hcju/isabelleps huggingface dataset:
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "int vec"   and y :: "int vec" assumes "gso_int_tail' x xa xb = y" obtains i :: "nat" and l :: "nat" and acc :: "int vec" where "x = i" and "xa = l" and "xb = acc" and "y = (if i \<le> l then acc else let v = \<mu>' l l \<cdot>\<^sub>v acc - \<mu>' i l \<cdot>\<^sub>v gso_int l l in Let (map_vec (\<lambda>k. k div \<mu>' (l - 1) (l - 1)) v) (gso_int_tail' i (l + 1)))"
path_component_refl: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "path_component S x x"
map_of_entries: fixes t :: "('a, 'b) RBT.rbt" shows "map_of (entries t) = lookup t"
SOME_new_thread_upd_invs: fixes Q :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool"   and t :: "'h"   and x :: "'i"   and m :: "'j"   and tas :: "('h, 'i, 'j) new_thread_action list"   and ts :: "'h \<Rightarrow> ('k \<times> 'l \<Rightarrow>f nat) option"   and I :: "'h \<Rightarrow> 'g option" assumes "Q (SOME i. Q i t x m) t x m"   and "NewThread t x m \<in> set tas"   and "thread_oks ts tas" shows "\<exists>i. upd_invs I Q tas t = \<lfloor>i\<rfloor> \<and> Q i t x m"
FAss_\<tau>red1r_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e\<bullet>F{D} := e2, xs) (e'\<bullet>F{D} := e2, xs')"
list_induct2': fixes P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" assumes "P [] []"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
hyps_5: shows "pred_option (is_valid_fun_call F1 gd n \<Sigma>2) opt"
card_even_odd_subset: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "card {B. B \<subseteq> A \<and> even (card B)} = card {B. B \<subseteq> A \<and> odd (card B)}"
abs_prod: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "abs (comm_monoid_mult.prod times one f A) = comm_monoid_mult.prod times one (\<lambda>x. abs (f x)) A"
preserves_reflects_arr: fixes f :: "'j" shows "C.arr (A'.F.map f) = J.arr f"
pwd_spy_enc_pubk_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_enc_pubk n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_enc_pubk_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
MkIde_Dom': fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "ide a" shows "MkIde (Dom a) = a"
assms_1: shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"
case_2: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Bal of Lh \<Rightarrow> f1 | Bal \<Rightarrow> f2 | Rh \<Rightarrow> f3) = f2"
proj_action_dom_eq_inter: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" shows "action_dom (fst (action_proj a vs)) (snd (action_proj a vs)) = action_dom (fst a) (snd a) \<inter> vs"
Max_in: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "Max A \<in> A"
rec: fixes n :: "nat" assumes "m \<le> n" shows "(\<Sum>k\<le>N. c k * f (n + k)) = g (n + N)"
order_trans: fixes less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x"   and y :: "'x"   and z :: "'x" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
cases: fixes x :: "('sequent, 'rule, 'link) ctree" obtains l :: "'link" where "x = Link l"   | s :: "'sequent" and r :: "'rule" and uu_ :: "('sequent, 'rule, 'link) ctree fset" where "x = cNode (s, r) uu_"
sorted_merge: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" assumes "P = (\<lambda>x y. f x \<le> f y)" shows "sorted_wrt P (Common.merge f xs ys) = (sorted_wrt P xs \<and> sorted_wrt P ys)"
Cons: fixes bs :: "'b list" shows "length (fst (filter2 P as bs)) = length (snd (filter2 P as bs))"
independent_is_basis: fixes B :: "('b, 'c) vec set" shows "(vec.independent B \<and> card B = CARD('c)) = is_basis B"
Sup_exp_least: fixes S :: "('a \<Rightarrow> real) set"   and Q :: "'a \<Rightarrow> real" assumes "\<forall>P\<in>S. P \<tturnstile> Q"   and "nneg Q" shows "Sup_exp S \<tturnstile> Q"
dsumMhom_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'k, 'l) Module_scheme"   and B :: "'h \<Rightarrow> ('j, 'k, 'm) Module_scheme"   and S :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" shows "dsumMhom I A B S = (\<lambda>f\<in>carr_dsumag I A. \<lambda>k\<in>I. S k (f k))"
