hcju/isabelleps huggingface dataset:
quotient_eq_iff_same_a_r_cos: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal I R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<ominus> b \<in> I) = (I +> a = I +> b)"
cobounded1: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "semilattice_order f less_eq less" shows "less_eq (f a b) a"
root_nat_def: fixes p :: "nat"   and x :: "nat" shows "root_nat p x = map nat (take 1 (root_int p (int x)))"
complete_latticeI: assumes "\<And>A. A \<subseteq> carrier thy_order \<Longrightarrow> \<exists>s. is_lub thy_order s A"   and "\<And>A. A \<subseteq> carrier thy_order \<Longrightarrow> \<exists>i. is_glb thy_order i A" shows "complete_lattice thy_order"
fps_power_nth_Suc: fixes a :: "'a fps"   and m :: "nat"   and n :: "nat" shows "fps_nth (a ^ Suc m) n = (\<Sum>v\<in>natpermute n (m + 1). \<Prod>j = 0..m. fps_nth a (v ! j))"
compact_rel_frontier_bounded: fixes S :: "'a set" assumes "bounded S" shows "compact (rel_frontier S)"
dm_eq: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset" assumes "irrefl r"   and "trans r" shows "r \<down>m M = r \<down>m (M -s r \<down>m M)"
assms_2: shows "e \<in> UNIV \<rightarrow> I"
prems_1: shows "x \<^sup>@ a \<cdot> y \<^sup>@ b = z \<^sup>@ c"
simps: fixes a1 :: "'b set"   and a2 :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" shows "((a1, a2) \<in> foldSetD D f e) = (a1 = {} \<and> a2 = e \<and> e \<in> D \<or> (\<exists>x A y. a1 = insert x A \<and> a2 = f x y \<and> x \<notin> A \<and> f x y \<in> D \<and> (A, y) \<in> foldSetD D f e))"
globally_consistent_var_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option"   and mds :: "Mode \<Rightarrow> 'b set"   and x :: "'b" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.globally_consistent_var dma A mds x \<equiv> case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> x \<notin> mds AsmNoWrite \<and> (dma x = Low \<longrightarrow> v = v')"
inorder_join_all: fixes ts :: "'a tree23s" shows "inorder (join_all ts) = inorder2 ts"
sshiftr_eq: fixes w :: "'b word"   and n :: "nat" shows "w >>> n = signed_drop_bit n w"
l3_inv6_derived: fixes implem :: "chan \<Rightarrow> msg" assumes "dhlvl3 implem" shows "dhlvl3.l3_inv3 implem \<inter> l3_inv4 \<subseteq> dhlvl3.l3_inv6 implem"
carrots_total_def: fixes c :: "nat" shows "carrots_total c \<equiv> sum ((^) 2) {..<lg c}"
pot_comp: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "pot_comp cmp = (\<lambda>x y. case comparator_of (snd (rep_nat_term x)) (snd (rep_nat_term y)) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
H_def: shows "H \<equiv> rec_nat (0, {0<..}, 0) (\<lambda>l (m, M, j). G l m M)"
corthogonal_vec_of_basis_enum: fixes S :: "'b list" shows "corthogonal (map vec_of_basis_enum S) = (is_ortho_set (set S) \<and> distinct S)"
hom: shows "\<forall>x\<in>[H]\<^bsup>\<gamma>\<^esup>. Suc (f x) = i"
set_iterator_genord_foldli_correct: fixes xs :: "'a list"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "distinct xs"   and "sorted_wrt R xs" shows "set_iterator_genord (foldli xs) (set xs) R"
Ball_starI: fixes as :: "'c list"   and A :: "'c list set" assumes "\<forall>a\<in>set as. [a] \<in> A" shows "as \<in> star A"
finite_has_minimal2: fixes A :: "'t set"   and a :: "'t" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. m \<preceq>\<^sub>t a \<and> (\<forall>b\<in>A. b \<preceq>\<^sub>t m \<longrightarrow> m = b)"
dims_matricize_1: fixes rmodes :: "nat set"   and T :: "'a tensor" shows "dim_row (matricize rmodes T) = prod_list (nths (dims T) rmodes)"
timpl_closure_set_timpls_trancl_eq': fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and c :: "('c set \<times> 'c set) set" shows "timpl_closure_set M {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b} = timpl_closure_set M c"
smooth_on_uminus_fun: fixes k :: "enat"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S g"   and "open S" shows "k-smooth_on S (- g)"
H_inc: fixes l :: "nat" shows "l \<le> \<nu> l"
