hcju/isabelleps huggingface dataset:
is_ft_dghm_axioms_def: fixes \<AA> :: "V"   and \<FF> :: "V" shows "is_ft_dghm_axioms \<AA> \<FF> \<equiv> \<forall>a b. a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<longrightarrow> v11 (\<FF>\<lparr>ArrMap\<rparr> \<restriction>\<^sup>l\<^sub>\<circ> Hom \<AA> a b)"
M1: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and p :: "'b"   and q :: "'b"   and s :: "'b"   and r :: "'b" assumes "class.interval meets \<I>"   and "meets p q"   and "meets p s"   and "meets r q" shows "meets r s"
rel_S_witness'2: fixes L1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and L2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Co1 :: "'e \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'g \<Rightarrow> bool"   and Co2 :: "'h \<Rightarrow> 'p \<Rightarrow> bool"   and Co2' :: "'p \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1 :: "'j \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1' :: "'q \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'r \<Rightarrow> bool"   and Contra2' :: "'r \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "('a, 'c, 'e, 'h, 'j, 'l, 'n) S"   and y :: "('b, 'd, 'g, 'i, 'k, 'm, 'n) S"   and tytok :: "('a \<times> ('a \<times> 'b) \<times> 'b \<times> 'c \<times> ('c \<times> 'd) \<times> 'd \<times> 'n) itself" assumes "rel_S L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y"   and "rel_S_neg_distr_cond' Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok" shows "rel_S (\<lambda>(x, y') y. y' = y \<and> L1 x y) (\<lambda>(x, y') y. y' = y \<and> L2 x y) Co1' Co2' Contra1' Contra2' (rel_S_witness' L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y)) y"
image_constant: fixes x :: "'d"   and A :: "'d set"   and c :: "'e" assumes "x \<in> A" shows "(\<lambda>x. c) ` A = {c}"
acyclic_subcls1: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "acyclicP (subcls1 P)"
weak_reduction_simulations_impl_bisimulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_simulation Rel Cal"   and "weak_reduction_simulation (Rel\<inverse>) Cal" shows "weak_reduction_bisimulation Rel Cal"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> min b c" obtains "a \<le> b" and "a \<le> c"
N_2: fixes x :: "'b" shows "N x \<in> null_sets M"
lattice_of_swap: fixes fs :: "rat vec list"   and i :: "nat"   and j :: "nat"   and gs :: "rat vec list" assumes "set fs \<subseteq> Rn"   and "i < length fs"   and "j < length fs"   and "i \<noteq> j"   and "gs = fs[i := fs ! j, j := fs ! i]" shows "gs.lattice_of gs = gs.lattice_of fs"
same_append_eq: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
rbt_inter_def: shows "rbt_inter = rbt_inter_with_key (\<lambda>_ _ rv. rv)"
comp_ide_self: fixes a :: "'b" assumes "Adj.D.ide a" shows "a \<cdot>\<^sub>D a = a"
BIGTHETA_FUN_def: fixes f :: "'a \<Rightarrow> 'b" shows "BIGTHETA_FUN f = \<Theta>(f)"
eq_refl: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x = y" shows "x \<subseteq># y"
smult_l_distr: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M1" shows "(a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M1\<^esub> x = a \<odot>\<^bsub>M1\<^esub> x \<oplus>\<^bsub>M1\<^esub> b \<odot>\<^bsub>M1\<^esub> x"
cls: shows "class P C = \<lfloor>a\<rfloor>"
length_insort: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "length (insort_key f x xs) = Suc (length xs)"
assms: fixes x :: "'c" shows "monotone (\<sqsubseteq>) (\<le>) (\<lambda>xs. f x xs (LCons x xs))"
