hcju/isabelleps huggingface dataset:
safe_reach_to_safe_reach_upto: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'a \<Rightarrow> 'a \<times> instr list \<Rightarrow> bool"   and safe :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and c\<^sub>0 :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" assumes "ini.safe_reach_direct program_step safe c\<^sub>0" shows "ini.safe_reach_upto program_step n safe c\<^sub>0"
atLeast_subset_iff: fixes x :: "'b"   and y :: "'b" shows "({x..} \<subseteq> {y..}) = (y \<le> x)"
prob_space_distr: fixes f :: "('c \<Rightarrow> 'a) \<Rightarrow> 'd"   and M' :: "'d measure" assumes "P.random_variable M' f" shows "prob_space (distr (Pi\<^sub>M UNIV (\<lambda>_. M)) M' f)"
norm_bound_def: fixes A :: "'b mat"   and b :: "real" shows "norm_bound A b \<equiv> \<forall>i j. i < dim_row A \<longrightarrow> j < dim_col A \<longrightarrow> norm (A $$ (i, j)) \<le> b"
canonical_retraction_comp_apartment_endomorphism: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<in> A"   and "D \<in> B" shows "ChamberComplexEndomorphism A (restrict1 (canonical_retraction A C \<circ> canonical_retraction B D) (\<Union> A))"
err_mode_val_def: fixes state :: "'b sparc_state" shows "err_mode_val state \<equiv> get_err_mode (state_var state)"
mmore: fixes x :: "'b"   and y :: "'a" shows "mmore \<lparr>unit_sys = x, \<dots> = y\<rparr> = y"
\<Delta>\<Phi>_del_min: fixes hs :: "'b tree"   and x :: "'b" shows "\<Phi> (del_min \<langle>hs, x, \<langle>\<rangle>\<rangle>) - \<Phi> \<langle>hs, x, \<langle>\<rangle>\<rangle> \<le> 2 * log 2 (real (size hs + 1)) - real (len hs) + 2"
cross_ratio_cvec_def: fixes z :: "complex \<times> complex"   and u :: "complex \<times> complex"   and v :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "cross_ratio_cvec z u v w = (let (z', z'') = z; (u', u'') = u; (v', v'') = v; (w', w'') = w; n1 = z' * u'' - u' * z''; n2 = v' * w'' - w' * v''; d1 = z' * w'' - w' * z''; d2 = v' * u'' - u' * v'' in if n1 * n2 \<noteq> 0 \<or> d1 * d2 \<noteq> 0 then (n1 * n2, d1 * d2) else (1, 1))"
irrefl_def: fixes r :: "('k \<times> 'k) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
assms_4: shows "val_process Mkt pf n (spick w n True) \<le> 0"
NegChecks: shows "a = \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle>"
rev_concat: fixes xs :: "'a list list" shows "rev (concat xs) = concat (map rev (rev xs))"
wqo_on_UNIV_conv: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wqo_on P UNIV = class.wqo P (strict P)"
Seed_holds_faces_subset: fixes p :: "nat" shows "faces_subset Seed\<^bsub>p\<^esub>"
entries_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "j < length \<HH>" shows "\<HH> ! i \<subseteq> \<HH> ! j"
decomp_1: shows "trace (S t n) tr (S t (Suc n))"
sum_list_mono: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x" shows "sum_list (map f xs) \<le> sum_list (map g xs)"
extended_ord_def: fixes s :: "'a option \<Rightarrow>\<^sub>0 nat"   and t :: "'a option \<Rightarrow>\<^sub>0 nat" shows "extended_ord s t = (restrict_indets_pp s \<prec> restrict_indets_pp t \<or> restrict_indets_pp s = restrict_indets_pp t \<and> lookup s None \<le> lookup t None)"
collinear_3_expand: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "collinear {a, b, c} = (a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c))"
simps_46: fixes x5 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Read x5 \<noteq> Ref x4"
exhaust: fixes y :: "'d node_action" obtains x11 :: "nat set" and x12 :: "'d" where "y = x11:*cast(x12)"   | x21 :: "nat" and x22 :: "nat" where "y = x21:deliver(x22)"   | x31 :: "nat set" and x32 :: "nat set" and x33 :: "'d" where "y = x31\<not>x32:arrive(x33)"   | x41 :: "nat" and x42 :: "nat" where "y = connect(x41, x42)"   | x51 :: "nat" and x52 :: "nat" where "y = disconnect(x51, x52)"   | x61 :: "nat" and x62 :: "nat" and x63 :: "nat" where "y = x61:newpkt(x62, x63)"   | "y = \<tau>\<^sub>n"
