hcju/isabelleps huggingface dataset:
prefix_fininfI: fixes u :: "'a list"   and w :: "'a stream"   and v :: "'a stream" assumes "u @- w = v" shows "u \<le>\<^sub>F\<^sub>I v"
map_disc_iff: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e list" shows "(map f a = []) = (a = [])"
halfspace_simps_11: shows "le_halfspace \<equiv> \<lambda>sctn x. x \<bullet> normal sctn \<le> pstn sctn"
eqpoll_iff_bijections: fixes A :: "'b set"   and B :: "'c set" shows "(A \<approx> B) = (\<exists>f g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
convert_allZero: fixes var :: "nat"   and p :: "real mpoly"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and p' :: "real \<times> real \<times> real" assumes "convert_poly var p (xs' @ x # xs) = Some p'"   and "length xs' = var" shows "eval (allZero p var) (xs' @ x # xs) = evalUni (allZero' p') x"
expand_new_name__less_zero: fixes nm :: "nat" shows "0 < expand_new_name nm"
wf_ci_def: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool" shows "wf_ci ci = (ci_stk_offer ci \<and> ci \<le> check_instr' \<and> (\<forall>ins P h stk loc C M pc pc' frs. ci ins P h stk loc C M pc frs \<longrightarrow> ins_jump_ok ins pc' \<longrightarrow> ci ins P h stk loc C M pc' frs))"
is_shadow_root_ptr_implies_kind: fixes ptr :: "(_) document_ptr" assumes "is_shadow_root_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" shows "is_shadow_root_ptr_kind\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"
back_inj: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and II1 :: "nat set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and II2 :: "nat set" assumes "PL_Indis cval indis"   and "II1 \<subseteq> {PL.brnL cl n..<PL.brnL cl n + PL.brn (cl ! n)}"   and "II2 \<subseteq> {PL.brnL cl n..<PL.brnL cl n + PL.brn (cl ! n)}" shows "(PL_Indis.back cl n II1 = PL_Indis.back cl n II2) = (II1 = II2)"
head_\<omega>_lt_imp_lt: fixes m :: "hmultiset"   and n :: "hmultiset" assumes "head_\<omega> m < head_\<omega> n" shows "m < n"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (prod = (fst prod, snd prod) \<longrightarrow> P (f (fst prod) (snd prod)))"
shift_map_keys_def: fixes t :: "'a"   and f :: "'d \<Rightarrow> 'c"   and m :: "('t, 'd) fmap" shows "shift_map_keys t f m = fmmap f (shift_keys t m)"
front_tickFree_def: fixes s :: "'c event list" shows "front_tickFree s = (s = [] \<or> tickFree (tl (rev s)))"
distrib_imp2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x + y \<sqinter> z = (x + y) \<sqinter> (x + z)" shows "x \<sqinter> (y + z) = x \<sqinter> y + x \<sqinter> z"
simps_2: fixes e :: "'b equation" shows "vars_of_lit (Neg e) = vars_of_eq e"
cases: fixes A :: "('c, 'd) nba"   and a1 :: "('c \<times> 'd) stream"   and a2 :: "'d" assumes "run A a1 a2" obtains a :: "'c \<times> 'd" and p :: "'d" and r :: "('c \<times> 'd) stream" where "a1 = a ## r" and "a2 = p" and "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p" and "run A r (snd a)"
a: shows "gc'' \<rightarrow>\<^bsub>ns\<^esub> gc'"
vec_set_uminus: fixes A :: "'b vec" shows "set\<^sub>v (- A) = uminus ` set\<^sub>v A"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c" assumes "Digraph G"   and "v0_v1_Digraph_axioms G v0 v1" shows "v0_v1_Digraph G v0 v1"