negatively_invariant_eq_rev: fixes M :: "'a set" shows "rev.negatively_invariant M = positively_invariant M"
complete_finite: fixes V :: "nat set" assumes "finite V" shows "finite_graph (Ugraph_Lemmas.complete V)"
flows_reverse: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> rev.existence_ivl0 x0" shows "rev.flow0 (rev.flow0 x0 t) (0 - t) = x0"
normalize_preserves_generators: fixes l :: "(bool \<times> 'd) list"   and gens :: "'d set" assumes "l \<in> lists (UNIV \<times> gens)" shows "Cancelation.normalize l \<in> lists (UNIV \<times> gens)"
complement_bibd_index: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and ps :: "'a set" assumes "bibd point_set block_collection u_block_size index"   and "ps \<subseteq> point_set"   and "card ps = 2" shows "int (incidence_system.complement_blocks point_set block_collection index ps) = int (size block_collection) + index - 2 * (index * (int (card point_set) - 1) div (u_block_size - 1))"
exec_bind_SE_success': fixes M :: "'d \<Rightarrow> (bool \<times> 'd) option"   and \<sigma> :: "'d"   and f :: "'d \<Rightarrow> bool" assumes "M \<sigma> = Some (f \<sigma>, \<sigma>)" shows "(\<sigma> \<Turnstile> M) = f \<sigma>"
zero_nd_fun_def: shows "0 = \<zeta>\<^sup>\<bullet>"
no_\<sigma>\<omega>: fixes x :: "\<sigma>"   and y :: "\<omega>" shows "\<sigma>\<upsilon> x \<noteq> \<omega>\<upsilon> y"
zero_compare_simps_12: fixes a :: "'b" shows "(0::'b) \<le> a\<^sup>2"
degree_head: fixes p :: "poly" shows "Reflected_Multivariate_Polynomial.degree (head p) = 0"
Range_rtrancl: fixes R :: "('b \<times> 'b) set" shows "Range (R\<^sup>*) = UNIV"
finite_Union: fixes A :: "'a set set" assumes "finite A"   and "\<And>M. M \<in> A \<Longrightarrow> finite M" shows "finite (\<Union> A)"
simps_3: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set" shows "min_maxsimpchain (x # xs @ [y]) = (x \<noteq> y \<and> is_arg_min length (\<lambda>zs. maxsimpchain (x # zs @ [y])) xs)"
f_closed: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a \<^bold>*\<^sub>o\<^sub>w b \<in> U"
wf_map_prod_image_Dom_Ran: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "wf r"   and "\<And>a a'. \<lbrakk>a \<in> Domain r; a' \<in> Range r; f a = f a'\<rbrakk> \<Longrightarrow> a = a'" shows "wf (map_prod f f ` r)"
scaleR_right_distrib: fixes a :: "real"   and x :: "'c"   and y :: "'c" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"
ccExp_pap: fixes e :: "exp" assumes "isVal e" shows "\<G>\<^bsub>0\<^esub> e = fv e\<^sup>2"
qequiv_sym: fixes a :: "'b['c, 'd]"   and b :: "'b['e, 'd]" assumes "a \<cong>\<^sub>Q b" shows "b \<cong>\<^sub>Q a"
ideal_nprod_inc: fixes A :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "\<forall>i\<le>n. f i \<in> carrier R"   and "\<exists>l\<le>n. f l \<in> A" shows "e\<Pi>\<^bsub>R,n\<^esub> f \<in> A"
Singleton: fixes x :: "'a" shows "lsorted (LCons x LNil)"
redT_updWs_None_implies_None: fixes t' :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and t :: "'a" assumes "redT_updWs t' ws was ws'"   and "t \<noteq> t'"   and "ws t = None" shows "ws' t = None"
simps_1: fixes b1 :: "bool"   and b2 :: "bool" shows "local.less_gexp_aux (Bc b1) (Bc b2) = (b1 < b2)"
FunctorM_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "FunctorM F \<equiv> PreFunctor F \<and> FunctorM_axioms F"
lemma_4_5: fixes a :: "'a"   and b :: "'a" shows "a \<squnion>1 b = a \<squnion>2 b"