assms_2: fixes r :: "int poly" assumes "q =m p * Mp r" shows "thesis"
conn: fixes T :: "'a set" assumes "closed T"   and "T \<subset> S" shows "connected_component (- T) a b"
finite_measure_def: fixes M :: "'c measure" shows "finite_measure M \<equiv> sigma_finite_measure M \<and> finite_measure_axioms M"
llast_worker_llast_body: shows "llast_worker = unwrap oo llast_body oo wrap"
arg_rcis_cis: fixes x :: "real"   and y :: "real" assumes "0 < x" shows "arg (rcis x y) = arg (cis y)"
glb_def: fixes S :: "'b set"   and po :: "'b potype" shows "glb S po \<equiv> greatest (\<lambda>x. \<forall>y\<in>S. (x, y) \<in> order po) po"
aclassical2: fixes v :: "i" shows "[\<^bold>\<exists>x y. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y \<^bold>& (\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) in v]"
deg_le_sectD_1: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> deg_le_sect X d" shows "t \<in> .[X]"
eval_ps': shows "P \<turnstile> \<langle>blocks (ps', Ts', vs', e),(h, l(p \<mapsto> v), sh)\<rangle> \<Rightarrow> \<langle>e',(h', l''', sh')\<rangle>"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "a\<^bsup>n\<^esup> \<cdot> a = a \<cdot> a\<^bsup>n\<^esup>"
verticesFrom_between: fixes v' :: "nat"   and f :: "face"   and u :: "nat"   and v :: "nat" assumes "v' \<in> \<V> f"   and "pre_between (vertices f) u v" shows "between (vertices f) u v = between (verticesFrom f v') u v"
scene_union_unit: fixes X :: "'a scene" shows "X \<squnion>\<^sub>S \<bottom>\<^sub>S = X"
rep_eq: fixes x :: "('a, 'b) mapping"   and xa :: "'b"   and xb :: "'b \<Rightarrow> 'b"   and xc :: "'a set" shows "mapping.rep (upd_set' x xa xb xc) = (\<lambda>a. if a \<in> xc then case Mapping.lookup x a of None \<Rightarrow> Some xa | Some b \<Rightarrow> Some (xb b) else Mapping.lookup x a)"
simps_2: fixes uv :: "'b infty" shows "(uv \<le> Infty) = True"
integrableI_nonneg: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>" shows "integrable M f"
ground_sub_of_denott: fixes t :: "fterm"   and E :: "char list \<Rightarrow> hterm" shows "ground\<^sub>t (t \<cdot>\<^sub>t sub_of_denot E)"
independent_bound_general: fixes S :: "'b set" assumes "independent S" shows "finite S \<and> card S \<le> dim S"
mul_ext_unfold: fixes x :: "'a list"   and y :: "'a list"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" shows "((x, y) \<in> {(a, b). fst (mul_ext g a b)}) = ((mset x, mset y) \<in> s_mul_ext {(a, b). snd (g a b)} {(a, b). fst (g a b)})"
ctr_1: fixes xs :: "'b llist" assumes "lnull xs" shows "lalternate xs = LNil"
favorites_altdef: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R" shows "favorites R i = Max_wrt_among (R i) alts"
Id_compose: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "f \<in> extensional A" shows "compose A (identity B) f = f"
comp_ide_arr: fixes b :: "'b"   and f :: "'b" assumes "D.ide b"   and "D.seq b f" shows "b \<cdot>\<^sub>D f = f"
PFadjoint_def: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'a \<Rightarrow> ('sf, 'sc, 'sz) formula" shows "PFadjoint I \<sigma> = \<lparr>Functions = Functions I, Predicates = Predicates I, Contexts = \<lambda>f. case f of Inl f' \<Rightarrow> Contexts I f' | Inr f' \<Rightarrow> \<lambda>_. fml_sem I (\<sigma> f'), Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
sum_fun_distr: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "finite X" shows "\<Sum> (f ` X) \<cdot> y = \<Sum> {f x \<cdot> y |x. x \<in> X}"
topology_inverse': fixes U :: "'a set \<Rightarrow> bool" assumes "istopology U" shows "openin (topology U) = U"
rep_eq: fixes x :: "('d, 'c) sq_matrix" shows "to_fun (Square_Matrix.cofactor x) = (\<lambda>i j. Square_Matrix.det (minor x i j))"
minus_leq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a - b \<le> (0::'a))"
comparator_option_simps_1: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order" shows "comparator_option comp\<^sub>'\<^sub>a None None = Eq"
A'_def: shows "A' = Mod_Type_Connect.from_hma\<^sub>m A"
