hcju/isabelleps huggingface dataset:
subst_subst3: fixes a :: "coname"   and P :: "trm"   and N :: "trm"   and c :: "coname"   and M :: "trm"   and x :: "name"   and y :: "name" assumes "a \<sharp> (P, N, c)"   and "c \<sharp> (M, N)"   and "x \<sharp> (y, P, M)"   and "y \<sharp> (P, x)"   and "M \<noteq> Ax y a" shows "N{x:=<a>.M}{y:=<c>.P} = N{y:=<c>.P}{x:=<a>.M{y:=<c>.P}}"
wm_to_state_inv: fixes s :: "predicate \<times> object list \<Rightarrow> bool" shows "wm_to_state (state_to_wm s) = s"
MGF_def: fixes Z :: "bit list"   and l :: "nat" shows "MGF Z l = (if l = 0 \<or> 2 ^ 32 * length (sha1 Z) < l then [] else MGF1 Z (roundup l (length (sha1 Z)) - 1) l)"
accmethd_declC: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "accmethd G S C sig = Some m"   and "ws_prog G"   and "is_class G C" shows "(\<exists>d. class G (declclass m) = Some d \<and> table_of (methods d) sig = Some (mthd m)) \<and> G\<turnstile>C\<preceq>\<^sub>C declclass m \<and> methd G (declclass m) sig = Some m \<and> G \<turnstile>Methd sig m of C accessible_from S"
continuous_map_pullback: fixes T1 :: "'a topology"   and T2 :: "'b topology"   and g :: "'a \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "continuous_map T1 T2 g" shows "continuous_map (pullback_topology A f T1) T2 (g \<circ> f)"
simps_1: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and A :: "'a" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.validate_literal I (Pos A) = (A \<in> I)"
even_not_iff: fixes a :: "'a" shows "even (NOT a) = odd a"
valuate_depend: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a"   and v' :: "nat \<Rightarrow> 'a" assumes "\<forall>x\<in>vars p. v x = v' x" shows "p \<lbrace> v \<rbrace> = p \<lbrace> v' \<rbrace>"
star_left_induct_mult_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y \<le> y) = (top \<sqinter> y \<le> y)"
smap_szip_fst: fixes f :: "'d \<Rightarrow> 'c"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "smap (\<lambda>x. f (fst x)) (szip s1 s2) = smap f s1"
vector_derivative_chain_at_general: fixes f :: "real \<Rightarrow> 'a"   and x :: "real"   and g :: "'a \<Rightarrow> 'a" assumes "f differentiable at x"   and "g field_differentiable at (f x)" shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) * deriv g (f x)"
simps_1: fixes f :: "'b \<Rightarrow> 'a option" shows "Preliminaries.find f [] = None"
minjec_inj: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "injec\<^bsub>M,N\<^esub> f" shows "inj_on f (carrier M)"
simps: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "('e, 'g, 'i) Vardi.var"   and a2 :: "('f, 'h, 'i) Vardi.var" shows "rel_var R S a1 a2 = (\<exists>x. a1 = map_var fst fst x \<and> a2 = map_var snd snd x \<and> set1_var x \<subseteq> {(x, y). R x y} \<and> set2_var x \<subseteq> {(x, y). S x y})"
hyps_3: shows "remove1 c w\<^sub>1 =\<^sub>F remove1 c w\<^sub>2"
bottomfloat_eq_m_largest: assumes "1 < LENGTH('a)" shows "valof (- topfloat) = - largest TYPE(('a, 'b) IEEE.float)"
simps_1: fixes carrier :: "'a set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
intro: fixes T :: "'b \<Rightarrow> 'b" assumes "\<And>x y. dist (T x) (T y) \<le> dist x y" shows "Kohlberg_Neyman_Karlsson T"