hyps_1: fixes G :: "'a pair_pre_digraph" assumes "n = card (pverts G - V)"   and "pair_pre_digraph.is_slim G V"   and "pair_pseudo_graph G" shows "subdivision_pair (gen_contr_graph (with_proj G) V) G"
invar_insert: fixes t :: "'b tree list"   and x :: "'b" assumes "invar t" shows "invar (insert x t)"
span_unique: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "subspace T"   and "\<And>T'. \<lbrakk>S \<subseteq> T'; subspace T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'" shows "span S = T"
fresh_set: fixes y :: "'a"   and xs :: "'b list" shows "y \<sharp> xs = (\<forall>x\<in>set xs. y \<sharp> x)"
compare_strict_2: fixes compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b" assumes "class.Ord_strict compare" shows "compare\<cdot>x\<cdot>\<bottom> = \<bottom>"
x_coord_inv_1: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real"   and x :: "real" assumes "circle i j d" shows "x_coord_inv (x_coord x) = x"
Length_Extend: fixes k :: "'k"   and i :: "nat"   and \<AA> :: "'i"   and P :: "'v" shows "Length (Extend k i \<AA> P) = max (Length \<AA>) (len P)"
eval_ternary_simps_simple_2: fixes x :: "ternaryvalue" shows "eval_ternary_And x TernaryTrue = x"
LeadsTo_Diff: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "F \<in> A - B \<longmapsto>w C"   and "F \<in> A \<inter> B \<longmapsto>w C" shows "F \<in> A \<longmapsto>w C"
tensor_vec_def: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" shows "u \<otimes> v \<equiv> vec_of_list (mult.vec_vec_Tensor (*) (list_of_vec u) (list_of_vec v))"
mbisim_wset_thread_ok2: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set" assumes "FWbisimulation_base.mbisim bisim bisim_wait s1 s2" shows "wset_thread_ok (wset s2) (thr s2)"
simps_1: shows "\<guillemotleft>Zero\<guillemotright> = Zero"
assms_5: shows "y \<noteq> \<infinity>\<^sub>h"
d_def: fixes t :: "real" shows "d t = (if t \<le> 0 then s\<^sub>o - s\<^sub>e else if t \<le> ego.t_stop \<and> t \<le> other.t_stop then ego_other.p t else if ego.t_stop \<le> t \<and> t \<le> other.t_stop then other.p t - ego.s_stop else if other.t_stop \<le> t \<and> t \<le> ego.t_stop then other.s_stop - ego.p t else other.s_stop - ego.s_stop)"
inv_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<oplus> y' = \<zero>"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "y' \<in> carrier R" shows "y = y'"
match_prx: shows "(\<exists>pt\<in>set prx. repNodes_eq pt p low high repa) \<longrightarrow> repa p = hd (filter (\<lambda>sn. repNodes_eq sn p low high repa) prx)"
le_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D" shows "B A Le D C"
simps_1: fixes a :: "'a"   and as :: "'a list"   and b :: "'a"   and bs :: "'a list" shows "sorted_list_subset (a # as) (b # bs) = (if a = b then sorted_list_subset as (b # bs) else if b < a then sorted_list_subset (a # as) bs else Some a)"
divide_fps_const': fixes f :: "'a fps"   and c :: "'a" shows "f / fps_const c = f * fps_const (inverse c)"
F: fixes a :: "'a"   and b :: "'b"   and a' :: "'a" assumes "a \<in> basisA'"   and "b \<in> basisB"   and "a' \<in> basisA'" shows "F a b \<in> basis \<and> F a b *\<^sub>V a' = (if a' = a then b else (0::'b))"
Abs_NF: fixes x :: "name"   and t :: "lam"   and ts :: "lam list" assumes "NF ((Lam [x].t) \<degree>\<degree> ts)" shows "ts = []"
A3_reversed: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2" assumes "a a \<congruent>\<^sub>K b c" shows "b = c"
vwb_lens_iff_mwb_UNIV_src: fixes X :: "'b \<Longrightarrow> 'c" shows "vwb_lens X = (mwb_lens X \<and> \<S>\<^bsub>X\<^esub> = UNIV)"