induct: fixes x1 :: "char list \<Rightarrow> Types.ty"   and x2 :: "com"   and P :: "(char list \<Rightarrow> Types.ty) \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>\<Gamma>. P \<Gamma> SKIP"   and "\<And>\<Gamma> a x. \<Gamma> \<turnstile> a : \<Gamma> x \<Longrightarrow> P \<Gamma> (x ::= a)"   and "\<And>\<Gamma> c1 c2. \<lbrakk>\<Gamma> \<turnstile> c1; P \<Gamma> c1; \<Gamma> \<turnstile> c2; P \<Gamma> c2\<rbrakk> \<Longrightarrow> P \<Gamma> (c1;; c2)"   and "\<And>\<Gamma> b c1 c2. \<lbrakk>\<Gamma> \<turnstile> b; \<Gamma> \<turnstile> c1; P \<Gamma> c1; \<Gamma> \<turnstile> c2; P \<Gamma> c2\<rbrakk> \<Longrightarrow> P \<Gamma> (IF b THEN c1 ELSE c2)"   and "\<And>\<Gamma> b c. \<lbrakk>\<Gamma> \<turnstile> b; \<Gamma> \<turnstile> c; P \<Gamma> c\<rbrakk> \<Longrightarrow> P \<Gamma> (WHILE b DO c)" shows "P x1 x2"
firstPosAppendNonMemberFirstMemberSecond: fixes a :: "'b"   and l1 :: "'b list"   and l2 :: "'b list" assumes "a \<notin> set l1"   and "a \<in> set l2" shows "firstPos a (l1 @ l2) = length l1 + firstPos a l2"
z'_3: shows "infinite {z'' \<in> {z \<in> S. (case z of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D)) = k}. (case z'' of (t, x) \<Rightarrow> (t mod \<bar>k\<bar>, int x mod \<bar>k\<bar>)) = (case z' of (t, x) \<Rightarrow> (t mod \<bar>k\<bar>, int x mod \<bar>k\<bar>))}"
K_I_K_I: shows "K \<circ> I \<circ> K \<circ> I = K \<circ> I"
Fourier_sum_limit_Fejer_kernel_half: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. (\<Sum>r<n. \<Sum>k\<le>2 * r. Fourier_coefficient f k * trigonometric_set k t) / real n) \<longlonglongrightarrow> l = (\<lambda>n. LINT x|lebesgue_on {0..pi}. Fejer_kernel n x * (f (t + x) + f (t - x) - 2 * l)) \<longlonglongrightarrow> 0"
top_apply: fixes x :: "'e" shows "\<top> x = \<top>"
ga_run_eq: fixes r :: "nat \<Rightarrow> (nat, local_state, global_state) Pid_Scheduler.pid_global_config" shows "pid.is_run r = ga.is_run r"
cmp\<^sub>D\<^sub>N_in_hom_1: fixes \<mu>\<nu> :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "arr (fst \<mu>\<nu>)"   and "arr (snd \<mu>\<nu>)"   and "src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>)" shows "\<guillemotleft>cmp\<^sub>D\<^sub>N \<mu>\<nu> : DN (src (snd \<mu>\<nu>)) \<rightarrow>\<^sub>B DN (trg (fst \<mu>\<nu>))\<guillemotright>"
open_contains_cball_eq: fixes S :: "'d set" assumes "open S" shows "\<forall>x. (x \<in> S) = (\<exists>e>0. cball x e \<subseteq> S)"
sel_1: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "old_side_prems_of (old_Infer x1 x2 x3) = x1"
e_def: shows "e = min (e' / 2) (e' / 2 / bnd)"
elims_2: fixes x :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and xa :: "'a match_expr list"   and xb :: "action"   and xc :: "'b" assumes "match_list x xa xb xc" obtains \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)" and m :: "'a match_expr" and ms :: "'a match_expr list" and a :: "action" and p :: "'b" where "x = \<gamma>" and "xa = m # ms" and "xb = a" and "xc = p" and "if matches \<gamma> m a p then True else match_list \<gamma> ms a p"
inf_right_idem: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y \<squnion> y = x \<squnion> y"
no_zero_genset: shows "(0::'w) \<notin> S"
fls_deriv_zero: shows "fls_deriv 0 = 0"
zdiff: fixes z :: "int"   and w :: "int" shows "z - w = z + - w"
inf_uset: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "islin p"   and "\<not> Ifm vs (x # bs) (minusinf p)"   and "\<not> Ifm vs (x # bs) (plusinf p)"   and "\<exists>x. Ifm vs (x # bs) p" shows "\<exists>(c, t)\<in>set (uset p). \<exists>(d, s)\<in>set (uset p). Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) p"
