hcju/isabelleps huggingface dataset:
up_a_inv_closed: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> up R" shows "(\<lambda>i. \<ominus> p i) \<in> up R"
sp_in_extends_not_using_sub: fixes v :: "'a \<times> nat"   and v' :: "'a \<times> nat"   and subs' :: "(('a \<times> nat) \<times> 'a \<times> nat) set"   and v1 :: "'a \<times> nat"   and es :: "('a \<times> nat) edge list"   and v2 :: "'a \<times> nat" assumes "extends g subs (v, v') subs'"   and "SubRel.subpath g v1 es v2 subs'"   and "\<not> uses_sub v1 es v2 (v, v')" shows "SubRel.subpath g v1 es v2 subs"
symmetric_point_uniqueness: fixes P :: "'p"   and A :: "'p"   and P1 :: "'p"   and P2 :: "'p" assumes "P Midpoint A P1"   and "P Midpoint A P2" shows "P1 = P2"
subspace_mul: fixes S :: "'c set"   and x :: "'c"   and c :: "real" assumes "subspace S"   and "x \<in> S" shows "c *\<^sub>R x \<in> S"
eq_\<I>_converter_trans: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and conv' :: "('h, 'i, 'j, 'k) converter"   and conv'' :: "('h, 'i, 'j, 'k) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv''" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv''"
pi_approx2': fixes n :: "nat"   and k :: "nat" assumes "even n"   and "0 < n"   and "k \<le> 46 * n - 1" shows "\<bar>pi - pi_approx2 n\<bar> \<le> inverse (2 ^ k)"
perm_7: fixes pi :: "(name \<times> name) list"   and x2 :: "pi"   and x1 :: "pi" shows "pi \<bullet> x2 \<oplus> x1 = (pi \<bullet> x2) \<oplus> (pi \<bullet> x1)"
sigma_fprod_algebra_sigma_eq: fixes I :: "'d set"   and S :: "'d \<Rightarrow> nat \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure"   and E :: "'d \<Rightarrow> 'e set set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> \<Union> (range (S i)) = space (M i)"   and "\<And>i. i \<in> I \<Longrightarrow> range (S i) \<subseteq> E i"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (space (M i))"   and "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sigma_sets (space (M i)) (E i)" shows "sets (Pi\<^sub>F {I} M) = sigma_sets (space (Pi\<^sub>F {I} M)) {Pi' I F |F. \<forall>i\<in>I. F i \<in> E i}"
closedin_self: fixes S :: "'c set" shows "closedin (top_of_set S) S"
fps_zero_nth: fixes n :: "nat" shows "0 $ n = (0::'a)"
pubK_neq_shrK: fixes A :: "agent"   and b :: "keymode"   and C :: "agent" shows "shrK A \<noteq> publicKey b C"
even_Suc: fixes n :: "nat" shows "even (Suc n) = odd n"
endpoints_shiftpath_2: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "pathstart (shiftpath a g) = g a"
emeasure_pair_measure_Times: fixes A :: "'c set"   and N :: "'c measure"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> events" shows "emeasure (N \<Otimes>\<^sub>M M) (A \<times> B) = emeasure N A * emeasure M B"
Gcd_minors_dvd_diag_PAQ: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "Smith_normal_form_mat (P * A * Q)"   and "k \<le> min m n" shows "Gcd (minors A k) dvd (\<Prod>i = 0..<k. (P * A * Q) $$ (i, i))"
equiv_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
app\<^sub>i_Putfield: fixes F :: "char list"   and C :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Putfield F C, P, pc, mxs, T\<^sub>r, T\<^sub>1 # T\<^sub>2 # ST, LT) = (\<exists>T\<^sub>f. P \<turnstile> C sees F,NonStatic:T\<^sub>f in C \<and> subtype P T\<^sub>2 (Class C) \<and> subtype P T\<^sub>1 T\<^sub>f)"
mustT_Par_L: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state" assumes "mustT (c1 | c2) s" shows "mustT c1 s"