point_equations_2: fixes p :: "'a" assumes "is_point p" shows "1 ; p = 1"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q \<and> Q \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> R))"
A1: shows "H = {x. (x, f x) \<in> r \<and> x \<in> A}"
fold_union_ffUnion: fixes l :: "'a fset list" shows "fold (|\<union>|) l {||} = ffUnion (fset_of_list l)"
induct: fixes x :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite x"   and "P {}"   and "\<And>A a. \<lbrakk>finite A; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P x"
is_fl_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
s_2: shows "(u \<circ> r \<circ> s) \<longlonglongrightarrow> limsup (u \<circ> r)"
fifo_discharge_def: fixes f\<^sub>0 :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and Q :: "nat list" shows "fifo_discharge f\<^sub>0 l Q \<equiv> assert (Q \<noteq> []) \<bind> (\<lambda>_. let u = hd Q; Q = tl Q in assert (u \<in> V \<and> u \<noteq> s \<and> u \<noteq> t) \<bind> (\<lambda>_. FOREACH\<^sub>C {v. (u, v) \<in> cfE_of f\<^sub>0} (\<lambda>(f, l, Q). excess f u \<noteq> (0::'capacity)) (\<lambda>v (f, l, Q). if l u = l v + 1 then fifo_push f l Q (u, v) \<bind> (\<lambda>(f', Q). assert (\<forall>v'. v' \<noteq> v \<longrightarrow> cf_of f' (u, v') = cf_of f (u, v')) \<bind> (\<lambda>_. return (f', l, Q))) else return (f, l, Q)) (f\<^sub>0, l, Q) \<bind> (\<lambda>(f, l, Q). if excess f u \<noteq> (0::'capacity) then fifo_gap_relabel f l Q u \<bind> (\<lambda>(l, Q). return (f, l, Q)) else return (f, l, Q))))"
snoc_outer_4: shows "\<S>\<bullet>\<C> \<turnstile> vs : [] _> ts1 @ ts2"
fps_XD'_fps_const: fixes a :: "'b fps"   and b :: "'b" shows "fps_XD' a (fps_const b) = 0"
closure_ofE2: fixes a :: "'a"   and S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" assumes "a \<in> closure_of\<^bsub>S\<^esub> A" obtains a' :: "'a" where "a \<in> carrier S" and "a' \<in> A" and "a .=\<^bsub>S\<^esub> a'"
leg0_simps_5: shows "cod s\<^sub>0 = s\<^sub>0"
finite_imp_finite_prob_proj: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" assumes "finite PROB" shows "finite (prob_proj PROB vs)"
if_split_eq1: fixes Q :: "bool"   and x :: "'a"   and y :: "'a"   and b :: "'a" shows "((if Q then x else y) = b) = ((Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b))"
kat_3: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and z :: "('b \<times> 'b) set" assumes "(rel_dbq.d x O y) O rel_dbq.a z = {}" shows "rel_dbq.d x O y = (rel_dbq.d x O y) O rel_dbq.d z"
lossless_S1_14_OT: fixes m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool"   and m4 :: "bool" shows "lossless_spmf (S1_14_OT (m1, m2, m3, m4) ())"
inj_on_funpow_dist: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "inj_on (\<lambda>n. (f ^^ n) x) {0..funpow_dist f x y}"
induct: fixes x :: "expr"   and P :: "expr \<Rightarrow> bool" assumes "lc x"   and "\<And>xT. P \<langle>xT\<rangle>"   and "\<And>b. P b"   and "\<And>e1 e2. \<lbrakk>lc e1; P e1; lc e2; P e2\<rbrakk> \<Longrightarrow> P (e1 ? e2)"   and "\<And>e1 e2. \<lbrakk>lc e1; P e1; lc e2; P e2\<rbrakk> \<Longrightarrow> P (e1 \<cdot> e2)"   and "\<And>T X e. \<forall>x. (x, T) |\<notin>| X \<longrightarrow> lc (open0_Var (x, T) e) \<and> P (open0_Var (x, T) e) \<Longrightarrow> P (\<Lambda>\<langle>T\<rangle> e)" shows "P x"