frestrictionD_2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<bullet> A" shows "b \<in>\<^sub>\<circ> A"
closure_subset: fixes S :: "'a set" shows "S \<subseteq> closure S"
iso_runit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "category.iso C (monoidal_category.runit C T \<alpha> \<iota> a)"
mem_Affine_project_coord_aformI: fixes x :: "'a"   and X :: "'a \<times> 'a pdevs"   and sctn :: "'a sctn" assumes "x \<in> Affine X"   and "x \<in> plane_of sctn" shows "x \<in> Affine (project_coord_aform sctn X)"
inv_mult_on_rcosets_is_bij: fixes H :: "'a set"   and g :: "'a" assumes "subgroup H G"   and "g \<in> carrier G" shows "(\<lambda>U\<in>rcosets H. U #> inv g) \<in> Bij (rcosets H)"
Sup_inf_eq_bot_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'b set"   and a :: "'b" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "(inf (Sup B) a = bot) = (\<forall>b\<in>B. inf b a = bot)"
assms_3: assumes "{#a#} + {#b#} \<subseteq># B" shows "P"
VLambda_iff1: fixes x :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" shows "(x \<in>\<^sub>\<circ> VLambda A f) = (\<exists>a\<in>elts A. x = \<langle>a, f a\<rangle>)"
flowD_source_IN: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "flow \<Delta> f" shows "d_IN f (source \<Delta>) = 0"
Zset_def: shows "Zset = {x. \<exists>n. x = n}"
distinct_17: fixes x6 :: "'b rexp" shows "One \<noteq> Star x6"
assms: shows "F = (\<mu> F. \<Lambda> x. \<^ps> g\<cdot>x \<union> F\<cdot>(\<^ps> R\<cdot>x))"
neutral: fixes A :: "'a multiset" assumes "\<forall>x\<in>#A. x = (1::'a)" shows "\<Prod>\<^sub># A = (1::'a)"
equality: fixes r :: "('a, 'b) monoid_scheme"   and r' :: "('a, 'b) monoid_scheme" assumes "carrier r = carrier r'"   and "(\<otimes>\<^bsub>r\<^esub>) = (\<otimes>\<^bsub>r'\<^esub>)"   and "\<one>\<^bsub>r\<^esub> = \<one>\<^bsub>r'\<^esub>"   and "monoid.more r = monoid.more r'" shows "r = r'"
unapp_rel: fixes t :: "'a"   and u :: "'b" assumes "P t u" shows "rel_option (rel_prod P P) (unapp t) (unapp u)"
intruder_synth_induct: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term"   and P :: "('a, 'b) Term.term set \<Rightarrow> ('a, 'b) Term.term \<Rightarrow> bool" assumes "intruder_model arity Ana"   and "intruder_model.intruder_synth arity public M t"   and "\<And>t. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> intruder_model.intruder_synth arity public M t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P M t"
unit_circle_fix_preserve_is_poincare_line_iff: fixes M :: "moebius"   and H :: "circline" assumes "unit_circle_fix M" shows "is_poincare_line (moebius_circline M H) = is_poincare_line H"
HOLCF_trans_rules_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
Nmlize_Hcomp_Hcomp: fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term" assumes "E.Arr t"   and "E.Arr u"   and "E.Arr v"   and "E.Src t = E.Trg u"   and "E.Src u = E.Trg v" shows "\<^bold>\<lfloor>(t \<^bold>\<star> u) \<^bold>\<star> v\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>v\<^bold>\<rfloor>\<^bold>\<rfloor>"
right_hcomp_closed: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b"   and \<mu> :: "'b"   and x :: "'b"   and y :: "'b"   and \<nu> :: "'b"   and c :: "'b"   and d :: "'b" assumes "left_hom V H \<omega>"   and "partial_magma.in_hom (subcategory.comp V (weak_composition.left V H \<omega>)) \<mu> x y"   and "partial_magma.in_hom V \<nu> c d"   and "H \<mu> \<nu> \<noteq> partial_magma.null V" shows "partial_magma.in_hom (subcategory.comp V (weak_composition.left V H \<omega>)) (H \<mu> \<nu>) (H x c) (H y d)"
card_proots_within_leq: fixes p :: "'b poly"   and s :: "'b set" assumes "p \<noteq> 0" shows "card (proots_within p s) \<le> proots_count p s"
left_move_to_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq a (plus (uminus c) b)" shows "less_eq (plus c a) b"