II: fixes \<sigma> :: "'e" assumes "I S \<sigma>" shows "P \<sigma>"
odd: shows "odd (card {s. ksimplex p n s \<and> (reduced n \<circ> lab) ` s = {..n}})"
char_to_digit_simps_8: shows "char_to_digit CHR ''7'' = 7"
of_bool_eq_1: fixes one :: "'a"   and zero :: "'a" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero False = zero"
continuous_within_E: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "continuous (at x within s) f"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
position_positive: fixes a :: "'b"   and l :: "'b list" assumes "a \<in> set l" shows "0 < position a l"
rotate_def: fixes n :: "nat" shows "rotate n = rotate1 ^^ n"
P: shows "P \<equiv> \<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W, U]. G) then W\<oplus>U\<leftarrow> a ! 1 else (0::'v)"
restr_stack_restr_stack: fixes V :: "Vars.var set"   and V' :: "Vars.var set"   and S :: "stack_elem list" shows "restr_stack V (restr_stack V' S) = restr_stack (V \<inter> V') S"
simps_2: fixes i :: "nat"   and j :: "nat" shows "depends\<^sub>d\<^sub>l\<^sub>o (Less i j) = (i = 0 \<or> j = 0)"
acyclic_6_def: fixes x :: "'a" shows "acyclic_6 x \<equiv> \<exists>y. y \<squnion> y\<^sup>T = x \<and> pd_kleene_allegory_class.acyclic y \<and> injective y"
nth_image: fixes l :: "nat"   and xs :: "'b list" assumes "l \<le> length xs" shows "(!) xs ` {0..<l} = set (take l xs)"
bind_UNION: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b set" shows "Set.bind A f = \<Union> (f ` A)"
cases: fixes a :: "(('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent" assumes "a \<in> edges" obtains c :: "'form" and "is" :: "nat list" where "a = edge_at c__ is__" and "c__ \<in> set conclusions" and "is__ \<in> it_paths (it' c__)"   | c :: "'form" and "is" :: "nat list" and n :: "nat" and s :: "'subst" and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" where "a = hyp_edge_at c is n s" and "c \<in> set conclusions" and "is \<in> it_paths (it' c)" and "tree_at (it' c) is = HNode n s ants"
simps_3: fixes A :: "'b ptrm"   and B :: "'b ptrm" shows "ptrm_fvs (PApp A B) = ptrm_fvs A \<union> ptrm_fvs B"
Quotient_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)) \<and> T = (\<lambda>x y. R x x \<and> Abs x = y))"
edka5_refine: fixes am :: "nat \<Rightarrow> nat list" assumes "is_adj_map am" shows "edka5 am \<le> \<Down> Id edka4"
setcompr_eq_image: fixes f :: "'e \<Rightarrow> 'd"   and P :: "'e \<Rightarrow> bool" shows "{uu_. \<exists>x. uu_ = f x \<and> P x} = f ` {x. P x}"
holomorphic_point_small_triangle: fixes x :: "complex"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex"   and e :: "real" assumes "x \<in> S"   and "continuous_on S f"   and "f field_differentiable at x within S"   and "0 < e" shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and> x \<in> convex hull {a, b, c} \<and> convex hull {a, b, c} \<subseteq> S \<longrightarrow> cmod (contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f) \<le> e * (dist a b + dist b c + dist c a)\<^sup>2"
tfr\<^sub>s\<^sub>e\<^sub>t_def: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" shows "tfr\<^sub>s\<^sub>e\<^sub>t M \<equiv> \<forall>s\<in>SMP M - range Var. \<forall>t\<in>SMP M - range Var. (\<exists>\<delta>. Unifier \<delta> s t) \<longrightarrow> \<Gamma> s = \<Gamma> t"
neg_test: fixes x :: "'a" assumes "test x" shows "test (! x)"
