hcju/isabelleps huggingface dataset:
gen_dfs_def: fixes gds :: "('b, 'a) gen_dfs_struct"   and V0 :: "'b set" shows "gen_dfs_defs.gen_dfs gds V0 \<equiv> gds_init gds \<bind> WHILE (gen_dfs_defs.gen_cond gds V0) (gen_dfs_defs.gen_step gds V0)"
arg_min_ex: fixes P :: "'c \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> nat" assumes "\<exists>x. P x" obtains x :: "'c" where "P x" and "\<And>y. P y \<Longrightarrow> weight x \<le> weight y"
conversep_iff: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'d" shows "r\<inverse>\<inverse> a b = r b a"
P: fixes l :: "'a" assumes "is_lub L l {x, y}" shows "P l"
must_sync_def2: fixes t :: "'t"   and x :: "'x"   and m :: "'m" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong> = (\<exists>taa x' m' s. t \<turnstile> \<langle>x, m\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t taa)"
D_base_finite'': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {linordered_monoid.len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
the_trivial_def: fixes v :: "'a"   and vs :: "'a list" shows "the_trivial v vs = (case foldl (\<lambda>(good, v') w. if w = v then (good, v') else case v' of None \<Rightarrow> (good, Some w) | Some v' \<Rightarrow> (good \<and> w = v', Some v')) (True, None) vs of (True, x) \<Rightarrow> x | (False, x) \<Rightarrow> None)"
simps_4: fixes x21 :: "real"   and x22 :: "real list"   and x11 :: "real"   and x12 :: "real list" shows "Eq x21 x22 \<noteq> Less x11 x12"
A\<^sub>A_asm_dma: shows "\<forall>x\<^sub>A. dma\<^sub>A mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C] x\<^sub>A \<noteq> dma\<^sub>A (mem\<^sub>A_of mem\<^sub>1\<^sub>C) x\<^sub>A \<longrightarrow> \<not> var_asm_not_written (mds\<^sub>A_of mds\<^sub>C) x\<^sub>A"
bigo_mult6: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. f x \<noteq> (0::'b)" shows "O(f * g) = f *o O(g)"
is_lub_unique: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "po.is_lub below S x"   and "po.is_lub below S y" shows "x = y"
cdf_diff_eq: fixes x :: "real"   and y :: "real" assumes "x < y" shows "cdf M2 y - cdf M2 x = M2.prob {x<..y}"
dghm_ObjMap_in_Vset: fixes \<beta> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
measurable_Pair1: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and M1 :: "'c measure"   and M2 :: "'d measure"   and M :: "'e measure"   and y :: "'d" assumes "f \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M"   and "y \<in> space M2" shows "(\<lambda>x. f (x, y)) \<in> M1 \<rightarrow>\<^sub>M M"
similar_mat_wit_refl: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat_wit A A (1\<^sub>m n) (1\<^sub>m n)"
nb: shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). v \<in> dom (range_of \<Psi>)"
assms_1: shows "[:- a, 1::'a:] ^ n dvd p"
wtC_append: fixes c1 :: "('fsym, 'psym) lit list"   and c2 :: "('fsym, 'psym) lit list" shows "wtC (c1 @ c2) = (wtC c1 \<and> wtC c2)"
get_element_by_id_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and iden :: "char list"   and element_ptr :: "(_) element_ptr"   and to :: "(_) object_ptr list" assumes "h \<turnstile> get_element_by_id ptr iden \<rightarrow>\<^sub>r Some element_ptr"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^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 element_ptr \<in> set to"
nonneg_2: fixes i :: "'a" assumes "i \<in> I" shows "0 \<le> g i"
intersects_x_axis_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "intersects_x_axis_cmat H = (let (A, B, C, D) = H in A = 0 \<or> (Re A)\<^sup>2 < (Re B)\<^sup>2)"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> r\<^sup>*" obtains a :: "'a" where "a1 = aa__" and "a2 = aa__"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>*" and "(b, c) \<in> r"
igFreshIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igFreshIGOp MOD \<equiv> \<forall>ys y delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