exec_gpv_fixp_parallel_induct: fixes P :: "(('r, 's, 't) gpv \<Rightarrow> 'u \<Rightarrow> ('r \<times> 'u) spmf) \<Rightarrow> (('v, 'w, 'x) gpv \<Rightarrow> 'y \<Rightarrow> ('v \<times> 'y) spmf) \<Rightarrow> bool"   and calleea :: "'u \<Rightarrow> 's \<Rightarrow> ('t \<times> 'u) spmf"   and callee :: "'y \<Rightarrow> 'w \<Rightarrow> ('x \<times> 'y) spmf" assumes "ccpo.admissible (prod_lub spmf.lub_fun spmf.lub_fun) (rel_prod spmf.le_fun spmf.le_fun) (\<lambda>x. P (curry (fst x)) (curry (snd x)))"   and "P (\<lambda>_ y. return_pmf None) (\<lambda>_ y. return_pmf None)"   and "\<And>f g. P f g \<Longrightarrow> P (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (x, aa) | IO out c \<Rightarrow> calleea aa out \<bind> (\<lambda>a. case a of (x, y) \<Rightarrow> f (c x) y))) (\<lambda>a aa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (x, aa) | IO out c \<Rightarrow> callee aa out \<bind> (\<lambda>a. case a of (x, y) \<Rightarrow> g (c x) y)))" shows "P (exec_gpv calleea) (exec_gpv callee)"
sorted_wrt2: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and zs :: "'d list" assumes "transp P" shows "sorted_wrt P (x # y # zs) = (P x y \<and> sorted_wrt P (y # zs))"
grd_demonic: fixes Q :: "'c \<Rightarrow> 'd set" shows "grd [: Q :] = {s. \<exists>s'. s' \<in> Q s}"
mono_meet: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and meet :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and x :: "'b"   and x' :: "'b"   and y :: "'b"   and y' :: "'b" assumes "class.L_top_bot le join Top meet bot"   and "le x x'"   and "le y y'" shows "le (meet x y) (meet x' y')"
ereal_less_eq_2: fixes x :: "ereal" shows "- \<infinity> \<le> x"
simps_7: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case LT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f1"
kh: fixes z :: "complex" assumes "z \<in> S" shows "k (h z) = z"
mod_smult_left: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly" shows "Polynomial.smult a x mod y = Polynomial.smult a (x mod y)"
wf_INTERSECT: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and xs :: "'c rexp list" shows "alphabet.wf wf_atom n (INTERSECT xs) = (\<forall>r\<in>set xs. alphabet.wf wf_atom n r)"
assms: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
projEmpty: fixes qs :: "'a list"   and S :: "'a set"   and x :: "'a" assumes "Lxy qs S = []"   and "x \<in> S" shows "x \<notin> set qs"
pred_map: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "pred_option Q (map_option f x) = pred_option (Q \<circ> f) x"
d_gso_Ints: fixes i :: "nat"   and k :: "nat" assumes "i < n"   and "k < length fs" shows "(d k \<cdot>\<^sub>v gso k) $ i \<in> \<int>"
degrees_leD: fixes X :: "'a pdevs list"   and K :: "nat"   and x :: "'a pdevs" assumes "degrees X \<le> K"   and "x \<in> set X" shows "degree x \<le> K"
assms_9: shows "unique_factorization_m u (lu, mset vs)"
value_to_sterm: fixes v :: "value" assumes "pred v" shows "S (value_to_sterm v)"
elims_2: fixes x :: "('d, 'e, 'f, 'g, 'h) qTerm"   and xa :: "('d, 'e, 'f, 'g, 'h) qTerm" assumes "x #= xa" obtains xs :: "'f" and x :: "'g" and xs' :: "'f" and x' :: "'g" where "xa__ = qVar xs x" and "xa = qVar xs' x'" and "xs = xs' \<and> x = x'"   | delta :: "'h" and inp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option" and binp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" and delta' :: "'h" and inp' :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option" and binp' :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" where "xa__ = qOp delta inp binp" and "xa = qOp delta' inp' binp'" and "delta = delta' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp'"
