hcju/isabelleps huggingface dataset:
WTrtVal: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and T :: "ty"   and E :: "char list \<Rightarrow> ty option" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,E,h \<turnstile> Val v : T"
subst_typ'_incr_boundvars: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and lev :: "nat"   and t :: "Core.term" shows "subst_typ' insts (incr_boundvars lev t) = incr_boundvars lev (subst_typ' insts t)"
induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b list"   and x3 :: "'b"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> 'b \<Rightarrow> bool" assumes "rtrancl_path r x1 x2 x3"   and "\<And>x. P x [] x"   and "\<And>x y ys z. \<lbrakk>r x y; rtrancl_path r y ys z; P y ys z\<rbrakk> \<Longrightarrow> P x (y # ys) z" shows "P x1 x2 x3"
iter_dom: fixes ss0 :: "nat list list"   and w0 :: "nat list"   and ss' :: "nat list list"   and w' :: "nat list" assumes "wf_dom ss0 w0" shows "iter f step ss0 w0 = (ss', w') \<longrightarrow> wf_dom ss' w' \<and> stables r step ss' \<and> ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ss' \<and> (\<forall>ts\<in>list n A. ss0 [\<sqsubseteq>\<^bsub>r\<^esub>] ts \<and> stables r step ts \<longrightarrow> ss' [\<sqsubseteq>\<^bsub>r\<^esub>] ts)"
simps_2: fixes w :: "'c"   and ws :: "'c list"   and ps :: "('c \<times> 'd) list" shows "removeKeyList (w # ws) ps = removeKey w (removeKeyList ws ps)"
op_union_coll_def: shows "op_union_coll = (\<union>)"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'a list" assumes "i < |xs|" shows "xs ! i # drop (Suc i) xs = drop i xs"
derived_clauses_are_finite_def: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" shows "derived_clauses_are_finite R = (\<forall>P1 P2 C. finite P1 \<longrightarrow> finite P2 \<longrightarrow> R P1 P2 C \<longrightarrow> finite C)"
im_cball: shows "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` (cball (0::'a) 1 \<inter> affine hull S) = S"
bigI: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "0 < c"   and "\<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))" shows "f \<in> L F g"
distinct_names: shows "all_distinct (Node (Node (Node Tip B False Tip) C False (Node Tip X False Tip)) a False (Node (Node Tip b False Tip) c False (Node Tip f False Tip)))"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<Z> \<alpha>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
induct: fixes x :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "H.walk x"   and "P []"   and "\<And>v. v \<in> V\<^bsub>remove_vertex x\<^esub> \<Longrightarrow> P [v]"   and "\<And>v w vs. \<lbrakk>v \<rightarrow>\<^bsub>remove_vertex x\<^esub> w; H.walk (w # vs); P (w # vs)\<rbrakk> \<Longrightarrow> P (v # w # vs)" shows "P x"
abs_eq: fixes xa :: "'c"   and x :: "'c" assumes "eq_onp dense xa xa"   and "eq_onp dense x x" shows "(Abs_dense xa \<le> Abs_dense x) = (xa \<le> x)"
r_c_qt_imp_cf: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "finite A"   and "complete A r"   and "quasi_trans r"   and "refl_on A r" shows "choiceFn A r"
hom_one: shows "h \<one> = \<one>\<^bsub>S\<^esub>"
ennreal_of_enat_infty: shows "ennreal_of_enat \<infinity> = \<infinity>"
Red_F_of_subset: fixes N :: "'g set"   and N' :: "'g set" assumes "N \<subseteq> N'" shows "Red_F_G N \<subseteq> Red_F_G N'"
is_standard_operator_rem_implicit_pres_ops: fixes a :: "(char list \<times> nat option \<times> char list list) list"   and b :: "nat list"   and c :: "(nat \<times> nat) list"   and d :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> (a, b, c, d)) \<Longrightarrow> is_standard_operator op"   and "op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops (a, b, c, d)))" shows "is_standard_operator op"
map_converter_of_callee: fixes f :: "'h \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'i"   and h :: "'n \<Rightarrow> 'j"   and k :: "'k \<Rightarrow> 'o"   and callee :: "'p \<Rightarrow> 'l \<Rightarrow> ('m \<times> 'p, 'n, 'o) gpv"   and s :: "'p" shows "map_converter f g h k (CNV callee s) = CNV (map_fun id (map_fun f (map_gpv' (map_prod g id) h k)) callee) s"
same_size_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "invertible_mat (M_mat_R signs subsets)" shows "length subsets = length signs"