Enc: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "Enc X Y \<in> synth H"
invar'_5: shows "subprob_cexpr (set vs) (set vs') \<Gamma> \<delta>"
uv: shows "(u, v) \<in> (R \<union> E)\<^sup>*"
mult_right_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> (0::'a)" shows "a * c \<le> b * c"
document_ptr_kinds_eq_h3: shows "document_ptr_kinds h' = document_ptr_kinds h3"
pred_intros_logic_3: fixes Q :: "'c \<Rightarrow> bool"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "pred M Q"   and "pred M P" shows "pred M (\<lambda>x. Q x \<and> P x)"
above_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "above r a \<equiv> {b. (a, b) \<in> r}"
thetaSPar_Sbis: shows "thetaSPar \<subseteq> Sbis"
inject_3: fixes x1 :: "name"   and x2 :: "lam"   and y1 :: "name"   and y2 :: "lam" shows "((Lam [x1].x2) = (Lam [y1].y2)) = ([x1].x2 = [y1].y2)"
when_diff_distrib: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b"   and P :: "bool" assumes "class.cancel_comm_monoid_add plus minus zero" shows "zero.when zero (minus a b) P = minus (zero.when zero a P) (zero.when zero b P)"
c2: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
simps_7: fixes v :: "formula" shows "get_and_list (Neg v) = [Neg v]"
antisym_conv: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and y :: "'g"   and x :: "'g" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
subst_intro: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and u :: "expr" assumes "fresh xT t"   and "lc u" shows "open0 u t = subst xT u (open0_Var xT t)"
sep_disj_addD: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y + z"   and "y ## z" shows "x ## y \<and> x ## z"
\<K>_def: shows "\<K> = (\<Sum>i\<in>L. 1 / b ^ cw_len i)"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<squnion> b = \<bottom>) = (a = \<bottom> \<and> b = \<bottom>)"
elims_2: fixes x :: "rat \<times> rat"   and xa :: "rat \<times> rat" assumes "sub_interval x xa" obtains l :: "rat" and r :: "rat" and l' :: "rat" and r' :: "rat" where "x = (l, r)" and "xa = (l', r')" and "l' \<le> l \<and> r \<le> r'"
derivable_premisses: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and P :: "'b eclause set"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> k C'" shows "P \<subseteq> S"
ldropWhile_simps_1: fixes P :: "'c \<Rightarrow> bool" shows "ldropWhile P LNil = LNil"
gfull_add: fixes n :: "nat"   and t :: "'a tree23"   and k :: "int"   and y :: "'a" assumes "full n t" shows "gfull n (add k y t)"
clinear_continuous_at: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "bounded_clinear f" shows "isCont f x"
induct: fixes P :: "('b set \<times> 'b set) list \<Rightarrow> (('d, 'e, 'b) prot_fun, 'c) Term.term \<Rightarrow> (('d, 'e, 'b) prot_fun, 'c) Term.term \<Rightarrow> bool"   and a0 :: "('b set \<times> 'b set) list"   and a1 :: "(('d, 'e, 'b) prot_fun, 'c) Term.term"   and a2 :: "(('d, 'e, 'b) prot_fun, 'c) Term.term" assumes "\<And>TI x y. P TI (Var x) (Var y)"   and "\<And>TI f T g S. (\<And>z yb. \<lbrakk>z \<in> set T; yb \<in> set S\<rbrakk> \<Longrightarrow> P TI z yb) \<Longrightarrow> P TI (Fun f T) (Fun g S)"   and "\<And>uu_ v va vb. P uu_ (Fun v va) (Var vb)"   and "\<And>uu_ vb v va. P uu_ (Var vb) (Fun v va)" shows "P a0 a1 a2"
assms_4: shows "v \<in> set (add_effects_of op)"
sing_gen_power: fixes u :: "'b list"   and x :: "'b list" assumes "u \<in> \<langle>{x}\<rangle>" shows "\<exists>k. u = x \<^sup>@ k"