bin_lcp_snd_lcp: shows "symcode.\<alpha> \<le>p u\<^sub>1 \<cdot> symcode.\<alpha>"
map_le_SomeD: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" assumes "m \<subseteq>\<^sub>m m'"   and "m x = \<lfloor>y\<rfloor>" shows "m' x = \<lfloor>y\<rfloor>"
atS_simps_2: fixes p :: "'e"   and l :: "'f"   and s :: "('g, 'f, 'e, 'h, 'i) system_state" shows "atS p {l} s = at p l s"
top_greatest: fixes a :: "'b upred" shows "true \<sqsubseteq> a"
forest_def: fixes G :: "('c, 'd) graph" shows "forest G \<equiv> valid_graph G \<and> forest_axioms G"
ct_list_eq_refl: fixes xs :: "ct list" shows "ct_list_eq xs xs"
S_selects_subseteq: fixes C :: "'a literal multiset" shows "S C \<subseteq># C"
a1: shows "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1 \<and> Some t2 = memory_write 10 addr bm data s2"
inv: fixes i :: "nat" shows "k ^ i \<noteq> (0::'a)"
pmult_by_x: fixes t :: "'a list" shows "[0::'a, 1::'a] *** t = (0::'a) # t"
A7_DAcc_level0: shows "DAcc level0 sA7 = {sA8}"
simps: fixes bs :: "bool list"   and "is" :: "int list"   and i :: "int"   and d :: "int" shows "rderiv0 bs (presb.Eq is i d) = (let l = - sum_list (concat (map (\<lambda>i. if i < 0 then [i] else []) is)); h = - sum_list (concat (map (\<lambda>i. if 0 < i then [i] else []) is)); d' = scalar_product (map of_bool bs) is + 2 * d in if d' \<in> {min h i..max l i} then FBase (presb.Eq is i d') else FBool False)"
orbit_altdef: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "orbit f x = {(f ^^ n) x |n. 0 < n}"
intros_4: fixes \<C> :: "'a" shows "OclVoid \<sqsubset>\<^sub>B \<langle>\<C>\<rangle>\<^sub>\<T>"
map_cong: fixes x :: "'c set['d]"   and ya :: "'c set['d]"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "x = ya"   and "\<And>z. z \<in> set_bset ya \<Longrightarrow> f z = g z" shows "map_bset f x = map_bset g ya"
map_idx_no_idx: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and n :: "nat" shows "map_idx (\<lambda>x _. f x) xs n = map f xs"
cnj_def: shows "Octonions.cnj \<equiv> corec_octo Ree (\<lambda>z. - Im1 z) (\<lambda>z. - Im2 z) (\<lambda>z. - Im3 z) (\<lambda>z. - Im4 z) (\<lambda>z. - Im5 z) (\<lambda>z. - Im6 z) (\<lambda>z. - Im7 z)"
length_concat_ge: fixes as :: "'a list list" assumes "as \<in> lists (- {[]})" shows "length as \<le> length (concat as)"
lit_def: shows "lit \<equiv> map_fun id mk\<^sub>e (\<lambda>v b. v)"
inject: fixes x1 :: "tcp_flag set"   and x2 :: "tcp_flag set"   and y1 :: "tcp_flag set"   and y2 :: "tcp_flag set" shows "(TCP_Flags x1 x2 = TCP_Flags y1 y2) = (x1 = y1 \<and> x2 = y2)"
sat_precond_as_proj_4: fixes fm2 :: "('d, 'e) fmap"   and fm1 :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fm2 \<subseteq>\<^sub>f fm1" shows "fmrestrict_set vs fm2 \<subseteq>\<^sub>f fm1"
non_overlapping_app_constI: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and name :: "name" shows "non_overlapping (t\<^sub>1 $ t\<^sub>2) (Const name)"
simps_1: fixes e :: "'a"   and f :: "udom\<cdot>'a error" shows "plusU\<cdot>(Err\<cdot>e)\<cdot>f = f"
cupcake_list_state_preserve0: fixes env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list"   and res :: "'b state \<times> (v list, v) result" assumes "Big_Step_Unclocked_Single.evaluate_list (\<lambda>s e res. Big_Step_Unclocked_Single.evaluate env s e res \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> s = fst res)) s es res"   and "list_all is_cupcake_exp es"   and "is_cupcake_all_env env" shows "s = fst res"