slice_take: fixes n :: "nat"   and w :: "'c word" shows "slice n w = of_bl (take (LENGTH('c) - n) (to_bl w))"
eNorm_def: fixes N :: "'a quasinorm"   and x :: "'a" shows "eNorm N x = snd (Rep_quasinorm N) x"
set_inorder: fixes t :: "'b tree" shows "set (Tree.inorder t) = tree.set_tree t"
elims_3: fixes x :: "'a Point"   and xa :: "'a Line" assumes "\<not> inLine x xa" obtains p :: "'a Point" and l :: "'a Line" where "x = p" and "xa = l" and "\<not> collinear p (basepoint l) (basepoint l \<leadsto> direction l)"
order_less_imp_not_eq2: fixes x :: "'e"   and y :: "'e" assumes "x < y" shows "(y = x) = False"
edka5_run_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and am :: "nat \<Rightarrow> nat list" shows "edka5_run cf am \<equiv> while\<^sub>T (\<lambda>(cf, brk). \<not> brk) (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. bfs2_op am cf \<bind> (\<lambda>p. case p of None \<Rightarrow> return (cf, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (Graph.isShortestPath cf s p t) \<bind> (\<lambda>_. resCap_cf_impl cf p \<bind> (\<lambda>bn. augment_cf_impl cf p bn \<bind> (\<lambda>cf. assert (RGraph c s t cf) \<bind> (\<lambda>_. return (cf, False))))))))) (cf, False) \<bind> (\<lambda>(cf, uu_). compute_rflow cf \<bind> return)"
zip_map_fst_snd: fixes zs :: "('c \<times> 'd) list" shows "map fst zs || map snd zs = zs"
map_vec_def: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c Matrix.vec" shows "map_vec f v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. f (v $v i))"
y_def: shows "y = \<Sqinter>{residual f x |x. x \<in> X}"
assms_2: shows "\<forall>a. A.ide a \<longrightarrow> Limit.limit_cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a D) (A_B.Map x a) (at a \<chi>)"
Disj_EH_1: fixes A :: "fm"   and H :: "fm set"   and C :: "fm"   and B :: "fm" assumes "insert A H \<turnstile> C"   and "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
take_bit_bin_cat_eq: fixes n :: "nat"   and w :: "int"   and v :: "int" shows "take_bit n (concat_bit n w v) = take_bit n w"
bracket_app: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and i :: "nat"   and s :: "dB"   and s' :: "dB" assumes "bracket_abstraction term_bracket"   and "term_bracket i s = Some s'" shows "s' \<degree> Var i \<leftrightarrow> s"
scale_minus_right: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d" assumes "module scale" shows "scale a (- x) = - scale a x"
minus_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<ominus> g = (\<lambda>e. f e \<sqinter> - g e)"
set_l'_is: shows "set (List.product [0..<t] ops) = {0..<t} \<times> set ops"
\<psi>_nonneg: fixes x :: "real" shows "0 \<le> \<psi> x"
resE_nonNegative: fixes e :: "nat \<times> nat" shows "(0::'a) \<le> f.cf e"
find_first_append: fixes x :: "'b"   and ys :: "'b list"   and zs :: "'b list" shows "find_first x (ys @ zs) = (case find_first x ys of None \<Rightarrow> map_option (\<lambda>i. i + length ys) (find_first x zs) | Some a \<Rightarrow> Some a)"
components_are_iso: fixes a :: "'a \<times> 'a" assumes "C.VV.ide a" shows "D.iso (\<Phi> a)"
map_props_def: fixes f :: "'b \<Rightarrow> 'a option"   and APs :: "'b set" shows "map_props f APs \<equiv> {i. \<exists>p\<in>APs. f p = Some i}"
modular_2_aux: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "g (x \<cdot> - f y) \<cdot> y = 0"
x: shows "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f"