f_join_union: fixes A :: "nat set"   and B :: "nat set"   and xs :: "'b list" assumes "finite A"   and "Max A < iMin B" shows "xs \<Join>\<^sub> (A \<union> B) = xs \<Join>\<^sub> A @ xs \<Join>\<^sub> B"
OO_Grp_alt: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g" shows "(BNF_Def.Grp A f)\<inverse>\<inverse> OO BNF_Def.Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
surj_to_conjugate_space: shows "surj to_conjugate_space"
combine_rbt_less: fixes k :: "'a"   and l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt" assumes "l |\<guillemotleft> k"   and "r |\<guillemotleft> k" shows "combine l r |\<guillemotleft> k"
l1_inv3_derived: shows "l1_inv1 \<inter> l1_inv2 \<subseteq> l1_inv3"
not_psubset_empty: fixes A :: "'d set" shows "\<not> A \<subset> {}"
ccard_zero: fixes n :: "nat"   and p :: "nat \<Rightarrow> bool" shows "card {i. n < i \<and> i \<le> n \<and> p i} = 0"
assms: shows "TE.wt (Fn (Wit \<sigma>) Tl)"
measurable_sets: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and A :: "'d measure"   and S :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "S \<in> sets A" shows "f -` S \<inter> space M \<in> sets M"
r_cancel_one: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<otimes>\<^bsub>G\<^esub> x = x) = (a = \<one>\<^bsub>G\<^esub>)"
elims: fixes x :: "game option"   and xa :: "fml option"   and y :: "fml option" assumes "Diamondo x xa = y" obtains \<alpha> :: "game" and \<phi> :: "fml" where "x = Agame \<alpha>" and "xa = Afml \<phi>'__" and "y = Afml (\<langle> \<alpha> \<rangle> \<phi>'__)"   | \<phi> :: "fml option" where "x = undefg" and "xa = \<phi>" and "y = undeff"   | v :: "game" where "x = Agame v" and "xa = undeff" and "y = undeff"
set_dom_cod_object_subset_4: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "set_dom f \<subseteq> U"
common_matcher_SrcDst_defined_1: fixes m :: "'a ipt_iprange"   and p :: "('a, 'b) tagged_packet_scheme" shows "common_matcher (Src m) p \<noteq> TernaryUnknown"
TER_current_of_bipartite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal" assumes "current (bipartite_web_of \<Gamma>) f"   and "wave (bipartite_web_of \<Gamma>) f" shows "TER (current_of_bipartite f) = separating_of_bipartite (TER\<^bsub>bipartite_web_of \<Gamma>\<^esub> f)"
basic_trans_rules_30: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
nrows_transpose: fixes A :: "'b matrix" shows "nrows (transpose_matrix A) = ncols A"
ring_simprules_3: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
Mf_Mf: fixes m :: "int"   and x :: "int \<times> int poly multiset" shows "poly_mod.Mf m (poly_mod.Mf m x) = poly_mod.Mf m x"
p_minus_one_l: shows "(int p - 1) div 2 < int p"
resCases': fixes \<Psi> :: "'b"   and x :: "name"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> \<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "x \<sharp> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "\<And>M xvec yvec y N P'. \<lbrakk>\<Psi> \<rhd> [(x, y)] \<bullet> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'; y \<in> supp N; x \<sharp> N; x \<sharp> P'; x \<noteq> y; y \<sharp> xvec; y \<sharp> yvec; y \<sharp> M; distinct xvec; distinct yvec; xvec \<sharp>* \<Psi>; y \<sharp> \<Psi>; yvec \<sharp>* \<Psi>; xvec \<sharp>* P; y \<sharp> P; yvec \<sharp>* P; xvec \<sharp>* M; y \<sharp> M; yvec \<sharp>* M; xvec \<sharp>* yvec\<rbrakk> \<Longrightarrow> Prop (M\<lparr>\<nu>*(xvec @ y # yvec)\<rparr>\<langle>N\<rangle>) P'"   and "\<And>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> Prop \<alpha> (\<lparr>\<nu>x\<rparr>P')" shows "Prop \<alpha> P'"
log_eq_div_ln_mult_log: fixes a :: "real"   and b :: "real"   and x :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "log a x = ln b / ln a * log b x"
T: fixes x :: "'a" assumes "x \<in> topspace X" shows "openin X (T x) \<and> x \<in> T x \<and> finite {U \<in> \<U>. U \<inter> T x \<noteq> {}}"
B_def: shows "B = {C \<in> N. D \<in> \<G>_F C}"