reachable_def: fixes succs :: "'d \<Rightarrow> 'd list"   and isNode :: "'d \<Rightarrow> bool"   and invariant :: "'e \<Rightarrow> bool"   and ins :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and memb :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and empt :: "'e"   and nodeAbs :: "'d \<Rightarrow> 'f"   and xs :: "'d set" assumes "DFS succs isNode invariant ins memb empt nodeAbs" shows "DFS.reachable succs xs \<equiv> {(x, y). y \<in> set (succs x)}\<^sup>* `` xs"
eq: shows "(\<lambda>p. if prime p then \<Sum>i. f (p ^ i) else (1::'a)) = (\<lambda>p. if prime p then inverse ((1::'a) - f p) else (1::'a))"
narrow: shows "G\<turnstile>PrimT Boolean\<succ>T"
fun_upd_idem_iff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c" shows "(f(x := y) = f) = (f x = y)"
sub_eq_zero_iff: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h \<or> w \<noteq> \<infinity>\<^sub>h" shows "(z -\<^sub>h w = 0\<^sub>h) = (z = w)"
filter_mono: fixes F1 :: "'b filter"   and F2 :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "F1 \<le> F2"   and "f \<in> \<Omega>[F2](g)" shows "f \<in> \<Omega>[F1](g)"
dvd_1_iff_1: fixes m :: "nat" shows "(m dvd Suc 0) = (m = Suc 0)"
fcard_funion_fsubset: fixes B :: "'a fset"   and A :: "'a fset" assumes "B |\<subseteq>| A" shows "fcard (A |-| B) = fcard A - fcard B"
reduced_word_for_arg_min: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and as :: "'b list"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero"   and "as \<in> lists A"   and "monoid_add.sum_list plus zero as = a" shows "monoid_add.reduced_word_for plus zero A a (ARG_MIN length as. as \<in> lists A \<and> monoid_add.sum_list plus zero as = a)"
axioms_2: fixes L :: "('c, 'd) gorder_scheme" assumes "upper_semilattice L" shows "upper_semilattice_axioms L"
induced_play_infinite: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "\<not> lfinite (strategy_progression \<sigma> p)"
bdd_aboveI2: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e multiset"   and M :: "'e multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "subset_mset.bdd_above (f ` A)"
Stone_Weierstrass_basic: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "continuous_on S f"   and "0 < e" shows "\<exists>g\<in>R. \<forall>x\<in>S. \<bar>f x - g x\<bar> < e"
prems: shows "set (a # adjs) \<subseteq> {a. case a of (v, d) \<Rightarrow> (u, v) \<in> edges g \<and> w {u, v} = d}"
assms_2: shows "(c1, s) \<rightarrow>*t s'"
hull_mono: fixes s :: "'a set"   and t :: "'a set"   and S :: "'a set \<Rightarrow> bool" assumes "s \<subseteq> t" shows "S hull s \<subseteq> S hull t"
pj_Hom: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "pj R I \<in> rHom R (R /\<^sub>r I)"
distributed_finite_measure_density: fixes N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" assumes "distributed M N X f" shows "finite_measure (density N f)"
lfloat10_def: fixes r :: "real" shows "lfloat10 r = (let f = float_of r in case float2_float10 20 True (mantissa f) (exponent f) of (x, xa) \<Rightarrow> x \<e> xa)"
cardSuc_Well_order: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "Well_order (cardSuc r)"
lm082: fixes g :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'d set"   and x :: "'c" assumes "g \<in> injections X Y"   and "x \<in> Domain g" shows "g \<in> {Outside' {x} g \<union> {(x, y)} |y. y \<in> Y - Range (Outside' {x} g)}"
RSubmodule_RSpan: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "RSubmodule (RSpan ms)"
lvars_rvars: shows "lvars (\<T> s') \<inter> rvars (\<T> s') = {}"
mk_next_pow_until: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "suffix (min i j) w \<Turnstile>\<^sub>n X\<^sub>n' (i - j) \<phi> U\<^sub>n X\<^sub>n' (j - i) \<psi> = w \<Turnstile>\<^sub>n X\<^sub>n' i \<phi> U\<^sub>n X\<^sub>n' j \<psi>"
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P\<^sub>1 :: "('e, 'd, 'f) ann"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and P\<^sub>2 :: "('e, 'd, 'f) ann"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 (pre P\<^sub>2), A"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q, A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnComp P\<^sub>1 P\<^sub>2) (Seq c\<^sub>1 c\<^sub>2) Q, A"
assms_2: shows "fmdom' s1 \<subseteq> vs1"
bot_entailed: shows "fst ` lhd Ns \<Turnstile>\<inter>\<G> {B}"
natset_finite_max: fixes A :: "nat set" assumes "finite A" shows "Suc (Max A) \<notin> A"
is_UNIV_unfold_2: fixes A :: "'b set" shows "(UNIV = A) = is_UNIV A"
simps: fixes dmus :: "int iarray iarray"   and i :: "nat" shows "dmu_array_cost dmus i = (if m \<le> i then (dmus, 0) else let (dmus', cost_row) = dmu_array_row_cost dmus i; (res, cost_rec) = dmu_array_cost dmus' (Suc i) in (res, cost_row + cost_rec))"