upd_incoming__ident: fixes ns :: "('b, 'c) node_scheme set"   and P :: "('b, 'c) node_scheme \<Rightarrow> bool"   and n :: "('b, 'd) node_scheme" assumes "\<forall>n\<in>ns. P n"   and "\<And>n v. \<lbrakk>n \<in> ns; P n\<rbrakk> \<Longrightarrow> P (n\<lparr>incoming := v\<rparr>)" shows "\<forall>n\<in>upd_incoming n ns. P n"
weak_reduction_contrasimulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_contrasimulation Rel Cal" shows "weak_reduction_contrasimulation (Rel\<^sup>*) Cal"
poincare_slp_impl_def: fixes ode_ops :: "ode_ops \<times> ('a \<times> 'c \<times> 'd \<times> 'e \<times> 'f) \<times> floatarith list list option \<times> 'g" shows "poincare_slp_impl ode_ops = (case ode_ops of (ode_ops, (xa, xb, xc, xd, xe), x, xf) \<Rightarrow> Affine_Code.the_default (let _ = print_msg_impl ''ODE solver not initialized: pslp missing'' in solve_poincare_slp ode_ops) x)"
Proc_CFG_Call_Labels: fixes prog :: "cmd"   and n :: "label"   and p :: "char list"   and es :: "expr list"   and rets :: "char list list"   and n' :: "label" assumes "prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'" shows "\<exists>l. n = Label l \<and> n' = Label (Suc l)"
coprime_power_left_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (power.power one times a n) b = (algebraic_semidom.coprime times one a b \<or> n = 0)"
col123__nos: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col P Q A" shows "\<not> P Q OS A B"
HOMCollection_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" shows "HOMCollection C X Y \<equiv> {m2z\<^bsub>C\<^esub>f |f. f maps\<^bsub>C\<^esub> X to Y}"
fdisjnt_alt_def: fixes M :: "'c fset"   and N :: "'c fset" shows "fdisjnt M N = (M |\<inter>| N = {||})"
assms_3: fixes a :: "'a"   and b :: "'a" assumes "a \<in> K"   and "a \<noteq> \<zero>"   and "b \<in> K"   and "p = [a, b]" shows "P"
smallo: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and \<epsilon> :: "real" assumes "exp_asymp_equiv_linear f g"   and "0 < \<epsilon>" shows "g \<in> o(\<lambda>x. exp ((1 + \<epsilon>) * x))"
dfs_algo_def: fixes \<Sigma>i :: "'b set"   and R :: "('b \<times> 'b) set" shows "dfs_algo \<Sigma>i R \<equiv> \<lparr>wa_cond = dfs_cond, wa_step = dfs_step R, wa_initial = dfs_initial \<Sigma>i, wa_invar = dfs_invar \<Sigma>i R\<rparr>"
failures_def: fixes P :: "'c process" shows "failures P \<equiv> fst (Rep_process P)"
case_option_conv_if: fixes f :: "'f"   and g :: "'g \<Rightarrow> 'f"   and v :: "'g option" shows "(case v of None \<Rightarrow> f | \<lfloor>x\<rfloor> \<Rightarrow> g x) = (if \<exists>a. v = \<lfloor>a\<rfloor> then g (the v) else f)"
induct: fixes P :: "formula \<Rightarrow> bool"   and formula :: "formula" assumes "\<And>x1 x2 x3. P (FAtom x1 x2 x3)"   and "\<And>x1 x2 x3. \<lbrakk>P x2; P x3\<rbrakk> \<Longrightarrow> P (FConj x1 x2 x3)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (FAll x1 x2)" shows "P formula"
subspace_imp_cone: fixes S :: "'a set" assumes "subspace S" shows "cone S"
advantage_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a set spmf" shows "local.lcdh.advantage \<A> = spmf (local.lcdh.lcdh \<A>) True"
ordLess_not_embed: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r <o r'" shows "\<nexists>f'. embed r' r f'"
set_Keys_to_list: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ps :: "('c \<Rightarrow>\<^sub>0 'f) list" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "set (ordered_term.Keys_to_list ord_term ord_term_strict ps) = Keys (set ps)"