is_sigma_algebra: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "algebra \<Omega> M"   and "finite M" shows "sigma_algebra \<Omega> M"
canonize_canonical: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set"   and t :: "'b" assumes "s ~A~ t" shows "canonize A s = canonize A t"
ta_union_correct_aux2: fixes TA :: "('a, 'b) tree_automaton_rec"   and TA' :: "('a, 'b) tree_automaton_rec" assumes "tree_automaton TA"   and "tree_automaton TA'" shows "tree_automaton (ta_union TA TA')"
q'_def: fixes v :: "real"   and t :: "real" shows "movement.q' v t = v"
LN_5: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : [] _> ts"
ctor_fold: fixes s' :: "('d, 'c) F0 \<Rightarrow> 'c"   and x :: "('d, 'd F) F0" shows "ctor_fold_F s' (ctor_F x) = s' (map_pre_F id (ctor_fold_F s') x)"
move_exp: fixes p' :: "real mpoly"   and i :: "nat"   and L :: "(real mpoly \<times> nat) list"   and v :: "nat"   and p :: "real mpoly"   and f :: "nat \<Rightarrow> real" assumes "(p', i) # L = chop (to_list v p)" shows "insertion f p = (\<Sum>(term, d)\<leftarrow>chop (to_list v p). insertion f term * f v ^ (d - i)) * f v ^ i"
a: shows "(INF n. SUP m\<in>{n..}. u (m + 1)) = (INF n. Sup (u ` {n + 1..}))"
merge_simps_1: fixes qs :: "('c \<times> 'd) list" shows "merge qs [] = qs"
univalent_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.univalent f = (\<forall>i j k. f (i, j) \<noteq> bot \<and> f (i, k) \<noteq> bot \<longrightarrow> j = k)"
oneD_gauge_integral_lesbegue_integrable: shows "integrable lborel (\<lambda>x. integral UNIV (\<lambda>y. f (y, x)))"
\<delta>\<^sub>p_def_2: shows "(\<delta>\<^sub>p', p\<^sub>1') = find_closest_code p\<^sub>0 \<delta>' (p\<^sub>2 # ps)"
subgroup_indicator_le_ord: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and H :: "'c set" assumes "finite_comm_group G"   and "a \<in> carrier G"   and "subgroup H G" shows "subgroup_indicator G H a \<le> group.ord G a"
type_max_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set"   and mem :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.type_max ev\<^sub>B t mem \<equiv> if \<forall>p\<in>t. ev\<^sub>B mem p then Low else High"
cnt_at_most_1_D: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "cnt x (xs @ x # ys) \<le> 1" shows "x \<notin> set xs \<and> x \<notin> set ys"
prj_sprod_def: shows "PRJ('e \<otimes> 'f) = sprod_map\<cdot>PRJ('e)\<cdot>PRJ('f) oo sprod_prj"
conj_def: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<and>\<^sup>c \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<and> \<gamma>' t n"
observations_wellformed': shows "\<O>\<^sub>s \<noteq> []"
continuous_on_def: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
genby_Group: fixes S :: "'b set" shows "Group \<langle>S\<rangle>"
prob_le_1: fixes M :: "'b pmf"   and A :: "'b set" shows "measure_pmf.prob M A \<le> 1"
norm_bound_mult: fixes A1 :: "'a mat"   and nr :: "nat"   and n :: "nat"   and A2 :: "'a mat"   and nc :: "nat"   and b1 :: "real"   and b2 :: "real" assumes "A1 \<in> carrier_mat nr n"   and "A2 \<in> carrier_mat n nc"   and "local.norm_bound A1 b1"   and "local.norm_bound A2 b2" shows "local.norm_bound (A1 * A2) (b1 * b2 * real n)"
lossless1: fixes s1 :: "'s1"   and x :: "'a" assumes "bad1 s1"   and "I1 s1"   and "x \<in> outs_\<I> \<I>" shows "lossless_spmf (oracle1 s1 x)"
