hcju/isabelleps huggingface dataset:
not_le_real: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "(\<not> less_eq b a) = less a b"
writerT_eq_iff: fixes a :: "'e\<cdot>('f, 'g) writerT"   and b :: "'e\<cdot>('f, 'g) writerT" shows "(a = b) = (runWriterT\<cdot>a = runWriterT\<cdot>b)"
in_rtrancl_UnI: fixes x :: "'a \<times> 'a"   and R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>*" shows "x \<in> (R \<union> S)\<^sup>*"
wset_find_path'_def: fixes E :: "('b \<times> 'b) set"   and U0 :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "wset_find_path' E U0 P \<equiv> ASSERT (finite U0) \<bind> (\<lambda>_. s_init_foreach U0 \<bind> (\<lambda>s0. WHILE\<^sub>T (\<lambda>(res, V, ws). res = None \<and> ws \<noteq> Map.empty) (\<lambda>(res, V, ws). ASSERT (ws \<noteq> Map.empty) \<bind> (\<lambda>_. op_map_pick_remove ws \<bind> (\<lambda>((u, p), ws). if P u then RETURN (Some (rev p, u), V, ws) else ws_update_foreach E u p V ws \<bind> (\<lambda>(V, ws). RETURN (None, V, ws))))) s0 \<bind> (\<lambda>(res, uu_, uu_). RETURN res)))"
keys_entries: fixes k :: "'a"   and t :: "('a, 'b) RBT.rbt" shows "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
limit_inter_INF: fixes w :: "nat \<Rightarrow> 'a"   and S :: "'a set" assumes "limit w \<inter> S \<noteq> {}" shows "\<exists>\<^sub>\<infinity>n. w n \<in> S"
map_add_dom_app_simps_2: fixes m :: "'a"   and l1 :: "'a \<Rightarrow> 'b option"   and l2 :: "'a \<Rightarrow> 'b option" assumes "m \<notin> dom l1" shows "(l1 ++ l2) m = l2 m"
distinct_31: fixes x17 :: "hash" shows "Unit \<noteq> Hash x17"
acyclic_def: fixes r :: "('b \<times> 'b) set" shows "acyclic r = (\<forall>x. (x, x) \<notin> r\<^sup>+)"
f_int: fixes i :: "int" shows "f (real_of_int i) = real_of_int i + 1"
elims_1: fixes x :: "checker_type"   and xa :: "t list"   and y :: "bool" assumes "c_types_agree x xa = y" obtains ts :: "t list" and ts' :: "t list" where "x = Type ts__" and "xa = ts'__" and "y = (ts__ = ts'__)"   | ts :: "ct list" and ts' :: "t list" where "x = TopType ts" and "xa = ts'" and "y = ct_suffix ts (to_ct_list ts')"   | uu_ :: "t list" where "x = Bot" and "xa = uu_" and "\<not> y"
QAT_Arrow: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "x \<sharp> (\<Gamma>, s, t)"   and "(x, T\<^sub>1) # \<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1\<rightarrow>T\<^sub>2"
dependent_inj_imageD: fixes s :: "'b set" assumes "m2.dependent (f ` s)"   and "inj_on f (m1.span s)" shows "m1.dependent s"
card_of_Pow: fixes A :: "'a set" shows "|A| <o |Pow A|"
succ_tran_def: fixes G :: "'b list_graph"   and v :: "'b" shows "FiniteListGraph.succ_tran G v = trancl_list_impl (edgesL G) [v]"
lsu_lsl_prop: shows "\<nu> \<le> \<nu>\<^sup>\<natural> \<circ> \<nu>"
smult_0_left: fixes p :: "'b poly" shows "Polynomial.smult (0::'b) p = 0"
aux: fixes S :: "('a \<times> 'b \<times> 'c) set"   and va :: "'a" assumes "finite S" shows "finite {n. \<exists>w. (va, w, n) \<in> S}"
hyps: fixes fs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and vs :: "'t list" shows "set fs \<subseteq> set (snd (sym_preproc_addnew gs vs fs v))"
cases: fixes x :: "('a\<^sub>m \<times> 'b\<^sub>m) \<times> 'a\<^sub>m \<times> 'b\<^sub>m" obtains xa :: "'a\<^sub>m" and y :: "'b\<^sub>m" and x' :: "'a\<^sub>m" and y' :: "'b\<^sub>m" where "x = ((xa, y), x', y')"