update_def: fixes f :: "'s \<Rightarrow> 's"   and m :: "'m" shows "update f m = get (\<lambda>s. put (f s) m)"
above_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "above r a \<equiv> {b. (a, b) \<in> r}"
distinct_23: fixes x3 :: "'b formula"   and x61 :: "'b formula"   and x62 :: "'b formula" shows "\<^bold>\<not> x3 \<noteq> x61 \<^bold>\<rightarrow> x62"
applyDecideEffect: fixes Vbl :: "nat set"   and state :: "State" assumes "\<not> Vbl \<subseteq> vars (elements (getM state))"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)" shows "let literal = selectLiteral state Vbl; state' = applyDecide state Vbl in var literal \<notin> vars (elements (getM state)) \<and> var literal \<in> Vbl \<and> getM state' = getM state @ [(literal, True)] \<and> getF state' = getF state"
fresh_fmap_update: fixes a :: "atom"   and f :: "('a, 'b) fmap"   and x :: "'a"   and y :: "'b" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f(x $$:= y)"
stateful_typed_model_axioms: shows "stateful_typed_model arity public Ana \<Gamma> Pair"
mono_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<sqsubseteq> f y)"
SN_on_imp_normalizability: fixes r :: "('b \<times> 'b) set"   and a :: "'b" assumes "SN_on r {a}" shows "\<exists>b. (a, b) \<in> r\<^sup>!"
cases: fixes a :: "(name \<times> ty) list" assumes "validn a" obtains "a = []"   | x :: "name" and \<Gamma> :: "(name \<times> ty) list" and T :: "ty" where "a = (x, T) # \<Gamma>" and "x \<sharp> \<Gamma>" and "validn \<Gamma>"
Stable_def: fixes A :: "'d set" shows "Stable A \<equiv> A Co A"
codomains_comp: fixes g :: "'b"   and f :: "'b" assumes "\<phi>.B.seq g f" shows "\<phi>.B.codomains (g \<cdot>\<^sub>B f) = \<phi>.B.codomains g"
fps_mult_fps_X_power_commute: fixes k :: "nat"   and a :: "'a fps" shows "fps_X ^ k * a = a * fps_X ^ k"
lm103: fixes Y :: "'c set"   and X :: "'c set" shows "sum (X <| Y) (X \<union> Y) = sum (chi X) (X \<union> Y)"
Fvars_Lq: shows "Fvars Lq = {xx, yy}"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure P A = ennreal (PS.prob A)"
STCal_divergent_2: fixes T :: "'procT" shows "TargetTerm T \<longmapsto>STCal Source Target\<omega> = T \<longmapsto>Target\<omega>"
roots2: fixes p :: "'a poly"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "p = [:c, b, a:]"   and "a \<noteq> (0::'a)" shows "{x. poly p x = (0::'a)} = {- (b / ((2::'a) * a)) + e |e. e\<^sup>2 = (b / ((2::'a) * a))\<^sup>2 - c / a}"
primes_M_floor_cancel: fixes y :: "real" shows "\<MM> (real_of_int \<lfloor>y\<rfloor>) = \<MM> y"
dom_union: shows "subst_domain (\<lambda>w. if \<sigma>1 w = Var w then \<theta>2 w else \<sigma>1 w) = subst_domain \<sigma>1 \<union> subst_domain \<theta>2"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_inverse) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_inverse \<and> a\<inverse>\<^sub>\<real> = b)"
mult_not_zero: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and a :: "'m"   and b :: "'m" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
atLeastAtMost_iff: fixes i :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and l :: "'g"   and u :: "'g" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
sel_1: fixes g :: "'a set" assumes "g \<in> G ` I" shows "sel g \<in> I"
Dom_in_Obj: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Dom f \<in> IDE"
defined_Integer_simps_1: fixes i :: "int" shows "defined (MkI\<cdot>i)"
not_ide_fromArr: fixes x :: "'arr" shows "\<not> ide (fromArr x)"
iso_verts_tail: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and e :: "'b" assumes "digraph_isomorphism hom"   and "e \<in> arcs G" shows "iso_tail hom (iso_arcs hom e) = iso_verts hom (tail G e)"
