valid_node_def: fixes n :: "'b LDCFG_node" shows "local.CFG.valid_node n \<equiv> \<exists>a. lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> (n = src a \<or> n = trg a)"
offs_suc: fixes i :: "nat"   and ns :: "nat list"   and k :: "nat" assumes "i < length ns" shows "offs ns (Suc k) ! i = Suc (offs ns k ! i)"
diff_le_self_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "a - b \<le> a"
assms_4: shows "poincare_collinear {u, v, w}"
interpret_floatarith_inner_eq: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and vs :: "real list" shows "interpret_floatarith (inner_floatariths xs ys) vs = sum_list (map2 (\<lambda>x y. interpret_floatarith x vs * interpret_floatarith y vs) xs ys)"
igSubstBinpIPresIGWlsBinpSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstBinpIPresIGWlsBinpSTR MOD \<equiv> \<forall>ys y Y delta binp. igWls MOD (asSort ys) Y \<longrightarrow> igWlsBinp MOD delta (igSubstBinp MOD ys Y y binp) = igWlsBinp MOD delta binp"
FEXISTS_1: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = lang n (rexp_of n \<phi>) - {[]}"
comm_add_vec: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 = v\<^sub>2 + v\<^sub>1"
exec_wfs_match: fixes \<Gamma> :: "char list set"   and s :: "stmt"   and \<sigma> :: "char list \<Rightarrow> nat option"   and \<sigma>' :: "char list \<Rightarrow> nat option" assumes "wfs \<Gamma> s"   and "match \<sigma> \<Gamma>"   and "s \<rhd> \<sigma> \<leadsto> Some \<sigma>'" shows "match \<sigma>' \<Gamma>"
one_mult_mat_vec: fixes v :: "'a Matrix.vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "1\<^sub>m n *\<^sub>v v = v"
multi_drop_mem_not_eq: fixes c :: "'c"   and B :: "'c multiset" assumes "c \<in># B" shows "remove1_mset c B \<noteq> B"
read_only_share_unowned': fixes \<O> :: "nat set"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "weak_sharing_consistent \<O> sb"   and "\<O> \<inter> read_only \<S> = {}"   and "a \<notin> \<O> \<union> all_acquired sb"   and "a \<in> read_only \<S>" shows "a \<in> read_only (share sb \<S>)"
refl: fixes x :: "'alt pmf" assumes "x \<in> lotteries_on alts" shows "Pareto (SD \<circ> R) x x"
osig_subclass_loc: fixes oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "wf_osig oss" shows "wf_subclass_loc (subclass oss)"
tight_1: shows "root_cond (q, l', r') (real_of_1 (q, l, r))"
R_OK: fixes xc1 :: "'b"   and xc2 :: "'b" assumes "invar xc1"   and "invar xc2"   and "Rc xc1 xc2" shows "Ra (\<alpha> xc1) (\<alpha> xc2)"
arrI: fixes f :: "'a" assumes "C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f)" shows "S.arr f"
func_lemma: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and I :: "('d, 'e, 'f) interp"   and args :: "'f \<Rightarrow> ('d, 'f) trm"   and \<nu> :: "(real, 'f) vec"   and f :: "'d" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4"   and "is_interp I"   and "\<And>\<theta>. \<theta> \<in> range args \<Longrightarrow> (sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)" shows "(sterm_sem I ($f f args) has_derivative frechet I ($f f args) \<nu>) (at \<nu>)"