edges_of_walk_empty_2: fixes v :: "'a" shows "H.edges_of_walk [v] = {}"
logically_equivalent_def: fixes P :: "'state"   and Q :: "'state" shows "P =\<cdot> Q \<equiv> \<forall>x. P \<Turnstile> x = Q \<Turnstile> x"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs H" shows "rev_H a \<noteq> a"
foldr_digit_correct: fixes f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and d :: "('e, 'f) Digit"   and \<sigma> :: "'d" shows "foldr_digit f d \<sigma> = List.foldr f (digitToList d) \<sigma>"
assms_4: shows "fst (rep_nat_term v) = fst (rep_nat_term b)"
ident_comp_subst_trm_if_disj: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a" assumes "subst_domain \<sigma> \<inter> range_vars \<theta> = {}"   and "v \<in> subst_domain \<theta>" shows "(\<theta> \<circ>\<^sub>s \<sigma>) v = \<theta> v"
PTrans: shows "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>y> \<prec> P'"
lcos_self: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H G" shows "x \<in> x <# H"
space_lim: shows "space local.lim = space (Pi\<^sub>M I (\<lambda>_. borel))"
const_deriv: shows "((\<lambda>v. axis x (1::'a)) has_derivative (\<lambda>v. 0)) (at \<nu>)"
callee': fixes x :: "'a"   and s :: "'d" assumes "x \<in> outs_\<I> \<I>" shows "results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"
ldropn_all: fixes xs :: "'b llist"   and m :: "nat" assumes "llength xs \<le> enat m" shows "ldropn m xs = <>"
set_rel_alt: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>set_rel = {(A, B). A \<subseteq> R\<inverse> `` B \<and> B \<subseteq> R `` A}"
InvariantQCharacterizationAfterAssertLiteralNotInQ: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "\<not> literal el getQ state" shows "let state' = assertLiteral literal decision state in InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
simps_8: fixes M :: "('fun, 'var) Term.term set"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F' :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) stateful_strand_step list" shows "\<lbrakk>M; D; \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> # S\<rbrakk>\<^sub>s = (\<lambda>\<I>. negchecks_model \<I> D X F F' \<and> \<lbrakk>M; D; S\<rbrakk>\<^sub>s \<I>)"
prebicategory_with_units_axioms: shows "prebicategory_with_units (\<cdot>) (\<star>) \<a> \<i>"
map_of_domAD: fixes \<Gamma> :: "('c \<times> 'b) list"   and x :: "'c"   and e :: "'b" assumes "map_of \<Gamma> x = Some e" shows "x \<in> domA \<Gamma>"
nofail_WHILEIT_wf_rel: assumes "nofail (gds_init gds \<bind> WHILE\<^sub>T\<^bsup>\<lambda>_. True\<^esup> gen_cond gen_step)" shows "wf ((rwof_rel (gds_init gds) gen_cond gen_step)\<inverse>)"
rank_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = row_rank A"
in_face_cycle_setD: fixes y :: "'b \<times> 'b"   and x :: "'b \<times> 'b" assumes "y \<in> face_cycle_set x"   and "x \<in> arcs (with_proj (list_digraph G_list))" shows "y \<in> arcs (with_proj (list_digraph G_list))"
f'_def: fixes n :: "nat"   and x :: "nat" shows "f' n x = (if x \<le> n then f x else if x = Suc n then \<Sum>k. f (k + (n + 1)) else 0)"
streams_sset: fixes s :: "'a stream"   and A :: "'a set" assumes "s \<in> streams A" shows "sset s \<subseteq> A"
arr_iff_in_hom: fixes f :: "'s" shows "S.arr f = \<guillemotleft>f : S.dom f \<rightarrow> S.cod f\<guillemotright>"