trancl_rtrancl_trancl: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>+"
boolean_semantics: fixes G :: "'e \<Rightarrow> 'f list \<Rightarrow> bool"   and E :: "nat \<Rightarrow> 'f"   and F :: "'d \<Rightarrow> 'f list \<Rightarrow> 'f" shows "boolean (\<lambda>a. G a \<circ> map \<lparr>E, F\<rparr>) \<lbrakk>E, F, G\<rbrakk> = \<lbrakk>E, F, G\<rbrakk>"
assms_1: fixes i :: "nat" shows "A i \<lless> A (Suc i)"
interp_ins_tail_unfold: fixes xs :: "('c \<times> 'c option) list"   and x :: "'c \<times> 'c option" shows "interp_ins (xs @ [x]) = insert_spec (interp_ins xs) x"
insert_fsetI1: fixes x :: "'b"   and S :: "'b fset" shows "x |\<in>| insert_fset x S"
Ord_Union: fixes A :: "hf" assumes "\<And>i. i \<^bold>\<in> A \<Longrightarrow> Ord i" shows "Ord (\<Squnion>A)"
balanced_net_shallow_model: fixes Y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat" shows "balanced_net (shallow_model Y Z M N)"
strong_induct: fixes x1 :: "lam"   and x2 :: "name list"   and x3 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> name list \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<mapsto> x2,x3"   and "\<And>a y. P y (Var a) [] (Var a)"   and "\<And>t1 t2 y. P y (App t1 t2) [] (App t1 t2)"   and "\<And>t xs t' x y. \<lbrakk>x \<sharp> y; t \<mapsto> xs,t'; \<And>z. P z t xs t'\<rbrakk> \<Longrightarrow> P y (Lam [x].t) (x # xs) t'" shows "P z x1 x2 x3"
simps_2: fixes p :: "prim" shows "|DP p| = {PP p}"
ex231b: fixes x :: "'a" shows "x * ((1::'a) \<sqinter> x\<^sup>T * x) = x"
sccs_verts_conv_scc_of: shows "H.sccs_verts = H.scc_of ` verts (del_vert u)"
absorb_iff2: fixes b :: "'a"   and a :: "'a" assumes "b \<in> U"   and "a \<in> U" shows "(b \<le>\<^sub>o\<^sub>w a) = (local.min a b = b)"
card_of_Sigma_cong1: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'c set" assumes "\<forall>i\<in>I. |A i| =o |B i|" shows "|Sigma I A| =o |Sigma I B|"
filter_disj_inds: fixes i :: "nat"   and ls :: "'b list"   and j :: "nat" assumes "i < length ls"   and "j < length ls"   and "i \<noteq> j" shows "filter (\<lambda>iaa. iaa \<noteq> j \<longrightarrow> iaa = i) [0..<length ls] = [i, j] \<or> filter (\<lambda>iaa. iaa \<noteq> j \<longrightarrow> iaa = i) [0..<length ls] = [j, i]"
remove_child_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
unwrapS'_unfold: fixes f :: "nat discr Box \<rightarrow> 'a" shows "unwrapS'\<cdot>f = f\<cdot>0 && smap\<cdot>(f oo (\<Lambda> x. 1 + x))\<cdot>nats"
rep_value_sym: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) = \<k>"
hyps: assumes "proper_signed_list xs"   and "proper_signed_list ys" shows "proper_signed_list (prappend_signed_list xs ys)"
lless_antisym: fixes L :: "('d, 'e) gorder_scheme"   and a :: "'d"   and b :: "'d"   and P :: "bool" assumes "weak_partial_order L"   and "a \<in> carrier L"   and "b \<in> carrier L"   and "a \<sqsubset>\<^bsub>L\<^esub> b"   and "b \<sqsubset>\<^bsub>L\<^esub> a" shows "P"
Fejer_kernel: fixes n :: "nat"   and x :: "real" shows "Fejer_kernel n x = (if n = 0 then 0 else if x = 0 then real n / 2 else (sin (real n / 2 * x))\<^sup>2 / (2 * real n * (sin (x / 2))\<^sup>2))"
hypnat_of_nat_one: shows "hypnat_of_nat (Suc 0) = 1"
state_transition_refl_def: fixes s :: "'b"   and s' :: "'b" shows "s \<rightarrow>\<^sub>i* s' \<equiv> (s, s') \<in> {(x, y). x \<rightarrow>\<^sub>i y}\<^sup>*"
cmod_diff_real_less: fixes z :: "complex"   and x :: "real" assumes "Im z \<noteq> 0"   and "x \<noteq> 0" shows "cmod (z - complex_of_real x) < cmod z + \<bar>x\<bar>"
dom_map_option_comp: fixes g :: "'f \<Rightarrow> 'e"   and m :: "'d \<Rightarrow> 'f option" shows "dom (map_option g \<circ> m) = dom m"