that: fixes i :: "'a" assumes "i \<in> I"   and "inv_into (carrier R) f (f a) = i \<oplus> a" shows "thesis"
HMA_M_1: shows "Mod_Type_Connect.HMA_M (1\<^sub>m CARD('c)) (mat (1::'b))"
GFP_lemma2: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "GFP f \<sqsubseteq> f (GFP f)"
declassified\<^sub>l\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'b) Term.term" shows "declassified\<^sub>l\<^sub>s\<^sub>t \<A> \<I> \<equiv> {t. (\<star>, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set \<A>} \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
LeafI: fixes a :: "'c" shows "Leaf a \<in> sexp"
rep_eq: fixes x :: "\<I>" shows "left x = fst (Rep_\<I> x)"
aux4lose2: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "3 + k + d \<le> 2 * d + (4 + k)" shows "lose (t + 3 + k + d) = [False]"
aux: fixes f :: "'u \<Rightarrow> 'v1 \<Rightarrow> 'v2 option" shows "invar2 (map_value_image_filter f m) \<and> \<alpha>2 (map_value_image_filter f m) = (\<lambda>k. \<alpha>1 m k \<bind> f k)"
primeideal_iff_prime: fixes p :: "'a" assumes "p \<in> carrier R - {\<zero>}" shows "primeideal (PIdl p) R = ring_prime p"
simps_2: fixes n :: "num" shows "Num.sqr (num.Bit0 n) = num.Bit0 (num.Bit0 (Num.sqr n))"
strict_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ordering less_eq less"   and "less a b"   and "less b c" shows "less a c"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
problem_plan_bound_charles_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "problem_plan_bound_charles PROB \<equiv> Sup (MPLS_charles PROB)"
subI: fixes f :: "'b \<Rightarrow> ('b, 'c, 'd) sp\<^sub>\<mu>"   and a :: "'b" shows "(f a, Get f) \<in> sub"
span_empty: shows "Ker.span {} = {\<zero>\<^bsub>V.vs kerT\<^esub>}"
map_add_dom_app_simps_1: fixes m :: "'a"   and l2 :: "'a \<Rightarrow> 'b option"   and l1 :: "'a \<Rightarrow> 'b option" assumes "m \<in> dom l2" shows "Map.map_add l1 l2 m = l2 m"
assm: shows "x \<sqinter> 1\<^sub>\<pi> = (0::'a)"
Phi_conv_map: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "is_map ((\<Phi> w x y z)\<^sup>\<smile>)"
ideduct_reduce: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "M \<union> M' \<turnstile> t"   and "\<And>t'. t' \<in> M' \<Longrightarrow> M \<turnstile> t'" shows "M \<turnstile> t"
find_outside_node: fixes u :: "'b"   and v :: "'b"   and E :: "('b \<times> 'b) set"   and U :: "'b set" assumes "(u, v) \<in> E\<^sup>*"   and "(u, v) \<notin> (Restr E U)\<^sup>*"   and "u \<in> U"   and "v \<in> U" shows "\<exists>u'. u' \<notin> U \<and> (u, u') \<in> E\<^sup>* \<and> (u', v) \<in> E\<^sup>*"
comp_def: fixes g :: "'a"   and f :: "'a" shows "Left_a.comp g f = (if left a f \<and> left a g \<and> cod f = local.dom g then g \<cdot> f else null)"
Lim_null: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and net :: "'b filter" shows "(f \<longlongrightarrow> l) net = ((\<lambda>x. f x - l) \<longlongrightarrow> (0::'c)) net"
ex_polyn_expr: fixes p :: "'a" assumes "p \<in> carrier R" shows "\<exists>c. pol_coeff S c \<and> p = polyn_expr R X (fst c) c"
lsided_bres: fixes x :: "'a" shows "lsd x = (x \<le> \<top> \<rightarrow> x)"
content_cf_pos_poly: fixes f :: "int poly" shows "content (cf_pos_poly f) = (if f = 0 then 0 else 1)"
induct: fixes P :: "('c \<times> ('c, 'd) RHS) list \<Rightarrow> (nat \<times> nat \<Rightarrow> 'c list) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "('c \<times> ('c, 'd) RHS) list"   and a1 :: "nat \<times> nat \<Rightarrow> 'c list"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "\<And>G T i k j. (k < j \<Longrightarrow> P G T i (k + 1) j) \<Longrightarrow> P G T i k j" shows "P a0 a1 a2 a3 a4"
arrE: fixes f :: "'a" assumes "S.arr f" shows "Arr f"
add_mset_not_empty: fixes a :: "'b"   and A :: "'b multiset" shows "a, A \<noteq> {#}"
edg_simps_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and v :: "'a antiparallel_edges.vertex" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.edg \<Delta> (antiparallel_edges.Vertex x) v = (\<exists>y. v = antiparallel_edges.Edge x y \<and> edge \<Delta> x y)"