node_net_state: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "(\<sigma>, s) \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U" shows "\<exists>\<zeta> R. s = NodeS i \<zeta> R"
osafe_Var: fixes c :: "'b" shows "osafe (OVar c)"
finalStateCharacterizationLemma: fixes state :: "State"   and decisionVars :: "nat set" assumes "InvariantUniq (getM state)"   and "InvariantConsistent (getM state)"   and "InvariantImpliedLiterals (getF state) (getM state)"   and "\<not> applicableDecide state decisionVars"   and "\<not> applicableBackjump state" shows "\<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> formulaFalse (getF state) (elements (getM state)) \<and> decisions (getM state) = []"
one_alg_dvd_iff: fixes x :: "'b" shows "((1::'b) alg_dvd x) = algebraic_int x"
lens_override_def: fixes X :: "'b \<Longrightarrow> 'a"   and S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 = put\<^bsub>X\<^esub> S\<^sub>1 (get\<^bsub>X\<^esub> S\<^sub>2)"
set_eq_insert: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "x .= x'" shows "insert x A {.=} insert x' A"
onode_tau: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and i :: "nat"   and R :: "nat set" assumes "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> S"   and "\<forall>j. j \<noteq> i \<longrightarrow> \<sigma>' j = \<sigma> j" shows "((\<sigma>, NodeS i s R), \<tau>, \<sigma>', NodeS i s' R) \<in> onode_sos S"
iT_0_4: fixes r :: "nat" shows "[ r, mod 0 ] = [r\<dots>,0]"
add1: fixes A :: "'k"   and k :: "'k"   and a :: "'k"   and b :: "'k" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
singleton_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<in> {a}) = (b = a)"
preserves_inv: fixes f :: "'c" assumes "C.iso f" shows "C.map (C.inv f) = C.inv (C.map f)"
anga_sym: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongAACute a"   and "a A B C" shows "a C B A"
has_next_def: fixes g :: "('b, 'a) generator" shows "list.has_next g = fst (generator g)"
simps_2: fixes d :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and n :: "nat" shows "matrix_sum d f (Suc n) = f n + matrix_sum d f n"
funpow_code_def: shows "Nat.funpow = funpow"
abs_eq: shows "(\<le>) = (\<lambda>X Y. Rep_upsets X \<subseteq> Rep_upsets Y)"
stab_rcosets_not_empty: fixes x :: "'c"   and R :: "'a set" assumes "x \<in> E"   and "R \<in> rcosets stabilizer G \<phi> x" shows "R \<noteq> {}"
simps_1: fixes G :: "'b list_graph" shows "FiniteListGraph.delete_edges G [] = G"
a0n_trans_def: shows "a0n_trans \<equiv> \<Union>h d ds. (a0i_running h d \<union> a0n_commit h d \<union> a0i_corrupt ds)\<^sup>="
aHom_0_0: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G" shows "f \<zero>\<^bsub>F\<^esub> = \<zero>\<^bsub>G\<^esub>"
balance_rbt_sorted: fixes l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted l"   and "rbt_sorted r"   and "l |\<guillemotleft> k"   and "k \<guillemotleft>| r" shows "rbt_sorted (balance l k v r)"
of_int_rat: fixes k :: "int" shows "rat_of_int k = Fract k 1"
entt_frame_fwd: fixes P :: "assn"   and Q :: "assn"   and A :: "assn"   and F :: "assn"   and B :: "assn" assumes "P \<Longrightarrow>\<^sub>t Q"   and "A \<Longrightarrow>\<^sub>t P * F"   and "Q * F \<Longrightarrow>\<^sub>t B" shows "A \<Longrightarrow>\<^sub>t B"
suffix_tl_subset: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys" shows "set (tl xs) \<subseteq> set (tl ys)"