LLL_reconstruction_inner_def: fixes p :: "int"   and l :: "nat"   and gs :: "int poly list"   and f :: "int poly"   and u :: "int poly"   and Degs :: "nat set"   and j :: "nat" shows "LLL_reconstruction_inner p l gs f u Degs j \<equiv> let j' = j - 1 in if j' \<notin> Degs then None else let ll = let n = sqrt_int_ceiling (\<parallel>f\<parallel>\<^sup>2 ^ (2 * j') * 2 ^ (5 * j' * j')); ll' = find_exponent p n in if ll' < l then ll' else l; pl = p ^ ll; g' = LLL_short_polynomial pl j u in if pl \<le> \<bar>lead_coeff g'\<bar> then None else let ppg = primitive_part g' in case div_int_poly f ppg of None \<Rightarrow> None | Some f' \<Rightarrow> Some (filter (\<lambda>gi. \<not> poly_mod.dvdm p gi ppg) gs, lead_coeff f', f', ppg)"
if_then_else_Error_2: fixes a :: "bool"   and c :: "('l, 'm) result"   and b :: "'m"   and d :: "'l" shows "((if a then c else Ok b) = Error d) = (a \<and> c = Error d)"
lins2D_3: fixes i :: "nat"   and j :: "nat"   and v :: "node"   and e :: "node fset \<times> command_gadget \<times> node fset" assumes "i < length (Inl v # \<pi>)"   and "j < length (Inl v # \<pi>)"   and "(Inl v # \<pi>) ! i = Inl v"   and "(Inl v # \<pi>) ! j = Inr e"   and "v |\<in>| fst3 e" shows "i < j"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_inverse" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse" and "a\<inverse>\<^sub>\<rat> = b"
path2_split_last_prop: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and P :: "'node \<Rightarrow> bool" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "\<exists>x\<in>set ns. P x" obtains n' :: "'node" and ns' :: "'node list" where "g \<turnstile> n'-ns'\<rightarrow>m" and "P n'" and "\<forall>x\<in>set (tl ns'). \<not> P x" and "suffix ns' ns"
nonzero_mult_divide_mult_cancel_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
evenperm_comp: fixes p :: "'c \<Rightarrow> 'c"   and q :: "'c \<Rightarrow> 'c" assumes "permutation p"   and "permutation q" shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)"
Well_order_isMinim_exists: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "\<exists>b. isMinim B b"
divide_divide_eq_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (divide b c) = divide (times a c) b"
cases: fixes a1 :: "('f, 'g, 'h) Resumption.resumption"   and a2 :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord a1 a2" obtains a :: "'f option" and a' :: "'f option" where "a1 = Resumption.resumption.Done a" and "a2 = Resumption.resumption.Done a'" and "option_ord a a'"   | out :: "'g" and c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" where "a1 = ABORT" and "a2 = Resumption.resumption.Pause out__ c__"   | c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" and c' :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" and out :: "'g" where "a1 = Resumption.resumption.Pause out c" and "a2 = Resumption.resumption.Pause out c'" and "rel_fun (=) resumption_ord c c'"
a: shows "i < dim_vec (B *\<^sub>v vec_last v (dim_col B))"
divide_right: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)"   and "f \<in> O[F](g)" shows "(\<lambda>x. f x / h x) \<in> O[F](\<lambda>x. g x / h x)"
assms: shows "intersect_segment_xline prec (p0, p1) x = Some (m, M)"
e_def: shows "e = Min ((ee \<circ> p) ` k)"
it_to_sorted_list_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
vreal_plus_def: shows "vreal_plus = (\<lambda>x\<in>\<^sub>\<circ>\<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (real_of_vreal (x\<lparr>0\<^sub>\<nat>\<rparr>) + real_of_vreal (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<real>)"
lcm_proj2_iff: fixes m :: "'a"   and n :: "'a" shows "(lcm m n = normalize n) = (m dvd n)"
pdevs_mapping_def: fixes xs :: "'b pdevs list" shows "pdevs_mapping xs = (let D = sorted_list_of_set (domain_pdevs xs); M = Mapping.tabulate D (pdevs_applys xs); zeroes = replicate (length xs) (0::'b) in Mapping.lookup_default zeroes M)"
uf_graph_invarD: fixes uf :: "('a \<times> 'a) set"   and T :: "'edge set" assumes "uf_graph_invar (uf, T)" shows "corresponding_union_find uf T"
conj_assoc: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "((P \<and> Q) \<and> R) = (P \<and> Q \<and> R)"