etc_2: shows "h\<^sub>0' \<bottom> h\<^sub>1'"
2: assumes "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t subst_range \<theta>" shows "\<forall>u\<in>set T. u \<notin> Var ` X"
transpD: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
path_component_subset: fixes S :: "'b set"   and x :: "'b" shows "path_component_set S x \<subseteq> S"
sat_push_action_bound': fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'" shows "length (filter is_SAT_PUSH' p) \<le> 4 * card V * card E"
poly_minus: fixes p :: "'c poly"   and x :: "'c" shows "poly (- p) x = - poly p x"
invrules_1: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"
vector_derivative_works: fixes f :: "real \<Rightarrow> 'a"   and net :: "real filter" shows "(f differentiable net) = (f has_vector_derivative vector_derivative f net) net"
cong: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "f = g"   and "x = y" shows "f x = g y"
simps_1: shows "extB.Bot + extB.Bot = extB.Val top"
vars0_def: shows "ps2_P.vars0 = vars1 \<union> vars2"
extended_distance_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c Bonk_Schramm_extension_unfolded"   and y :: "'c Bonk_Schramm_extension_unfolded" assumes "class.metric_space dist uniformity open" shows "metric_space.extended_distance dist x y = metric_space.extend_distance_fp dist (max2 metric_space.Bonk_Schramm_extension_unfolded_wo x y) x y"
unit_imp_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 b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times b a"
corthogonal_inv_result: fixes A :: "'c mat" assumes "corthogonal_mat A" shows "inverts_mat (corthogonal_inv A) A"
p_def: fixes t :: "real" shows "delayed_safe_distance.ego_other.p t = other.s \<delta> - ego.q \<delta> + (other.s' \<delta> - v\<^sub>e) * t + 1 / 2 * (a\<^sub>o - a\<^sub>e) * t\<^sup>2"
thetaParWL2_def: shows "thetaParWL2 \<equiv> {(c1 | c2, d) |c1 c2 d. discr c1 \<and> c2 \<approx>w d}"
intros_1: fixes u :: "'c \<Rightarrow> 'd"   and cc1 :: "('c, 'd) cconstraint"   and cc2 :: "('c, 'd) cconstraint" assumes "u \<turnstile> cc1"   and "u \<turnstile> cc2" shows "u \<turnstile> AND cc1 cc2"
sub_distrib_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "sub_distrib t = (\<forall>P s. sound P \<longrightarrow> t P s \<ominus> 1 \<le> t (\<lambda>s'. P s' \<ominus> 1) s)"
h1_var: shows "M_mat_R (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) (get_subsets_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) = M_mat_R (take_indices signs (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (take_indices subsets (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))))"
gcollapse_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat" shows "gcollapse_code node_eq_impl node_hash_impl \<equiv> \<lambda>x xa. let y' = xa in case y' of (xb, xc, xda, xe, xf) \<Rightarrow> idx_of_code node_eq_impl node_hash_impl (xc, xda, xe, xf) x \<bind> (\<lambda>xg. let y'a = as_take (xg + 1) xda in RETURN (Un_set_drop_tr (bs_empty ()) bs_union xg xb) \<bind> (\<lambda>xh. let y'b = as_push (as_take xg xb) xh in RETURN (y'b, xc, y'a, xe, xf)))"
Sbis_raw_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> Sretr theta" shows "theta \<subseteq> Sbis"
trivial_group: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (\<exists>a. carrier G = {a})"
of_int_of_int_mod_ring: shows "of_int = of_int_mod_ring"
trans: shows "root \<Midarrow>xs\<Rightarrow> root'"