v11_app_if_vconverse_app: fixes r :: "V"   and y :: "V"   and x :: "V" assumes "v11 r"   and "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"   and "r\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> = x" shows "r\<lparr>x\<rparr> = y"
is_ladder_prop2: fixes u :: "nat"   and v :: "nat" assumes "u < v"   and "v < length L'" shows "ladder_n L' u < ladder_n L' v"
assms_5: fixes l1 :: "'a list"   and l2 :: "'a list"   and l3 :: "'a list" assumes "x \<noteq> y"   and "l = l1 @ y # l2 @ x # l3" shows "P"
\<V>_def: shows "\<V> \<equiv> (\<lambda>U. U \<inter> S) ` \<U>"
sel_3: fixes x21 :: "'a process"   and x22 :: "'a process" shows "ch1Of (Choice x21 x22) = x21"
lift_to_metric_completion: fixes f :: "'b \<Rightarrow> 'c" assumes "isUCont f" shows "\<exists>g. isUCont g \<and> f = g \<circ> to_metric_completion \<and> (\<forall>x\<in>range to_metric_completion. g x = f (inv to_metric_completion x))"
enat_less_imp_le: fixes n :: "enat"   and m :: "enat" assumes "\<And>k. n < enat k \<Longrightarrow> m < enat k" shows "m \<le> n"
simps_15: fixes x1 :: "'a" shows "set_Literal (Pos x1) = {x1}"
d3_def: fixes a :: "'a"   and b :: "'a" shows "d3 a b = d1 b a"
Conj\<^sub>\<alpha>_eq_iff: fixes tset1\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]"   and tset2\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]" shows "(Conj\<^sub>\<alpha> tset1\<^sub>\<alpha> = Conj\<^sub>\<alpha> tset2\<^sub>\<alpha>) = (tset1\<^sub>\<alpha> = tset2\<^sub>\<alpha>)"
small_UN_V: fixes X :: "'a set"   and B :: "'a \<Rightarrow> V set" assumes "small X"   and "\<And>x. x \<in> X \<Longrightarrow> small (B x)" shows "small (\<Union> (B ` X))"
shiftr_less_t2n: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x < 2 ^ (n + m)" shows "x >> n < 2 ^ m"
monofun_LAM: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "cont f"   and "cont g"   and "\<And>x. f x \<sqsubseteq> g x" shows "(\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)"
rename_closed: fixes f :: "'form"   and p :: "'var \<Rightarrow> 'var" assumes "lconsts f = {}" shows "renameLCs p f = f"
rep_eq: fixes x :: "(nat, 'd) slist" shows "set_slist x = set (Rep_slist x)"
Nil2: shows "q\<^sub>0 = foldl \<delta> q\<^sub>0 []"
matrix_vector_mult_0: fixes w :: "('b, 'd) vec" shows "0 *v w = 0"
take_cols_from_matrix_def: fixes matr :: "'b mat"   and indices_to_keep :: "nat list" shows "take_cols_from_matrix matr indices_to_keep = mat_of_cols (dim_row matr) (take_indices (cols matr) indices_to_keep)"
intro: fixes letter_eq :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and alphabet1 :: "'b list"   and alphabet2 :: "'c list" assumes "\<And>a b. letter_eq a b \<Longrightarrow> (a \<in> set alphabet1) = (b \<in> set alphabet2)" shows "DAs_axioms alphabet1 alphabet2 letter_eq"
image_projection_PiE: fixes i :: "'b"   and I :: "'b set"   and S :: "'b \<Rightarrow> 'a set" shows "(\<lambda>f. f i) ` Pi\<^sub>E I S = (if Pi\<^sub>E I S = {} then {} else if i \<in> I then S i else {undefined})"
Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex_axioms: shows "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns"
add: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "additive f" shows "f (x + y) = f x + f y"
inject_2: fixes x2 :: "agent"   and y2 :: "agent" shows "(publK x2 = publK y2) = (x2 = y2)"
is_hom_ord_def: fixes x :: "'a option" shows "extended_ord.is_hom_ord x = (\<forall>s t. deg_pm s = deg_pm t \<longrightarrow> extended_ord s t = extended_ord (except s {x}) (except t {x}))"
sd_sa_sublinear: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "sub_distrib t"   and "sub_add t"   and "healthy t" shows "sublinear t"
fun_zero_def: shows "fun_zero \<equiv> \<lambda>v. 0::'a"
ProofStepInduct_axioms_def: fixes paths :: "'c list set"   and sep_size :: "nat" shows "ProofStepInduct_axioms paths sep_size \<equiv> sep_size \<noteq> 0 \<and> card paths = sep_size"