norm_f_n_x: fixes x :: "'a" shows "\<exists>M. \<forall>n. norm (f n *\<^sub>v x) \<le> M"
sets_scale_measure: fixes r :: "ennreal"   and M :: "'b measure" shows "sets (scale_measure r M) = sets M"
intro: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> less_eq x y \<or> less_eq y x" shows "connex A less_eq"
dw_def: shows "dw \<equiv> dropWhile ((\<noteq>) u) (tj_stack s)"
eval_wlp_SetPC: fixes f :: "'c \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" shows "wlp (SetPC f p) P = (\<lambda>s. \<Sum>a\<in>supp (p s). p s a * wlp (f a) P s)"
natural_eq_iff: fixes m :: "natural"   and n :: "natural" shows "(m = n) = (nat_of_natural m = nat_of_natural n)"
assms_1: shows "(_Entry_) -as\<rightarrow>* n'"
real_vector_affinity_eq: fixes m :: "real"   and x :: "'a"   and c :: "'a"   and y :: "'a" assumes "m \<noteq> 0" shows "(m *\<^sub>R x + c = y) = (x = y /\<^sub>R m - c /\<^sub>R m)"
finite_gen_over_ideal: fixes A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "'a" assumes "ideal R A"   and "h \<in> {j. j \<le> n} \<rightarrow> carrier M"   and "generator R M (h ` {j. j \<le> n})"   and "A \<odot>\<^bsub>R\<^esub> M = carrier M"   and "m \<in> carrier M" shows "\<exists>s\<in>{j. j \<le> n} \<rightarrow> A. m = l_comb R M n s h"
Rep_BinomialHeap: fixes x :: "('a, 'b) BinomialHeap" shows "Rep_BinomialHeap x \<in> {q. BinomialHeapStruc.invar q}"
elims_2: fixes x :: "'a \<times> 'b \<times> 'c"   and xa :: "'a \<times> 'd"   and xb :: "'d \<Rightarrow> 'b"   and xc :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" assumes "triple_eq_pair x xa xb xc" obtains t :: "'a" and a1 :: "'b" and a2 :: "'c" and ts' :: "'a" and tp' :: "'d" and f :: "'d \<Rightarrow> 'b" and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" where "x = (t, a1, a2)" and "xa = (ts', tp')" and "xb = f" and "xc = g" and "t = ts' \<and> a1 = f tp' \<and> a2 = g ts' tp'"
merge_overlap_length: fixes ss :: "('b word \<times> 'b word) list"   and A :: "'b word \<times> 'b word" assumes "\<exists>s'\<in>set ss. \<not> disjoint (interval_of A) (interval_of s')" shows "length (merge_overlap A ss) = length ss"
maxsimpchain_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "folding_g.gallery (xs @ ys)" shows "folding_g.gallery xs"
AE_restrict_space_iff: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "\<Omega> \<inter> space M \<in> sets M" shows "almost_everywhere (restrict_space M \<Omega>) P = set_almost_everywhere \<Omega> M P"
insert_before_list_in_set: fixes x :: "'m"   and v :: "'m"   and ref :: "'m option"   and xs :: "'m list" shows "(x \<in> set (insert_before_list v ref xs)) = (x = v \<or> x \<in> set xs)"
Category_axioms_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "Category_axioms C \<equiv> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>) \<and> (\<forall>X. X \<in> obj\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X)) \<and> ((\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f) \<and> (\<forall>f. f \<in> mor\<^bsub>C\<^esub> \<longrightarrow> f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f)) \<and> (\<forall>f g h. f \<approx>>\<^bsub>C\<^esub> g \<longrightarrow> g \<approx>>\<^bsub>C\<^esub> h \<longrightarrow> f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)) \<and> (\<forall>f X Y g Z. f maps\<^bsub>C\<^esub> X to Y \<longrightarrow> g maps\<^bsub>C\<^esub> Y to Z \<longrightarrow> f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z)"