commute: fixes a :: "'a"   and b :: "'a" shows "lcm a b = lcm b a"
mat_max_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat" shows "ord.mat_max less_eq A B = mat (dim_row A) (dim_col A) (\<lambda>ij. ord.max less_eq (A $$ ij) (B $$ ij))"
is_extensional: fixes f :: "'a1 \<times> 'a2" assumes "\<not> A1xA2.arr f" shows "G f = B.null"
le_sup_iff: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "(x \<union># y \<subseteq># z) = (x \<subseteq># z \<and> y \<subseteq># z)"
Sublists_star: fixes A :: "'a list set" assumes "A \<noteq> {}" shows "Sublists (star A) = Sublists A \<union> Suffixes A @@ star A @@ Prefixes A"
finite_set_rel_transfer: fixes s :: "'c set"   and s' :: "'d set"   and R :: "('c \<times> 'd) set" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "single_valued R"   and "finite s" shows "finite s'"
subspace_affine: fixes S :: "'c set" shows "subspace S = (affine S \<and> (0::'c) \<in> S)"
Min_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "linorder.Min (\<sqsubseteq>) A \<in> A"
Zp_x_is_UP_cring: fixes p :: "int" assumes "padic_integers p" shows "UP_cring (padic_int p)"
standard_top_not_bot: fixes C :: "'c set"   and G :: "('c Standard_Constant, 'c + 'd) labeled_graph" assumes "standard' C G" shows ":G:\<lbrakk>\<bottom>\<rbrakk> \<noteq> :G:\<lbrakk>\<top>\<rbrakk>"
induct: fixes P :: "tnam' \<Rightarrow> bool"   and tnam' :: "tnam'" assumes "P HasFoo'"   and "P Base'"   and "P Ext'"   and "P Main'" shows "P tnam'"
right_implies_left: assumes "\<exists>\<alpha>. x = finished_item \<alpha>" shows "is_finished x"
map_option_eq_Some: fixes f :: "'b \<Rightarrow> 'a"   and xo :: "'b option"   and y :: "'a" shows "(map_option f xo = \<lfloor>y\<rfloor>) = (\<exists>z. xo = \<lfloor>z\<rfloor> \<and> f z = y)"
find_closest_pair_c0_ne_c1: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int" assumes "c\<^sub>0 \<noteq> c\<^sub>1"   and "distinct ps"   and "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps" shows "C\<^sub>0 \<noteq> C\<^sub>1"
that: assumes "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" shows "thesis"
restrict_indets_dehomogenize: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "restrict_indets (dehomogenize None p) = restrict_indets p"
coeff_p: fixes k :: "nat" shows "coeff p k = (if k \<le> n then f k else (0::'a))"
atom_def: fixes x :: "'a" shows "atom x = (x \<noteq> \<bottom> \<and> \<not> (\<exists>y>\<bottom>. y < x))"
run_alt_option: fixes alt :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and m1 :: "('e, 'd) optionT"   and m2 :: "('e, 'd) optionT" shows "run_option (alt_option alt m1 m2) = alt (run_option m1) (run_option m2)"
merge_states_symmetry: fixes x :: "nat"   and y :: "nat"   and t :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "merge_states x y t = merge_states y x t"
4: shows "ks \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> [m] = ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m]"
linorder_linear: fixes x :: "'d"   and y :: "'d" shows "x \<le> y \<or> y \<le> x"
le_trans_refl: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "le_trans x x"
nonempty_single: fixes x :: "'a multiset" shows "{#x#} \<noteq> {#}"
eval_fds_eqD: fixes f :: "'a fds"   and g :: "'a fds" assumes "conv_abscissa f < \<infinity>"   and "conv_abscissa g < \<infinity>"   and "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> (1::'a)) going_to at_top. eval_fds f s = eval_fds g s" shows "f = g"