prod_add: fixes A :: "'c set"   and f1 :: "'c \<Rightarrow> 'd"   and f2 :: "'c \<Rightarrow> 'd" assumes "finite A" shows "(\<Prod>x\<in>A. f1 x + f2 x) = (\<Sum>X\<in>Pow A. prod f1 X * prod f2 (A - X))"
hv_vector: fixes x :: "real"   and y :: "real"   and z :: "real" shows "Hv (vector [x, y, z]) = Quat 0 x y z"
assms_4: shows "interp_rga xs = interp_ins ys"
mult_2_right: fixes one :: "'h"   and times :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and plus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and z :: "'h" assumes "class.semiring_numeral one times plus" shows "times z (numeral.numeral one plus (num.Bit0 num.One)) = plus z z"
foundation16'': fixes \<tau> :: "'b state \<times> 'b state"   and X :: "'b state \<times> 'b state \<Rightarrow> 'c" shows "(\<tau> \<Turnstile> \<delta> X) \<noteq> (\<tau> \<Turnstile> X \<triangleq> invalid \<or> \<tau> \<Turnstile> X \<triangleq> null)"
1: shows "IsProperInX (\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<rparr>)"
cod_def: fixes S :: "nat \<Rightarrow> typ" shows "cod S = (\<Union>m\<in>Type.dom S. free_tv (S m))"
gfp_eq_flatf_gfp: fixes B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "monotone flatf_ge flatf_ge B"   and "mono B" shows "gfp B = flatf_gfp B"
is_glb_glb: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_glb below M x" shows "po.is_glb below M (po.glb below M)"
MPoly_inject: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'q"   and y :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'q" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(MPoly x = MPoly y) = (x = y)"
digraph_subgraph: fixes H :: "('a, 'b) pre_digraph" assumes "subgraph H G" shows "digraph H"
bc: shows "b \<squnion>\<^bsub>f\<^esub> c \<in> A"
list_of_eucl_eucl_of_list: fixes xs :: "real list" assumes "length xs = DIM('a)" shows "list_of_eucl (eucl_of_list xs) = xs"
reduction_P1: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "\<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (funct_OT_14 M (c0, c1) \<bind> (\<lambda>(out1, out2). S1_14 M out1 \<bind> D)) True\<bar> \<le> 3 * adv_OT12"
hyps: fixes m :: "nat" assumes "Suc (bl2wc c) = 2 ^ m" shows "\<exists>rs n. c = Oc \<up> rs @ Bk \<up> n"
synthetic_div_correct: fixes p :: "'a poly"   and c :: "'a" shows "p + Polynomial.smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
gvut3: fixes s :: "('d, 'c) configuration stream"   and q :: "'d"   and t :: "'c" assumes "GlobVacantUpto (shd s) q t"   and "InvGlobVacantUptoImpliesNrec (shd s)"   and "InvGlobalRecordCount (shd s)"   and "InvGlobalIncomingInfoUpright (shd s)"   and "InvGlobalInfoAtBetaUpright (shd s)"   and "next_recvupd s" shows "GlobVacantUpto (shd (stl s)) q t"
hyps: fixes as :: "'a" shows "\<guillemotleft>lderiv as \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>lderiv as r\<guillemotright>"
TRANS: fixes X :: "freemsg"   and Y :: "freemsg"   and Z :: "freemsg" assumes "X \<sim> Y"   and "Y \<sim> Z" shows "X \<sim> Z"
stable_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "ManiPref.stable_on ds X = (ManiPref.individually_rational_on ds X \<and> ManiPref.stable_no_blocking_on ds X)"
parallel_oracle_split: fixes P :: "(('b + 'd) \<times> 's1 \<times> 's2) spmf \<Rightarrow> bool"   and s1s2 :: "'s1 \<times> 's2"   and lr :: "'a + 'c" shows "P (local.parallel_oracle s1s2 lr) = ((\<forall>s1 s2 x. s1s2 = (s1, s2) \<longrightarrow> lr = Inl x \<longrightarrow> P (map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 x))) \<and> (\<forall>s1 s2 y. s1s2 = (s1, s2) \<longrightarrow> lr = Inr y \<longrightarrow> P (map_spmf (map_prod Inr (Pair s1)) (right s2 y))))"
11: shows "bprv (imp (NN \<langle>\<phi>\<rangle> (Var yy)) (imp \<chi> (subst \<chi> \<langle>neg \<phi>\<rangle> yy)))"
H_unhalt_inv: fixes M :: "(action \<times> nat) list"   and ns :: "nat list" assumes "halts M ns" shows "{\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0>)}"