euler_float_increment: fixes e :: "int"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "real"   and t :: "real"   and x :: "'b" shows "euler_increment' e f h t x = eucl_down e (f t x)"
bottom_lower: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_partial_order_bottom L"   and "x \<in> carrier L" shows "\<bottom>\<^bsub>L\<^esub> \<sqsubseteq>\<^bsub>L\<^esub> x"
sup_left_isotone: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x" shows "y \<sqinter> z \<le> x \<sqinter> z"
P1: shows "M' Midpoint P' P \<and> Col A B M'"
openin_euclidean_subtopology_iff: fixes U :: "'c set"   and S :: "'c set" shows "openin (top_of_set U) S = (S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S))"
ds_ord_refl: fixes x :: "'a" shows "x \<sqsubseteq> x"
wadjust_on_left_moving_B_Bk_drop_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_on_left_moving_B m rs (c, Bk # list)"   and "hd c = Oc" shows "wadjust_on_left_moving_O m rs (tl c, Oc # Bk # list)"
pred_subset_eq2: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" shows "((\<lambda>x y. s2p R (x, y)) \<le> (\<lambda>x y. s2p S (x, y))) = (R \<subseteq> S)"
not_wordlength_too_small: shows "\<not> LENGTH('a) < 2"
edges_list_it_correct: fixes g :: "'G" assumes "invar g" shows "set_iterator (edges_list_it g) (edges (\<alpha> g))"
set_intros_2: fixes x2 :: "'c" shows "x2 \<in> set2_step (Skip x2)"
edge_succ_GM: shows "edge_succ GM = perm_swap uw uv (perm_swap vw (rev_G uv) (fold perm_rem [rev_H uw, rev_H vw] (edge_succ HM)))"
eval_zero: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and x :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval zero_expansion x = 0"
assms_3: fixes y :: "'a" assumes "y \<in> A"   and "\<And>x. \<lbrakk>x \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x" shows "Q y"
Pow_bottom: fixes B :: "'c set" shows "{} \<in> Pow B"
simps: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and ze :: "'a"   and on :: "'a"   and dv :: "nat"   and v :: "'a list"   and w :: "'a list"   and d :: "nat"   and res :: "(nat \<times> 'a list) list" shows "dist_degree_factorize_main_i p ff_ops ze on dv v w d res = (if v = [on] then res else if dv < d + d then (dv, v) # res else let w = power_poly_f_mod_i ff_ops (\<lambda>f. mod_field_poly_i ff_ops f v) w (nat p); d = Suc d; gd = gcd_poly_i ff_ops (minus_poly_i ff_ops w [ze, on]) v in if gd = [on] then dist_degree_factorize_main_i p ff_ops ze on dv v w d res else let v' = div_field_poly_i ff_ops v gd in dist_degree_factorize_main_i p ff_ops ze on (degree_i v') v' (mod_field_poly_i ff_ops w v') d ((d, gd) # res))"
projection_is_projection_on': fixes M :: "'b set" assumes "closed_csubspace M" shows "is_projection_on (projection M) M"
circle_eq_rot_circle: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "circle_cube (x, y) = rot_circle_cube (x_coord x / (2 * circle_y (x_coord (x_coord_inv ((2 * y - 1) * circle_y (x_coord x))))) + 1 / 2, x_coord_inv ((2 * y - 1) * circle_y (x_coord x)))"
map_index_cong: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>p<length xs. f p (xs ! p) = g p (xs ! p)" shows "map_index f xs = map_index g xs"
is_singletonE: fixes A :: "'d set" assumes "is_singleton A" obtains x :: "'d" where "A = {x}"
converse_Int: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
remove_vertex_Digraph: fixes v :: "'a" shows "Digraph (Q.remove_vertex v)"