c_def: shows "(\<Oplus>\<^bsub>V\<^esub>x\<in>{0..<l}. of_int (g x) \<cdot>\<^sub>v vs' ! x) = (\<Oplus>\<^bsub>V\<^esub>v\<in>set vs'. of_int (c v) \<cdot>\<^sub>v v)"
rest: shows "trace_rest_eq rest1 rest2 (outs_\<I> \<I>RA) (outs_\<I> \<I>RU) (return_spmf s_rest) (return_spmf s_rest')"
cos_tan_half: fixes x :: "'a" assumes "cos x \<noteq> (0::'a)" shows "cos ((2::'a) * x) = ((1::'a) - (tan x)\<^sup>2) / ((1::'a) + (tan x)\<^sup>2)"
perm_tmp4: fixes R :: "(formula list \<times> formula list set) set"   and A :: "formula list"   and a :: "formula"   and list :: "formula list" assumes "Perms \<subseteq> R"   and "A @ (a # list) @ a # list \<in> deductions R" shows "(a # a # A) @ list @ list \<in> deductions R"
klift_def: fixes f :: "'b \<Rightarrow> 'c set" shows "f\<^sup>\<dagger> = \<mu> \<circ> \<P> f"
set_add_def': fixes G :: "('a, 'b) ring_scheme"   and H :: "'a set"   and K :: "'a set" shows "H <+>\<^bsub>G\<^esub> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus>\<^bsub>G\<^esub> k}"
ordered_keys_def: fixes m :: "('a, 'b) fmap" shows "ordered_keys m = sorted_list_of_set (fset (fmdom m))"
denominator_nonzero: fixes t :: "real" shows "(f (2 - t) + f (t - 1) = 0) = False"
cyclic_group_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c" shows "cyclic_group_axioms G gen \<equiv> gen \<in> carrier G \<and> carrier G = generate G {gen}"
CondRedsThrow: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>"
Class_equivalence: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "(Class a = Class b) = ((a, b) \<in> E(\<alpha>))"
adv_P1_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'e spmf"   and S2 :: "'b \<Rightarrow> 'f \<Rightarrow> 'e spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf"   and m1 :: "'a"   and m2 :: "'b"   and D :: "'c \<Rightarrow> bool spmf" assumes "sim_det_def R1 S1 R2 S2 funct" shows "sim_det_def.adv_P1 R1 S1 funct m1 m2 D \<equiv> \<bar>spmf (R1 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> D)) True\<bar>"
domD: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd option" assumes "a \<in> dom m" shows "\<exists>b. m a = \<lfloor>b\<rfloor>"
inf_left_commute: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
gbinomial_ratio_limit: fixes a :: "'a" assumes "a \<notin> \<nat>" shows "(\<lambda>n. (a gchoose n) / (a gchoose Suc n)) \<longlonglongrightarrow> - (1::'a)"
simps_2: fixes Z :: "'a"   and M :: "nat"   and N :: "nat" shows "shallow_model' Z M (Suc N) = Pool (shallow_model' Z M 0) (shallow_model' Z M N)"
weakCongE: 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 P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and \<sigma> :: "(name list \<times> 'd list) list"   and \<Psi> :: "'e" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P Q"   and "wellFormedSubst \<sigma>" shows "env.weakPsiCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (substPsi.seqSubs substTerm substAssert substCond P \<sigma>) (substPsi.seqSubs substTerm substAssert substCond Q \<sigma>)"
coprime_pairs_asymptotics: shows "(\<lambda>N. real (card {(m, n). (m, n) \<in> {1..N} \<times> {1..N} \<and> coprime m n}) - 6 / pi\<^sup>2 * (real N)\<^sup>2) \<in> O(\<lambda>N. real N * ln (real N))"