resultant_map_poly: fixes p :: "int poly"   and q :: "int poly" assumes "degree (of_int_poly p) = degree p"   and "degree (of_int_poly q) = degree q" shows "resultant (of_int_poly p) (of_int_poly q) = of_int (resultant p q)"
card_of_Plus_empty2: fixes A :: "'a set" shows "|A| =o |{} <+> A|"
of_int_eq_id: shows "ring_1_class.of_int = id"
accept: shows "gba.is_acc_run \<sigma> \<and> (\<forall>i. local.gba.L (\<sigma> i) (\<xi> i))"
index_map_vec_1: fixes i :: "nat"   and v :: "'b vec"   and f :: "'b \<Rightarrow> 'c" assumes "i < dim_vec v" shows "map_vec f v $ i = f (v $ i)"
WellDefinedTyE: fixes S :: "('a, 'b) Signature"   and A :: "'a" assumes "Sig S \<triangleright> \<turnstile> A Type" obtains "A \<in> Ty\<^bsub>S\<^esub>"
A: shows "{x. x \<le> a \<and> poly p x = 0} = {x. x < a \<and> poly p x = 0} \<union> (if poly p a = 0 then {a} else {})"
continuous_on_closure_sequentially: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on (closure S) f = (\<forall>x a. a \<in> closure S \<and> (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
is_le_relE: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "is_le_rel rel" obtains "rel = (=)"   | "rel = (\<le>)"   | "rel = (<)"
distinct_167: fixes x6 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Arctan x6 \<noteq> Power x151 x152"
min_height_acomplete: fixes t :: "'b tree" assumes "acomplete t" shows "min_height t = nat \<lfloor>log 2 (real (size1 t))\<rfloor>"
ImpR: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>"
interval_extension_floor: shows "interval_extension1 (\<lambda>ivl. Some (floor_float_interval ivl)) (\<lambda>x. real_of_int \<lfloor>x\<rfloor>)"
Pair_le: fixes a :: "'d"   and b :: "'e"   and c :: "'d"   and d :: "'e" shows "((a, b) \<le> (c, d)) = (a \<le> c \<and> b \<le> d)"
chine_assoc_props_3: shows "trg_trg_f.Prj\<^sub>1\<^sub>0 \<cdot> trg_trg_f.chine_assoc = trg_trg_f.Prj\<^sub>0\<^sub>1"
mod_mult_self2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b * c) mod b = a mod b"
universe_eq: shows "u = (THE z. \<forall>w. P w z)"
ind_eq_fin: fixes u :: "'action list"   and v :: "'action list" assumes "Ind (set u) (set v)" shows "u @ v =\<^sub>F v @ u"
simps_1: fixes uu :: "'b"   and B :: "'b"   and uv :: "'c" shows "local.Cod (MkArr uu B uv) = B"
kcomp_klift: fixes f :: "'e \<Rightarrow> 'g set"   and g :: "'g \<Rightarrow> 'f set" shows "f \<circ>\<^sub>K g = g\<^sup>\<dagger> \<circ> f"
Quotient_refl1: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c"   and s :: "'c" assumes "Quotient R Abs Rep T"   and "R r s" shows "R r r"
vimageE: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "a \<in> f -` B" obtains x :: "'b" where "f a = x" and "x \<in> B"
path_image_part_circlepath_subset: fixes r :: "real"   and z :: "complex"   and st :: "real"   and tt :: "real" assumes "0 \<le> r" shows "path_image (part_circlepath z r st tt) \<subseteq> sphere z r"
assms_3: shows "i \<noteq> 1 \<or> j \<noteq> 1"
poly_inf_def: fixes p :: "'b poly" shows "poly_inf p \<equiv> sgn (lead_coeff p)"
select_convs_1: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "sccs \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = sccs"
simps_2: fixes s :: "'c stream"   and n :: "nat" shows "s !! Suc n = stl s !! n"
fold_Option_bind_eq_Some_at_each_point_Some: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd \<Rightarrow> 'd option"   and l1 :: "('e \<times> 'f) list"   and l2 :: "('e \<times> 'f) list"   and start :: "'d option"   and res :: "'d" assumes "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (l1 @ l2) start = Some res" shows "\<exists>point. fold (\<lambda>(x, y) option. Option.bind option (f x y)) l1 start = Some point \<and> fold (\<lambda>(x, y) option. Option.bind option (f x y)) l2 (Some point) = Some res"
g_orbit_def: fixes X :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> bool"   and U :: "'d set" shows "\<gamma> X G U = \<Union> {\<P> X (down U t) |t. \<P> X (down U t) \<subseteq> {s. G s}}"