wf_converse_matchrel: shows "wf (matchrel\<inverse>)"
cont_Atm: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and atm :: "'d"   and s :: "'b"   and i :: "nat" assumes "PL cval" shows "PL.cont aval tval cval (Atm atm) s i = Done"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "infinite A" shows "sum_0 g A = (0::'a)"
le_add_diff_inverse2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
dg_Set_Cod_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Set \<alpha>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> dg_Set \<alpha>\<lparr>Obj\<rparr>"
triangle: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "(a \<otimes> \<l>[b]) \<cdot> \<a>[a, \<I>, b] = \<r>[a] \<otimes> b"
exhaust: fixes y :: "fmla" obtains x1 :: "nat" where "y = Atom x1"   | x2 :: "fmla" where "y = Neg x2"   | x31 :: "fmla" and x32 :: "fmla" where "y = Conj x31 x32"
not_below_FF_iff: fixes x :: "tr" shows "(x \<notsqsubseteq> FF) = (x = TT)"
const: fixes name :: "name" assumes "True" shows "\<not> shadows_consts (const name)"
vmc_path_axioms: shows "vmc_path G (ldropn n P) v p \<sigma>"
bounds_of_interval: fixes x :: "'b interval" shows "bounds_of_interval x \<in> {(a, b). a \<le> b}"
algebraic_minus: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "algebraic y" shows "algebraic (x - y)"
lmap_eq_LCons_conv: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h llist"   and y :: "'g"   and ys :: "'g llist" shows "(lmap f xs = LCons y ys) = (\<exists>x xs'. xs = LCons x xs' \<and> y = f x \<and> ys = lmap f xs')"
\<gamma>'_def: shows "\<gamma>' \<equiv> \<lambda>t. 2 * z - \<gamma> t"
bounded_hashcode_bounds: fixes def_hashmap_size :: "'a itself \<Rightarrow> nat"   and n :: "uint32"   and hashcode :: "'a \<Rightarrow> uint32"   and a :: "'a" assumes "class.hashable def_hashmap_size"   and "1 < n" shows "hashable.bounded_hashcode hashcode n a < n"
farkas_minkowsky_weyl_theorem: fixes P :: "'a vec set" shows "(\<exists>X\<subseteq>carrier_vec (n + 1). finite X \<and> P = next_dim.cone X) = (\<exists>A nr. A \<in> carrier_mat nr (n + 1) \<and> P = next_dim.polyhedral_cone A)"
Compositionality_Spawn: fixes V :: "('exp, 'id) MWLsCom list"   and \<iota> :: "nat" assumes "WHATWHERE_Secure V"   and "unique_PPc (spawn\<^bsub>\<iota>\<^esub> V)" shows "WHATWHERE_Secure [spawn\<^bsub>\<iota>\<^esub> V]"
hX_def: fixes X :: "'a set"   and h :: "'c" shows "hX X h = {x \<in> X. Xh x = h}"
inj_map_strong: fixes x :: "'c list"   and xa :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and fa :: "'c \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> set x; za \<in> set xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map f x = map fa xa" shows "x = xa"
that: fixes k' :: "nat" assumes "fst (prod_dfs A B (s1, s2)) ! dfa_trans A i bs ! dfa_trans B j bs = Some k'"   and "dfa_is_node A (dfa_trans A i bs)"   and "dfa_is_node B (dfa_trans B j bs)"   and "k' < length (snd (prod_dfs A B (s1, s2)))" shows "thesis"
that: fixes x :: "'a" shows "0 \<le> x \<bullet>\<^sub>C (A *\<^sub>V x)"
1: shows "(\<lambda>x. x - y) \<circ> (\<lambda>x. x + y) = id"
finite_enumerate_Suc'': fixes S :: "'b set"   and n :: "nat" assumes "finite S"   and "Suc n < card S" shows "wellorder_class.enumerate S (Suc n) = (LEAST s. s \<in> S \<and> wellorder_class.enumerate S n < s)"