same_prefix_prefix: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "prefix (xs @ ys) (xs @ zs) = prefix ys zs"
union_prs: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c" assumes "Quotient3 R Abs Rep" shows "((-`) Abs ---> (-`) Abs ---> (-`) Rep) (\<union>) = (\<union>)"
assms: fixes n :: "nat" shows "A (Suc n) \<subseteq> A n"
augment_edge_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and e :: "nat \<times> nat"   and cap :: "'capacity" shows "augment_edge_impl cf e cap \<equiv> cf_get cf e \<bind> (\<lambda>v. cf_set cf e (v - cap) \<bind> (\<lambda>cf. let e = prod.swap e in cf_get cf e \<bind> (\<lambda>v. cf_set cf e (v + cap) \<bind> return)))"
col__refl: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet P A B" shows "Tarski_neutral_dimensionless.ReflectL Bet Cong P P A B"
gr_implies_not_zero_hmset: fixes M :: "hmultiset"   and N :: "hmultiset" assumes "M < N" shows "N \<noteq> 0"
is_minimal_basis_trd_distinct: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_minimal_basis (set (x # xs))"   and "distinct (x # xs)" shows "distinct (trd xs x # xs)"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (- x) = - lookup x"
convergentD: fixes X :: "nat \<Rightarrow> 'e" assumes "convergent X" shows "\<exists>L. X \<longlonglongrightarrow> L"
ABinds_strict: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "ABinds \<Gamma>\<cdot>\<bottom> = \<bottom>"
smult_map_poly: fixes a :: "'b" shows "smult a = map_poly ((*) a)"
lsu_conuc: shows "conucleus \<nu>\<^sup>\<natural>"
freshAbs_substAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "freshAbs zs z (A $[Y / y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> fresh zs z Y))"
8: shows "w \<sqinter> - d \<le> forest_components (w \<sqinter> - d \<squnion> e)"
extend_in_C: fixes C :: "'b fm set set"   and S :: "'b fm set"   and f :: "nat \<Rightarrow> 'b fm"   and n :: "nat" assumes "consistency C"   and "S \<in> C" shows "extend S C f n \<in> C"
FAccNone\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and h :: "heap"   and ls :: "val list"   and sh :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, ls, sh)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h, ls, sh)\<rangle>"
sign_inverse: fixes p :: "'c \<Rightarrow> 'c" assumes "permutation p" shows "sign (inv p) = sign p"
compl_inf: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
Cauchy_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'c" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
one_eq_divide_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "(one = divide a b) = (b \<noteq> zero \<and> a = b)"
sconst_conv_of_seq: fixes a :: "'c" shows "sconst a = of_seq (\<lambda>_. a)"
NormalizeI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Language.normalize c Q,A"
is_in_correct: fixes t :: "'a tree\<^sub>0"   and k :: "'a" assumes "is_ord t" shows "is_in\<^sub>0 k t = (k \<in> set_of t)"
exhaust: fixes y :: "'e \<times> 'f" obtains x1 :: "'e" and x2 :: "'f" where "y = (x1, x2)"
smult_distrib_left_minus_mat: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat"   and c :: "'c" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "c \<cdot>\<^sub>m (B - A) = c \<cdot>\<^sub>m B - c \<cdot>\<^sub>m A"
simps_4: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n\<^sub>s :: "nat"   and n' :: "nat" assumes "\<not> Option.is_none (devBC t n nid (n\<^sub>s + Suc n'))" shows "devExt t n nid n\<^sub>s (Suc n') = bc (\<sigma>\<^bsub>the (devBC t n nid (n\<^sub>s + Suc n'))\<^esub>t (n\<^sub>s + Suc n'))"
hom_dvd_simp: fixes x :: "'a"   and y' :: "'b" shows "(hom x dvd y') = (x dvd inv hom y')"
length_upto_enum_cases: fixes a :: "'b word"   and b :: "'b word" shows "length [a .e. b] = (if a \<le> b then Suc (unat b) - unat a else 0)"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