nAct_not_active: fixes i :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat i < llength t"   and "\<not> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "\<langle>c #\<^bsub>enat (Suc i)\<^esub>t\<rangle> = \<langle>c #\<^bsub>enat i\<^esub>t\<rangle>"
coefficients_head: fixes p :: "poly" shows "last (coefficients p) = head p"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Ide (t \<^bold>\<otimes> u) = (Ide t \<and> Ide u)"
prod_encode_gr1: fixes a :: "nat"   and x :: "nat" assumes "1 < a" shows "1 < prod_encode (a, x)"
iter_tensor_of_H_is_gate: fixes n :: "nat" assumes "1 \<le> n" shows "gate n (H \<otimes>\<^bsup>n\<^esup>)"
min_height_balanced: fixes kdt :: "'a kdt" assumes "balanced kdt" shows "min_height kdt = nat \<lfloor>log 2 (real (KD_Tree.size_kdt kdt))\<rfloor>"
singleton_conv: fixes a :: "'c" shows "{x. x = a} = {a}"
hypnat_omega_def: shows "whn = star_n (\<lambda>n. n)"
convex_sums: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "convex T" shows "convex (\<Union>x\<in>S. \<Union>y\<in>T. {x + y})"
IO: fixes x :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "IO x c \<in> set_spmf (the_gpv gpv)" shows "outsp_gpv \<I> x gpv"
norm_add_less: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and r :: "real"   and y :: "'a"   and s :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x < r"   and "norm y < s" shows "norm (plus x y) < r + s"
dirichlet_inverse_code: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b"   and n :: "nat" shows "dirichlet_inverse f i n = (if n = 0 then 0::'b else if n = 1 then i else - i * fold_atLeastAtMost_nat (\<lambda>d acc. if d dvd n then f (n div d) * dirichlet_inverse f i d + acc else acc) 1 (n - 1) (0::'b))"
comp_dist_mix: fixes X :: "'b set"   and Y :: "'b set" shows "\<Squnion> X \<cdot> \<Squnion> Y = \<Squnion> (X \<cdot> Y)"
3: shows "gusuccessors A w v \<subseteq> UNIV \<times> nodes A"
simps_2: fixes y :: "name"   and e :: "trm"   and \<theta> :: "(name \<times> trm) list"   and x :: "name" shows "lookup ((y, e) # \<theta>) x = (if x = y then e else lookup \<theta> x)"
simps_12: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (E p) = (\<exists>x. Ifm vs (x # bs) p)"
qfree_min_inf: fixes \<phi> :: "atom fm" assumes "qfree \<phi>" shows "qfree (inf\<^sub>- \<phi>)"
sum: shows "inner_prod v (A *\<^sub>v v) = (\<Sum>i = 0..<n. B $$ (i, i) * conjugate (w $ i) * w $ i)"
in_imp_ex_iprev_nth: fixes I :: "nat set"   and x :: "nat" assumes "finite I"   and "x \<in> I" shows "\<exists>n. x = I \<leftarrow> n"
simps_1: shows "consequent allI = [ForallX (P x)]"
cases: fixes a1 :: "cmd"   and a2 :: "nat"   and a3 :: "cmd" assumes "labels a1 a2 a3" obtains c :: "cmd" where "a1 = cb__" and "a2 = 0" and "a3 = cb__"   | V :: "char list" and e :: "expr" where "a1 = V:=e" and "a2 = 1" and "a3 = Skip"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = c\<^sub>1__;; c\<^sub>2__" and "a2 = lb__" and "a3 = cc__;; c\<^sub>2__" and "labels c\<^sub>1__ lb__ cc__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>1 :: "cmd" where "a1 = c\<^sub>1'__;; c\<^sub>2'__" and "a2 = lc__ + #:c\<^sub>1'__" and "a3 = cd__" and "labels c\<^sub>2'__ lc__ cd__"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>2 :: "cmd" where "a1 = if (ba__) c\<^sub>1''__ else c\<^sub>2''__" and "a2 = ld__ + 1" and "a3 = ce__" and "labels c\<^sub>1''__ ld__ ce__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>1 :: "cmd" where "a1 = if (bb__) c\<^sub>1 else c\<^sub>2" and "a2 = le__ + #:c\<^sub>1 + 1" and "a3 = cf__" and "labels c\<^sub>2 le__ cf__"   | c' :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" where "a1 = while (bc__) c'b__" and "a2 = l + 2" and "a3 = c;; while (bc__) c'b__" and "labels c'b__ l c"   | b :: "expr" and c' :: "cmd" where "a1 = while (b) c'" and "a2 = 1" and "a3 = Skip"
perm_rem_conv: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "bij f" shows "perm_rem x f y = (if x = y then x else if f y = x then f (f y) else f y)"
elims_1: fixes x :: "'b set"   and xa :: "'b \<Rightarrow> 'c"   and xb :: "nat"   and y :: "bool" assumes "higher_differentiable_on x xa xb = y" obtains S :: "'b set" and f :: "'b \<Rightarrow> 'c" where "x = S__" and "xa = f__" and "xb = 0" and "y = continuous_on S__ f__"   | S :: "'b set" and f :: "'b \<Rightarrow> 'c" and n :: "nat" where "x = S" and "xa = f" and "xb = Suc n" and "y = ((\<forall>x\<in>S. f differentiable at x) \<and> (\<forall>v. higher_differentiable_on S (\<lambda>x. frechet_derivative f (at x) v) n))"
length_1_last_hd: fixes ns :: "'a list" assumes "length ns = 1" shows "last ns = hd ns"
ln_inverse: fixes x :: "real" assumes "0 < x" shows "real_ln (inverse x) = - real_ln x"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "wave \<Gamma> f" obtains "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)" and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<Longrightarrow> d_OUT f x = 0"
interleave_rev_1: fixes w :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "w \<in> w1 \<otimes> w2" shows "rev w \<in> rev w1 \<otimes> rev w2"
parallel_converter2_comp1_out: fixes conv :: "('h, 'j, 'p, 'q) converter"   and conv' :: "('p, 'q, 'l, 'n) converter"   and conv'' :: "('i, 'k, 'm, 'o) converter" shows "conv \<odot> conv' |\<^sub>= conv'' = (conv |\<^sub>= 1\<^sub>C) \<odot> (conv' |\<^sub>= conv'')"
MkArr_in_hom: fixes A :: "'c \<Rightarrow> 's"   and B :: "'c \<Rightarrow> 's"   and f :: "'c \<Rightarrow> 's" assumes "A \<in> Collect (functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S))"   and "B \<in> Collect (functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S))"   and "f \<in> Collect (natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) A B)" shows "\<guillemotleft>Cop_S.MkArr A B f : Cop_S.MkIde A \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] Cop_S.MkIde B\<guillemotright>"
that: fixes g' :: "'a \<Rightarrow> 'b" assumes "homotopic_with_canon (\<lambda>x. True) T U f g'"   and "continuous_on T g'"   and "g' ` T \<subseteq> U"   and "\<And>x. x \<in> S \<Longrightarrow> g' x = g x" shows "thesis"
mat_diag_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "mat_diag n f = mat n n (\<lambda>(i, j). if i = j then f j else (0::'b))"
index_valid_indices_state: fixes as :: "('a \<times> 'b atom) set"   and s :: "('a, 'b) state" assumes "index_valid as s" shows "indices_state s \<subseteq> fst ` as"
prod_eq_iff: fixes s :: "'e \<times> 'f"   and t :: "'e \<times> 'f" shows "(s = t) = (fst s = fst t \<and> snd s = snd t)"
star_n_diff: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "star_n X - star_n Y = star_n (\<lambda>n. X n - Y n)"