poly_subst_plus: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (p + q) = poly_subst f p + poly_subst f q"
simps_1: fixes xs :: "'c list"   and oid :: "'c" shows "insert_spec xs (oid, None) = oid # xs"
pseudoAllocation_def: fixes allocation :: "('c \<times> 'd set) set" shows "pseudoAllocation allocation \<equiv> \<Union> (omega ` allocation)"
distrib_right_numeral: fixes a :: "'a"   and b :: "'a"   and v :: "num" shows "(a + b) * numeral v = a * numeral v + b * numeral v"
rtrancl_fold_product: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'e set"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "{((r, s), f a r, g b s) |a b r s. a \<in> A \<and> b \<in> B \<and> R a b}\<^sup>* = {((r, s), fold f w1 r, fold g w2 s) |w1 w2 r s. w1 \<in> lists A \<and> w2 \<in> lists B \<and> list_all2 R w1 w2}"
bs: fixes b :: "'a" assumes "b \<in> bs" shows "b \<le> p"
IH: fixes c :: "nat \<Rightarrow> 'a" assumes "\<exists>k\<le>n. c k \<noteq> (0::'a)" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<and> card {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)} \<le> n"
order_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
cong_solve: fixes a :: "nat"   and n :: "nat"   and b :: "nat" assumes "coprime a n" shows "\<exists>x. [a * x = b] (mod n)"
preserves_dom: fixes f :: "'a" assumes "C.arr f" shows "C.dom (trg\<^sub>C f) = trg\<^sub>C (C.dom f)"
tendsto_power_zero: fixes f :: "'a \<Rightarrow> nat"   and F :: "'a filter"   and x :: "'b" assumes "filterlim f sequentially F"   and "norm x < 1" shows "((\<lambda>y. x ^ f y) \<longlongrightarrow> (0::'b)) F"
openin_subtopology_self: fixes S :: "'c set" shows "openin (top_of_set S) S"
nonempty: shows "{} \<subset> {r. 0 < r \<and> r < q}"
aux5d: fixes b :: "'b"   and a :: "'b"   and po :: "'c"   and p :: "('b, 'c) Combinators list" assumes "AllowPortFromTo b a po \<in> set p" shows "a \<in> set (net_list p)"
rf_price: shows "prices Mkt risk_free_asset = disc_rfr_proc r"
i_cut_Un_4: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<union> B) \<down>\<ge> t = A \<down>\<ge> t \<union> B \<down>\<ge> t"
MapsToId: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X \<approx>>\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X"
of_int_uint: fixes w :: "'l word" shows "ring_1_class.of_int (uint w) = unsigned w"
assms_1: shows "w \<in> set_mode_locs element_ptr"
unsynlr_inject: fixes x :: "('a, 'b) synlr"   and y :: "('a, 'b) synlr" shows "(unsynlr x = unsynlr y) = (x = y)"
add_stalk_def: shows "stx.add_stalk \<equiv> stx.add_rel"
collect_interrupts_interrupted: fixes "is" :: "'a set"   and ias :: "'a interrupt_action list"   and t' :: "'a" assumes "interrupt_actions_ok is ias"   and "t' \<in> collect_interrupts ias" shows "t' \<in> is"
eval_real: fixes c :: "real"   and x :: "real" shows "eval c x = c"
encode2_def: fixes dims :: "nat list"   and vars :: "nat set"   and i :: "nat" shows "partial_state.encode2 dims vars i = digit_decode (partial_state.dims2 dims vars) (nths (digit_encode dims i) (- vars))"
bconf_FAss: fixes P :: "'a prog"   and sh :: "sheap"   and e1 :: "'b exp"   and F :: "char list"   and D :: "char list"   and e2 :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e1\<bullet>F{D} := e2,b) \<surd>) = (case val_of e1 of None \<Rightarrow> P,sh \<turnstile>\<^sub>b (e1,b) \<surd> | \<lfloor>v\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (e2,b) \<surd>)"
assms_2: shows "put \<sigma> v = put \<rho> v"