lincomb_empty: fixes a :: "'a vec \<Rightarrow> 'a" shows "lincomb a {} = 0\<^sub>v n"
inf_aci_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
emb_step_at_head: fixes uu :: "dir list"   and uv :: "dir"   and h :: "('c, 'd) hd" shows "emb_step_at uu uv (Hd h) = Hd h"
AE_support_countable: assumes "D.events = UNIV" shows "(AE x in distr (uniform_measure M {\<omega> \<in> space M. X n \<omega> = s}) (count_space UNIV) (X (Suc n)). D.prob {x} \<noteq> 0) = (\<exists>S. countable S \<and> (AE x in distr (uniform_measure M {\<omega> \<in> space M. X n \<omega> = s}) (count_space UNIV) (X (Suc n)). x \<in> S))"
disjointD: fixes A :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "disjoint A"   and "a \<in> A"   and "b \<in> A"   and "a \<noteq> b" shows "a \<inter> b = {}"
nxt_sdrop: fixes n :: "nat"   and \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" shows "(nxt ^^ n) \<phi> xs = \<phi> (sdrop n xs)"
hyps: fixes v :: "'a" assumes "valid_unMultigraph.is_trail G1 v xs v'" shows "valid_unMultigraph.is_trail G2 v xs v'"
filter_cong: fixes xs :: "'b List.list"   and ys :: "'b List.list"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> list.set ys \<Longrightarrow> P x = Q x" shows "List.filter P xs = List.filter Q ys"
rel_def: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c \<times> 'c"   and y :: "'c \<times> 'c" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit" shows "quotient_ring.rel S R addition multiplication zero x y \<equiv> \<exists>s1. s1 \<in> S \<and> multiplication s1 (addition (multiplication (snd y) (fst x)) (monoid.inverse R addition zero (multiplication (snd x) (fst y)))) = zero"
fps_exp_neg: fixes a :: "'a" shows "fps_exp (- a) = inverse (fps_exp a)"
simps_1: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "fold f [] = id"
equality_axioms_hold: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> equality_axioms" shows "eval_fm e A"
gSubstGVar1_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstGVar1 MOD \<equiv> \<forall>ys y Y' Y xs x. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> gSubst MOD ys Y' Y y (Var xs x) (gVar MOD xs x) = gVar MOD xs x"
h3: shows "\<forall>i<n. FlexRayController (nReturn i) rcv (nC i) (nStore i) (nSend i) (nGet i)"
Rep_Pg_inverse: fixes x :: "Pg" shows "Abs_Pg (Rep_Pg x) = x"
unit_invertible: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b" assumes "Group_Theory.monoid M composition unit" shows "monoid.invertible M composition unit unit"
iso_\<epsilon>': shows "D.iso (unit (src\<^sub>C g) \<cdot>\<^sub>D \<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>) \<cdot>\<^sub>D D.inv (\<Phi> (f, g)))"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>" obtains "S.arr f" and "S.dom f = a" and "S.cod f = b"
split_paired_All: fixes P :: "'b \<times> 'c \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>a b. P (a, b))"
cring_simprules_6: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
prime_imp_fermat_liar: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "a \<in> {1..<p}" shows "fermat_liar a p"
simps_1: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c" shows "Leg0 \<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = Leg0"
spectrum_root_char_poly: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "spectrum A = {k. poly (char_poly A) k = (0::'a)}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) Cones.map Cones.map Cones.map"
add_pos_left: 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 b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq zero a" shows "less_eq b (plus a b)"
z_def: shows "z \<equiv> getFr [xx, yy, zz] [t1, t1', t2, t2'] []"
li_le_dim_1: fixes A :: "'c set" assumes "V.fin_dim"   and "A \<subseteq> carrier V"   and "V.module.lin_indpt A" shows "finite A"
weakenBisimE_3: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and \<Psi>' :: "'e" assumes "weak substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> weak.weakenBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(SCompose' \<Psi> \<Psi>', P, Q) \<in> weak.weakenBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