rep_eq: fixes x :: "float" shows "real_of_float (sgn x) = sgn (real_of_float x)"
convol_def: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" shows "BNF_Def.convol f g \<equiv> \<lambda>a. (f a, g a)"
proj_iff: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "equiv A r"   and "{x, y} \<subseteq> A" shows "(proj r x = proj r y) = ((x, y) \<in> r)"
cases: fixes a1 :: "nat \<Rightarrow> nat"   and a2 :: "((nat \<times> nat) \<times> transition) fset"   and a3 :: "nat"   and a4 :: "nat \<Rightarrow>f value option"   and a5 :: "((nat \<times> nat) \<times> transition) fset"   and a6 :: "nat"   and a7 :: "nat \<Rightarrow>f value option"   and a8 :: "(String.literal \<times> value list) list" assumes "execution_simulation a1 a2 a3 a4 a5 a6 a7 a8" obtains s2 :: "nat" and f :: "nat \<Rightarrow> nat" and s1 :: "nat" and e1 :: "((nat \<times> nat) \<times> transition) fset" and r1 :: "nat \<Rightarrow>f value option" and e2 :: "((nat \<times> nat) \<times> transition) fset" and r2 :: "nat \<Rightarrow>f value option" where "a1 = f__" and "a2 = e1__" and "a3 = s1__" and "a4 = r1__" and "a5 = e2__" and "a6 = s2__" and "a7 = r2__" and "a8 = []" and "s2__ = f__ s1__"   | s2 :: "nat" and f :: "nat \<Rightarrow> nat" and s1 :: "nat" and e1 :: "((nat \<times> nat) \<times> transition) fset" and r1 :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and e2 :: "((nat \<times> nat) \<times> transition) fset" and r2 :: "nat \<Rightarrow>f value option" and es :: "(String.literal \<times> value list) list" where "a1 = f" and "a2 = e1" and "a3 = s1" and "a4 = r1" and "a5 = e2" and "a6 = s2" and "a7 = r2" and "a8 = (l, i) # es" and "s2 = f s1" and "\<forall>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> execution_simulation f e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es"
Ch_domain: fixes h :: "'c"   and A :: "'a set" shows "Ch h (A \<inter> {x. Xh x = h}) = Ch h A"
strip_guards_simps_6: fixes F :: "'f set"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "strip_guards F (While b c) = While b (strip_guards F c)"
nonneg: fixes f :: "real \<Rightarrow> real"   and x :: "real" assumes "antimono_fun_sum_integral_diff f"   and "0 \<le> x" shows "0 \<le> f x"
diff_divide_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "minus x (divide y z) = divide (minus (times x z) y) z"
null_eqI: fixes n :: "'a \<times> 'a" assumes "\<And>f. VoV.comp n f = n \<and> VoV.comp f n = n" shows "n = VoV.null"
comp_null_1: fixes f :: "'c" shows "J' J'.null f = J'.null"
simps_1: fixes s :: "'s"   and t_end :: "'s" shows "viterbi_prob s t_end [] = indicator {t_end} s"
RrF_V\<alpha>: shows "\<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> F) \<in>\<^sub>\<circ> Vset \<alpha>"
simps_6: fixes v :: "poly"   and va :: "poly" shows "isnpolyh (Mul v va) = (\<lambda>k. False)"
applicable_def: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma> = (C\<^sub>1 \<noteq> {} \<and> C\<^sub>2 \<noteq> {} \<and> L\<^sub>1 \<noteq> {} \<and> L\<^sub>2 \<noteq> {} \<and> vars\<^sub>l\<^sub>s C\<^sub>1 \<inter> vars\<^sub>l\<^sub>s C\<^sub>2 = {} \<and> L\<^sub>1 \<subseteq> C\<^sub>1 \<and> L\<^sub>2 \<subseteq> C\<^sub>2 \<and> mgu\<^sub>l\<^sub>s \<sigma> (L\<^sub>1 \<union> L\<^sub>2\<^sup>C))"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. C.tensor f (C.cod \<iota>))"
nth_rev: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - 1 - n)"