Dom_cod: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Dom (cod f) = Cod f"
root_self: fixes w :: "'b list" assumes "w \<noteq> \<epsilon>" shows "w \<le>p w\<^sup>\<omega>"
T_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "tensor f g = f \<otimes>\<^sub>S g"
left_minus_one_mult_self: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (times (power.power one times (uminus one) n) a) = a"
induct: fixes S :: "'d set set"   and x :: "'d set"   and P :: "'d set \<Rightarrow> bool" assumes "generate_topology S x"   and "P UNIV"   and "\<And>a b. \<lbrakk>generate_topology S a; P a; generate_topology S b; P b\<rbrakk> \<Longrightarrow> P (a \<inter> b)"   and "\<And>K. \<lbrakk>\<And>k. k \<in> K \<Longrightarrow> generate_topology S k; \<And>k. k \<in> K \<Longrightarrow> P k\<rbrakk> \<Longrightarrow> P (\<Union> K)"   and "\<And>s. s \<in> S \<Longrightarrow> P s" shows "P x"
refl: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" shows "P c c"
wf_below_pointwise: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "wf_below r A = (\<forall>a. a \<in> A \<longrightarrow> wf_below r {a})"
xy: shows "(x, y) \<in> succsr\<^sup>*"
set_divide_inverse: fixes A :: "('a \<Rightarrow> 'b) set"   and B :: "('a \<Rightarrow> 'b) set" shows "set_divide A B = set_mult A (set_inverse B)"
gt_irrefl: fixes s :: "('s, 'v) tm" assumes "wary s" shows "\<not> local.kbo_std.gt s s"
filter_is_Write\<^sub>s\<^sub>b_Cons_Write\<^sub>s\<^sub>b: fixes xs :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and ys :: "'b memref list" assumes "filter is_Write\<^sub>s\<^sub>b xs = Write\<^sub>s\<^sub>b volatile a sop v A L R W # ys" shows "\<exists>rs rws. (\<forall>r\<in>set rs. is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r) \<and> xs = rs @ Write\<^sub>s\<^sub>b volatile a sop v A L R W # rws \<and> ys = filter is_Write\<^sub>s\<^sub>b rws"
eq_iff: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
Amortized_def: fixes init :: "'a"   and nxt :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and inv :: "'a \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and \<Phi> :: "'a \<Rightarrow> real"   and U :: "'b \<Rightarrow> 'a \<Rightarrow> real" shows "Amortized init nxt inv T \<Phi> U \<equiv> (inv init \<and> (\<forall>s f. inv s \<longrightarrow> inv (nxt f s))) \<and> (\<forall>s. inv s \<longrightarrow> 0 \<le> \<Phi> s) \<and> \<Phi> init = 0 \<and> (\<forall>s f. inv s \<longrightarrow> T f s + \<Phi> (nxt f s) - \<Phi> s \<le> U f s)"
N\<V>_inter_E2_is_N\<V>2: shows "N\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES2\<^esub> = N\<^bsub>\<V>2\<^esub>"
assoc_inv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "inverse_arrows \<a>[a, b, c] \<a>\<^sup>-\<^sup>1[a, b, c]"
hyps_2: fixes i :: "nat" shows "subst_db (Suc i) x s2 \<ge>\<^sub>t subst_db i x s2"
hyps_7: shows "n' = Formal_in (targetnode a, x)"
bij_betw_idx: shows "bij_betw idx agents {..<length xs}"
simps_1: fixes uu :: "('a, 'b) simple_packet_scheme" shows "simple_fw [] uu = Undecided"