s_def: fixes t :: "real" shows "delayed_safe_distance.other.s t = (if t \<le> 0 then other.s \<delta> else if t \<le> delayed_safe_distance.other.t_stop then delayed_safe_distance.other.p t else delayed_safe_distance.other.p_max)"
fds_nth_0: fixes f :: "'b fds" shows "fds_nth f 0 = (0::'b)"
alt_writer_def: fixes alt :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and m :: "('d, 'e, 'f) writerT"   and m' :: "('d, 'e, 'f) writerT" shows "alt_writer alt m m' = WriterT (alt (run_writer m) (run_writer m'))"
fv\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map fv\<^sub>s\<^sub>t\<^sub>p S))"
prod_sum_distrib_lists: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and P :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite S" shows "(\<Sum>ms | set ms \<subseteq> S \<and> length ms = n \<and> (\<forall>i<n. P i (ms ! i)). \<Prod>x<n. f (ms ! x)) = (\<Prod>i<n. \<Sum>m | m \<in> S \<and> P i m. f m)"
rcons_inv: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.rcons t a)"
unrest_as_exists: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" assumes "vwb_lens x" shows "(x \<sharp> P) = ((\<exists> x \<bullet> P) = P)"
agree_on_then_eq: fixes G :: "('e, 'f) labeled_graph"   and f\<^sub>1 :: "('f \<times> 'g) set"   and f\<^sub>2 :: "('f \<times> 'g) set" assumes "agree_on G f\<^sub>1 f\<^sub>2"   and "Domain f\<^sub>1 = vertices G"   and "Domain f\<^sub>2 = vertices G" shows "f\<^sub>1 = f\<^sub>2"
pick_asAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs qA" shows "pick (asAbs qA) $= qA"
a2_def: shows "a2 = Complex (Re a3) (Im a1)"
region_not_empty: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set" assumes "finite X"   and "valid_region X k I r" shows "\<exists>u. u \<in> region X I r"
segment_minimum_minimum: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set"   and d :: "'c"   and m :: "'c" assumes "Torder D"   and "X \<subseteq> carrier D"   and "d \<in> X"   and "minimum_elem (Iod D (segment D d)) (X \<inter> segment D d) m" shows "minimum_elem D X m"
K_imply_Cons: fixes ps :: "'b fm list"   and q :: "'b fm"   and p :: "'b fm" assumes "\<turnstile> imply ps q" shows "\<turnstile> imply (p # ps) q"
prime_elem_linear_field_poly: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "prime_elem [:a, b:]"
interrupts_conv: fixes ls :: "'h"   and ts :: "'i"   and m :: "'j"   and ws :: "'k"   and "is" :: "'g" shows "interrupts (ls, (ts, m), ws, is) = is"
rep_eq: fixes x :: "('b, 'a) aezfun"   and xa :: "('b, 'a) aezfun" shows "aezfun (x - xa) = aezfun x - aezfun xa"
countable_image_inj_gen: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and A :: "'c set" assumes "inj_on f S"   and "countable A" shows "countable {x \<in> S. f x \<in> A}"
has_integral_reflect: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and b :: "'b"   and a :: "'b" shows "((\<lambda>x. f (- x)) has_integral i) (cbox (- b) (- a)) = (f has_integral i) (cbox a b)"
Bij_compose_restrict_eq: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f \<in> Bij S" shows "compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"
set_intros_3: fixes x23 :: "'b"   and x21 :: "'c"   and x22 :: "'c" shows "x23 \<in> set2_arr (MkArr x21 x22 x23)"
i: shows "atom i \<sharp> (k1, k2, k, x, A)"
abs_eq: fixes xa :: "int"   and x :: "int" shows "(integer_of_int xa < integer_of_int x) = (xa < x)"
jordan_block_index_2: fixes n :: "nat"   and k :: "'a" shows "dim_row (jordan_block n k) = n"
zero_TC_le: fixes y :: "V" shows "0 \<sqsubseteq> y"
mk_all_list'_preserves_term_ok_typ_of: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and B :: "Core.term"   and vs :: "(variable \<times> typ) list" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> B"   and "typ_of B = Some propT"   and "\<forall>(idn, ty)\<in>set vs. typ_ok \<Theta> ty" shows "term_ok \<Theta> (mk_all_list vs B) \<and> typ_of (mk_all_list vs B) = Some propT"