strict_trans2: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "c \<le> b" shows "c < a"
simps_86: fixes f :: "'b \<Rightarrow> 'a"   and x6 :: "'b poly" shows "map_poly f (Neg x6) = Neg (map_poly f x6)"
iff_def: fixes p :: "fm"   and q :: "fm" shows "MIR.iff p q \<equiv> if p = q then T else if p = not q \<or> not p = q then F else if p = F then not q else if q = F then not p else if p = T then q else if q = T then p else Iff p q"
ApplicationTr4: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "\<forall>j\<in>{h. h \<le> Suc n}. x j \<in> carrier K" shows "\<exists>l. \<forall>N>l. \<forall>j\<le>Suc n. an m \<le> vv j (\<Sigma>\<^sub>e K (\<lambda>j\<in>{h. h \<le> Suc n}. x j \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a (\<Omega>\<^bsub>K vv Suc n\<^esub>) j^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)) Suc n \<plusminus> -\<^sub>a x j)"
components_def: fixes S :: "'c set" shows "components S \<equiv> connected_component_set S ` S"
HomFtorContraCod: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "CatCod (Hom\<^bsub>C\<^esub>[\<midarrow>,X]) = SET"
stutter_sampled_in_range: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and s :: "'b" assumes "stutter_sampler f \<sigma>"   and "s \<in> range \<sigma>" shows "s \<in> range (\<sigma> \<circ> f)"
assms: shows "\<Theta>,\<Gamma> \<turnstile> p"
lformula_RESTRICT: fixes \<phi> :: "('a, 'k) aformula" assumes "lformula \<phi>" shows "lformula (RESTRICT \<phi>)"
lex_prodp_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "local.lex_prodp x y \<equiv> r1 (fst x) (fst y) \<or> fst x = fst y \<and> r2 (snd x) (snd y)"
less_set_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subset> B) = ((\<lambda>x. x \<in> A) < (\<lambda>x. x \<in> B))"
assms: shows "\<forall>a. A.ide a \<longrightarrow> limit_cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (\<lambda>j. D (j, a)) (l a) (\<chi> a)"
trancl_collect: fixes x :: "'d"   and y :: "'d"   and R :: "('d \<times> 'd) set"   and S :: "'d" assumes "(x, y) \<in> R\<^sup>*"   and "S \<notin> Domain R"   and "y \<noteq> S" shows "(x, y) \<in> {p. fst p \<noteq> S \<and> snd p \<noteq> S \<and> p \<in> R}\<^sup>*"
binomial_n0: fixes n :: "nat" shows "Suc n choose 0 = n choose 0"
assms_2: shows "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"
singleton_inject: fixes a :: "'q"   and b :: "'q" assumes "{a} = {b}" shows "a = b"
simps_4: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "dnf (\<phi> or\<^sub>n \<psi>) = dnf \<phi> \<union> dnf \<psi>"
binarysum_div: fixes B :: "nat set"   and j :: "nat" assumes "finite B" shows "sum ((^) 2) B div 2 ^ j = (\<Sum>i\<in>B. if i < j then 0 else 2 ^ (i - j))"
freeword_funlift_Abs_freelist: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "freeword_funlift f (Abs_freelist xs) = sum_list (map f xs)"
sel_1: fixes x21 :: "'f"   and x22 :: "'f list" shows "hd (x21 # x22) = x21"
submonoid_subgroupI: fixes H :: "'a set" assumes "submonoid H G"   and "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H" shows "subgroup H G"
assms_4: shows "prv (imp (cnj \<phi> \<chi>) \<psi>)"
fun_mono: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and D :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "C \<le> A"   and "B \<le> D" shows "rel_fun A B \<le> rel_fun C D"
simI: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
loc_confluent_WCR: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "relation.is_loc_confluent r = WCR {(x, y). r x y}"
c_A_def: fixes n :: "nat" shows "c_A n = int (index (swaps (sw_A ! n) (s_A n)) (rs ! n) + 1)"
GroupHom_composite_left: fixes H :: "'n set"   and S :: "'n \<Rightarrow> 'b" assumes "ImG \<subseteq> H"   and "GroupHom H S" shows "GroupHom M (S \<circ> T)"
helper: assumes "derivation_ge F' (length (take (Suc j) \<beta>))" shows "LeftDerivation (drop (Suc i) \<alpha>) (derivation_shift F' (length (take (Suc j) \<beta>)) 0) (drop (length (take (Suc j) \<beta>)) \<beta>)"