HomFtorMor_1: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,f] \<in> mor\<^bsub>SET\<^esub>"
reds_lcl_add_aux: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and es :: "'addr expr list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and es' :: "'addr expr list"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and l0 :: "String.literal \<Rightarrow> 'addr val option" assumes "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>" shows "extTA,P,t \<turnstile> \<langle>es,(hp s, l0 ++ lcl s)\<rangle> [-ta\<rightarrow>] \<langle>es',(hp s', l0 ++ lcl s')\<rangle>"
cases: fixes r :: "'b list_graph" obtains nodesL :: "'b list" and edgesL :: "('b \<times> 'b) list" where "r = \<lparr>nodesL = nodesL, edgesL = edgesL\<rparr>"
le_imp_erfc_real_le: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "erfc b \<le> erfc a"
list_all: fixes x :: "nat \<times> event_data option list set \<times> event_data option list set" assumes "x \<in> set (take (length done) auxlist)" shows "check_before I nt x"
step_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Skip, s) \<rightarrow> u" shows "P"
representation_add: fixes basis :: "'b set"   and v :: "'b"   and u :: "'b" assumes "cindependent basis"   and "v \<in> cspan basis"   and "u \<in> cspan basis" shows "crepresentation basis (u + v) = (\<lambda>b. crepresentation basis u b + crepresentation basis v b)"
assms_1: shows "interval_lebesgue_integrable lborel a b g"
exhaust: fixes y :: "W" obtains "y = W1"   | "y = W2"   | "y = W3"
qGoodBinp_def: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" shows "qGoodBinp qbinp \<equiv> liftAll qGoodAbs qbinp \<and> |{i. qbinp i \<noteq> None}| <o |UNIV|"
Quotient3_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b" shows "Quotient3 R Abs Rep = ((\<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)))"
abs_eq: fixes x :: "('b \<times> 'b) list" shows "($) (abs_prm x) = preprm_apply x"
simps_3: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" shows "approximating_bigstep_fun \<gamma> p (Rule m a # rs) Undecided = (if \<not> matches \<gamma> m a p then approximating_bigstep_fun \<gamma> p rs Undecided else case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny | Log \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided | Reject \<Rightarrow> Decision FinalDeny | Empty \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided)"
p_zero: shows "ego.p 0 = s\<^sub>e"
inf_sup_ord_3: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
in_num: fixes n :: "'b" assumes "n \<in> num" shows "n \<in> trm"
seq1: fixes c1 :: "('a, 'b) MWLsCom"   and m :: "'b \<Rightarrow> 'c"   and \<alpha> :: "('a, 'b) MWLsCom list"   and m' :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and c2 :: "('a, 'b) MWLsCom" assumes "((c1, m), \<alpha>, None, m') \<in> MWLs_semantics.MWLsSteps_det E BMap" shows "((c1;c2, m), \<alpha>, Some c2, m') \<in> MWLs_semantics.MWLsSteps_det E BMap"
succ_of_pred_def: fixes s :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "succ_of_pred s \<equiv> \<lambda>a. {b. s a b}"
shrK_injective: fixes x :: "agent"   and y :: "agent" shows "(shrK x = shrK y) = (x = y)"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> G"   and "v \<in> G" shows "inverse u \<cdot> (u \<cdot> v) = v"
sup_commute: fixes x :: "'d upred"   and y :: "'d upred" shows "(x \<or> y) = (y \<or> x)"
subst_apply_fv_subset: fixes t :: "('b, 'a) Term.term"   and V :: "'a set"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "fv t \<subseteq> V" shows "fv (t \<cdot> \<delta>) \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)"
some_eq_trivial: fixes x :: "'c" shows "(SOME y. y = x) = x"