vector_smult_lzero: fixes x :: "('c, 'd) vec" shows "(0::'c) *s x = 0"
qq_def: shows "qq = (\<lambda>x. poly q x / x ^ degree q)"
LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m_is_LPHIL0: shows "LPHIL0 = LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m 0"
distr_return: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and x :: "'c" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "x \<in> space M" shows "distr (return M x) N f = return N (f x)"
simps_1: fixes uu :: "linear_poly"   and uv :: "rat" shows "is_le (LT uu uv) = True"
prems_4: shows "list_all (\<lambda>Q. index_free Q i) Qs"
pair_spmf_alt_def: fixes p :: "'g spmf"   and q :: "'h spmf" shows "pair_spmf p q = p \<bind> (\<lambda>x. q \<bind> (\<lambda>y. return_spmf (x, y)))"
CD_on_Chernoff: fixes ds :: "'d set" shows "Chernoff (CD_on ds)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>n. \<lbrakk>\<And>x. x \<in> {..n} \<Longrightarrow> P x; \<And>x. x \<in> {..n} \<Longrightarrow> P (n - x)\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P a0"
Rxa_def: fixes R :: "('d, 'e) Ring_scheme"   and a :: "'d" shows "R \<diamondsuit>\<^sub>p a = {x. \<exists>r\<in>carrier R. x = r \<cdot>\<^sub>r\<^bsub>R\<^esub> a}"
simps: fixes x :: "'a ext0" shows "x\<^sup>T = x"
holds_only_current: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and k :: "nat" assumes "A.reachable (c, n, ts)"   and "t < N"   and "locks_ticket (ts t) k" shows "k = c"
matrix_row_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "int poly fract list list" assumes "mat nr nc M" shows "mat (rat_poly.row_length M) (length M) M"
length_concat_map_split_zip: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'c list" shows "length (map2 f xs ys) = min (length xs) (length ys)"
mrbst_delete_altdef: fixes x :: "'a"   and t :: "'a tree" shows "mrbst_delete x t = mrbst_diff t \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>"
gcd_dvd1: fixes a :: "'a"   and b :: "'a" shows "gcd a b dvd a"
exec_while\<^sub>C: fixes \<sigma> :: "'a control_state_scheme"   and b :: "'a control_state_scheme \<Rightarrow> bool"   and c :: "'a control_state_scheme \<Rightarrow> (unit \<times> 'a control_state_scheme) option"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" shows "(\<sigma> \<Turnstile> _while_C b c ;- M) = (\<sigma> \<Turnstile> _if_SECLEAN b (c ;- _while_C b c ;- unset_break_status) skip\<^sub>S\<^sub>E ;- M)"
CAS_\<tau>red1t_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 D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e\<bullet>compareAndSwap(D\<bullet>F, e2, e3), xs) (e'\<bullet>compareAndSwap(D\<bullet>F, e2, e3), xs')"
grid_union_dims: fixes ds :: "nat set"   and ds' :: "nat set"   and p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list" assumes "ds \<subseteq> ds'"   and "p \<in> grid b ds" shows "p \<in> grid b ds'"
comp_char: fixes g :: "'d \<times> 'd"   and f :: "'d \<times> 'd" shows "D.VV.comp g f = (if D.VV.arr f \<and> D.VV.arr g \<and> D.VxV.seq g f then D.VxV.comp g f else D.VxV.null)"
lprod_3_2: fixes z' :: "'a"   and z :: "'a"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "(z', z) \<in> R" shows "([z', x, y], [x, y, z]) \<in> lprod R"
simps_26: fixes x4 :: "('a, 'b, 'c) expr"   and x21 :: "('a, 'b, 'c) expr"   and x22 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Apply x21 x22"