long_division_theorem: fixes K :: "'a list set"   and p :: "'a list list"   and b :: "'a list list" assumes "subring K (poly_ring R)"   and "polynomial\<^bsub>poly_ring R\<^esub> K p"   and "polynomial\<^bsub>poly_ring R\<^esub> K b"   and "b \<noteq> []"   and "lead_coeff b \<in> Units (poly_ring R\<lparr>carrier := K\<rparr>)" shows "\<exists>q r. polynomial\<^bsub>poly_ring R\<^esub> K q \<and> polynomial\<^bsub>poly_ring R\<^esub> K r \<and> p = b \<otimes>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> q \<oplus>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> r \<and> (r = [] \<or> degree r < degree b)"
encode1_lt: fixes i :: "nat" assumes "i < st.d" shows "st.encode1 i < st.d1"
alloc_def: fixes x :: "'d"   and h :: "heap" shows "Ref.alloc x h = (let l = lim h; r = Ref l in (r, Ref.set r x (h\<lparr>lim := l + 1\<rparr>)))"
rel_FGcontra_def: fixes L1 :: "'y \<Rightarrow> 'al \<Rightarrow> bool"   and Co1 :: "'z \<Rightarrow> 'am \<Rightarrow> bool"   and Co2 :: "'ah \<Rightarrow> 'au \<Rightarrow> bool"   and Co3 :: "'aa \<Rightarrow> 'an \<Rightarrow> bool"   and Co4 :: "'ab \<Rightarrow> 'ao \<Rightarrow> bool"   and Co5 :: "'ac \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra1 :: "'ad \<Rightarrow> 'aq \<Rightarrow> bool"   and Contra2 :: "'ae \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra3 :: "'af \<Rightarrow> 'as \<Rightarrow> bool"   and Contra4 :: "'ag \<Rightarrow> 'at \<Rightarrow> bool"   and Contra5 :: "'aj \<Rightarrow> 'av \<Rightarrow> bool" shows "rel_FGcontra L1 Co1 Co2 Co3 Co4 Co5 Contra1 Contra2 Contra3 Contra4 Contra5 = rel_F L1 Co1 Co3 Co1 Co4 Co5 (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) Contra1 Contra5"
hoareQ_Seq_assoc: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and A :: "com"   and B :: "com"   and C :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "(\<turnstile>\<^sub>2\<^sub>' {P} A;; B;; C {Q}) = (\<turnstile>\<^sub>2\<^sub>' {P} A;; (B;; C) {Q})"
relH_refl: fixes h :: "heap"   and as :: "nat set" assumes "in_range (h, as)" shows "relH as h h"
diff_axioms: shows "diff k (charts_submanifold S) charts (\<lambda>x. x)"
allconstvar: shows "\<forall>p\<in>set qs. \<forall>x y. poly p x = poly p y"
extension_maximal: fixes v :: "'a" shows "maximal_path (LCons v P)"
comp_cod_arr: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and b :: "'d" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.cod C f = b" shows "C b f = f"
eq_states_def: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" shows "s \<approx> s' \<equiv> \<exists>\<alpha> \<beta>. bij \<alpha> \<and> bij \<beta> \<and> \<R>\<^sub>G \<alpha> \<beta> s = s'"
sg2: shows "st (t + m) = hd (ack (t + m))"
negation_type_to_match_expr_simps_2: fixes e :: "'b" shows "negation_type_to_match_expr (Neg e) = MatchNot (Match e)"
solved_def: shows "task2_11.solved = (\<forall>c. c |\<in>| task2_11.conc_forms \<longrightarrow> (\<exists>\<Gamma> t. fst (tree.root t) = (\<Gamma> \<turnstile> c) \<and> \<Gamma> |\<subseteq>| task2_11.ass_forms \<and> task1_1.wf t \<and> tfinite t))"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S1 S2. S1 \<in>S a1 \<and> S2 \<in>S a2 \<and> (S1, S2) \<in> SRel) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q \<and> Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> R))"
cong_exp_trans_2: fixes c :: "'b"   and a :: "'b"   and b :: "nat"   and n :: "'b"   and d :: "'b" assumes "[c = a ^ b] (mod n)"   and "[a = d] (mod n)" shows "[c = d ^ b] (mod n)"