Lcm_fract_def: shows "Lcm \<equiv> normalization_euclidean_semiring_class.Lcm"
poly_mult_zero_2: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R"   and "set p \<subseteq> carrier R" shows "ring.poly_mult R p [] = []"
zero_on_circline: fixes H :: "circline" shows "on_circline H 0\<^sub>h = circline_D0 H"
Lcm_dvd_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and x :: "'a" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (Lcm A) x = (\<forall>y\<in>A. dvd.dvd times y x)"
borel_integrable_atLeastAtMost: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont f x" shows "integrable lborel (\<lambda>x. f x * indicat_real {a..b} x)"
enum_all_bit0_def: fixes P :: "'a bit0 \<Rightarrow> bool" shows "enum_class.enum_all P = (\<forall>b\<in>set enum_class.enum. P b)"
remove_vertex_V: fixes x :: "'a" shows "V\<^bsub>H.remove_vertex x\<^esub> = V\<^bsub>H\<^esub> - {x}"
f'_inv: fixes y :: "'a" assumes "y \<in> src.carrier" shows "f' (f y) = y"
weakScope: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>" shows "\<Psi> : \<lparr>\<nu>x\<rparr>Q \<rhd> \<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
strict_weak_trans: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "a \<prec>[le] b"   and "le b c" shows "a \<prec>[le] c"
transp_relcompp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transp r = (r OO r \<le> r)"
bounded_cbilinear_cblinfun_compose: shows "bounded_cbilinear (o\<^sub>C\<^sub>L)"
WHILEIT_refine: fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I' x'\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f x \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I'\<^esup> b' f' x')"
dvd_modulo_to_lincomb: fixes u :: "int poly"   and k :: "int"   and g :: "int vec" assumes "degree u < n"   and "poly_mod.dvdm k u (poly_of_vec g)"   and "k \<noteq> 0"   and "monic u"   and "dim_vec g = n"   and "0 < degree u" shows "\<exists>c. lincomb_list c (factorization_lattice u (n - degree u) k) = g"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "lcm b (lcm a c) = lcm a (lcm b c)"
rep_transfer: shows "rel_fun cr_uint8 (=) (\<lambda>x. x) Rep_uint8"
Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. eSuc (P s)} SKIP {P}"
strong_reduction_simulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal" shows "strong_reduction_simulation (Rel\<^sup>+) Cal"
gtrI_s: fixes s :: "'a"   and c :: "'a multiset"   and e :: "'b"   and s' :: "'c"   and c' :: "'c multiset"   and gtrs :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'c \<times> 'c multiset) set" assumes "((s, c), e, s', c') \<in> gtrs" shows "(add_mset s c, e, add_mset s' c') \<in> gtr gtrs"
inv_solve_left: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = inv\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c) = (c = b \<otimes>\<^bsub>G\<^esub> a)"
mult_poly_p'_def: fixes p' :: "(char list multiset \<times> int) multiset"   and q' :: "(char list multiset \<times> int) multiset" shows "mult_poly_p' p' q' = SPEC (\<lambda>r. (mult_poly_p q')\<^sup>*\<^sup>* (p', {#}) ({#}, r)) \<bind> (\<lambda>pq. SPEC (normalize_poly_p\<^sup>*\<^sup>* pq))"
algebra_iff_Un: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<Omega> - a \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M))"
pmf_embed_pmf: fixes x :: "'a" shows "pmf local.embed_pmf x = f x"
r2f_def: fixes R :: "('a \<times> 'b) set" shows "\<F> R = (``) R \<circ> \<eta>"
assign_post_valid: fixes P :: "'BExp set"   and mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and e :: "'AExp" assumes "pred P mem" shows "pred (assign_post P x e) (mem(x := ev\<^sub>A mem e))"
dvd_div_mult_self: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m"   and b :: "'m" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times (divide b a) a = b"
relcompp_apply: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and c :: "'d" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
join_return: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N"   and "subprob_space M" shows "join (return (subprob_algebra N) M) = M"
simps_1: fixes ys :: "['b]" shows "[] ++ ys = ys"
suffix_drop: fixes n :: "nat"   and as :: "'c list" shows "suffix (drop n as) as"
ASSUME_\<Theta>_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and f\<^sub>0 :: "'b \<Rightarrow> 'c"   and s\<^sub>0 :: "'b"   and R :: "('d \<times> 'c) set"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'd) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "ASSUME_\<Theta> \<pi> f\<^sub>0 s\<^sub>0 R \<Theta> \<equiv> HT'set_r (\<lambda>f' s'. (f' s', f\<^sub>0 s\<^sub>0) \<in> R) \<pi> \<Theta>"
preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "S.dom (F f) = F (Cop.dom f)"
up_def: shows "up = lift (\<lambda>d lm p \<alpha>. snd (up' d lm p \<alpha>))"
bij_betw_apply: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and a :: "'c" assumes "bij_betw f A B"   and "a \<in> A" shows "f a \<in> B"
wfrestrictedvars\<^sub>s\<^sub>t_def: fixes S :: "('b, 'a) strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfrestrictedvars\<^sub>s\<^sub>t\<^sub>p S))"
DERIV_shift: fixes f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" shows "(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"
set_remove1_eq: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs" shows "set (remove1 x xs) = set xs - {x}"
out_one_side: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B X \<or> \<not> Tarski_neutral_dimensionless.Col Bet A B Y"   and "Tarski_neutral_dimensionless.Out Bet A X Y" shows "Tarski_neutral_dimensionless.OS Bet A B X Y"
analz_trans: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> analz G"   and "G \<subseteq> analz H" shows "X \<in> analz H"
intros_2: fixes n :: "nat"   and l :: "'a tree23"   and r :: "'a tree23"   and p :: "int \<times> 'a" assumes "full n l"   and "full n r" shows "full (Suc n) (Branch2 l p r)"
cnorm: shows "c \<in> {normal_vector D, - normal_vector D}"
spmf_lub_spmf: fixes Y :: "'c spmf set"   and x :: "'c" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "spmf (lub_spmf Y) x = (\<Squnion>p\<in>Y. spmf p x)"
before_in_index2: fixes l :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 1 else 0) = index l y"
appL: fixes s :: "trm"   and u :: "trm"   and t :: "trm" assumes "s \<longlonglongrightarrow> u" shows "s \<degree> t \<longlonglongrightarrow> u \<degree> t"
indep_vars_def: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and I :: "'b set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
bi_total_alt_def: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
all_finite_subset_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) = (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c heap list \<Rightarrow> 'b"   and heap :: "'c heap" shows "P (case heap of heap.Empty \<Rightarrow> f1 | Hp x xa \<Rightarrow> f2 x xa) = ((heap = heap.Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22. heap = Hp x21 x22 \<longrightarrow> P (f2 x21 x22)))"
pa_step_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa"   and S :: "('a \<times> 'b, unit) HashMap.hashmap \<times> ('a \<times> 'b) list \<times> ('a \<times> 'b, 'c) ta_rule dlist" shows "pa_step H1 H2 S \<equiv> let (Q, W, \<delta>d) = S; (q1, q2) = hd W in ls.iterate (hta_lookup_s q1 H1) (\<lambda>r1. ls.iterate (hta_lookup_sf q2 (rhsl r1) H2) (\<lambda>r2 res. if length (rhsq r1) = length (rhsq r2) then let rp = r_prod r1 r2; (Q, W, \<delta>d) = res; (Q', W') = pa_upd_rule Q W (rhsq rp) in (Q', W', ls.ins_dj rp \<delta>d) else res)) (Q, tl W, \<delta>d)"
simps_1: fixes a :: "'c" shows "anno a com.SKIP = SKIP {a}"
dest_port_ipv4_TCPUDP_def: fixes x :: "int \<times> ((int \<times> int \<times> int \<times> int) \<times> int \<times> protocol) \<times> ((int \<times> int \<times> int \<times> int) \<times> int \<times> protocol) \<times> 'a" shows "dest_port x \<equiv> (fst \<circ> snd \<circ> fst \<circ> snd \<circ> snd) x"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'b llist" assumes "enat m < n" shows "lnth (ltake n xs) m = lnth xs m"
case_3: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x31 :: "('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool"   and x32 :: "'d"   and x33 :: "'e"   and x34 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list" shows "(case x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f3 x31 x32 x33 x34"
ZObis_pres_discr_L: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and d :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "(c, d) \<in> PL_Indis.ZObis aval tval cval indis"   and "PL_Indis.discr aval tval cval indis d" shows "PL_Indis.discr aval tval cval indis c"
branch_prob_altdef: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure (set vs') \<Gamma>)" shows "branch_prob \<Y> \<rho> = \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge (set vs) (set vs') (x, \<rho>)) \<delta>)) \<partial>state_measure (set vs) \<Gamma>"
c: shows "inf_concat_simple f (k + l) = (i, j + l)"