hcju/isabelleps huggingface dataset:
abs_ast_goal_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.abs_ast_goal problem \<equiv> map_of (ast_problem.astG problem)"
rBang: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "Prop \<Psi> (P \<parallel> !P) M N P'"   and "guarded P" shows "Prop \<Psi> (!P) M N P'"
interaction_bounded_by_map_gpv_id: fixes P :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and f :: "'b \<Rightarrow> 'd" assumes "interaction_bounded_by P gpv n" shows "interaction_bounded_by P (map_gpv f id gpv) n"
fresh_psubst: fixes z :: "name"   and t :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "z \<sharp> t"   and "z \<sharp> \<theta>" shows "z \<sharp> \<theta><t>"
scale: fixes r :: "real"   and b :: "(real, 'n) vec" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
prime_one_plus_i_gauss_int: shows "prime (1 + \<i>\<^sub>\<int>)"
nat_filtration_borel_measurable_init: fixes f :: "bool stream \<Rightarrow> 'a" assumes "f \<in> borel_measurable (nat_filtration 0)" shows "f = (\<lambda>w. f (sconst True))"
The_on_I2: fixes a :: "'c"   and U :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "a \<in> U"   and "P a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> x = a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> Q x" shows "Q (the (THE x on U. P x))"
degree_div_mod_smult: fixes g :: "int poly"   and j :: "nat"   and r :: "int poly"   and d :: "nat"   and u :: "int poly"   and q :: "int poly"   and m :: "int" assumes "degree g < j"   and "degree r < d"   and "degree u = d"   and "g = q * u + smult m r"   and "q \<noteq> 0"   and "m \<noteq> 0" shows "degree q < j - d"
nodes_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.nodes prob = ifc_problem.nodes prob"
is_pd_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a" shows "IFC_def.is_pd prob y x = (x \<in> IFC_def.nodes prob \<and> (\<forall>\<pi> n. IFC_def.is_path prob \<pi> \<and> \<pi> 0 = x \<and> \<pi> n = IFC_def.return prob \<longrightarrow> (\<exists>k\<le>n. \<pi> k = y)))"
blinding_of_G_unfold: fixes rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "blinding_of_G rha boa rhb bob = vimage2p the_G\<^sub>m the_G\<^sub>m (blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob)"
simps_1: fixes n :: "nat" shows "string_to_nat_aux n [] = n"
assms_3: shows "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
can_syncE: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>" obtains taa :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "t \<turnstile> \<langle>x, m\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle>" and "LT = collect_waits taa"
delete_edges_list_wf: fixes G :: "'b graph"   and E :: "('b \<times> 'b) list" assumes "wf_graph G" shows "wf_graph (delete_edges_list G E)"
Deduct2_axioms_def: fixes fmla :: "'a set"   and Fvars :: "'a \<Rightarrow> 'b set"   and prv :: "'a \<Rightarrow> bool"   and bprv :: "'a \<Rightarrow> bool" shows "Deduct2_axioms fmla Fvars prv bprv \<equiv> \<forall>\<phi>. \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {} \<longrightarrow> bprv \<phi> \<longrightarrow> prv \<phi>"
leD: fixes y :: "'b list"   and x :: "'b list" assumes "prefix y x" shows "\<not> strict_prefix x y"
le_INF_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and u :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq u (Inf (f ` A)) = (\<forall>i\<in>A. less_eq u (f i))"
basis_def: fixes X :: "nat uprod set" shows "s.SpanningForest X = (subforest X \<and> (\<forall>x\<in>E - X. \<not> subforest (insert x X)))"
uminus_sum_list_order2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and ss :: "'b list" assumes "class.group_add minus plus zero uminus"   and "\<forall>s\<in>set ss. plus s s = zero" shows "uminus (monoid_add.sum_list plus zero ss) = monoid_add.sum_list plus zero (rev ss)"
mix_of_preferred_is_preferred: fixes p :: "'a pmf"   and w :: "'a pmf"   and q :: "'a pmf"   and \<alpha> :: "real" assumes "p \<succeq>[\<R>] w"   and "q \<succeq>[\<R>] w"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> p q \<succeq>[\<R>] w"
set_bset_inverse: fixes x :: "'d set['e]" shows "Abs_bset (set_bset x) = x"
sup_assoc_1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.pseudo_hoop_lattice sup times inf imp_l less_eq less one imp_r" shows "sup a (sup b c) = sup (sup a b) c"
psen_ex: fixes s :: "'state"   and a :: "'action"   and p :: "'process" assumes "s \<in> nodes"   and "a \<in> {a. en a s} - pac p" shows "psen p (ex a s) = psen p s"
map_SSup_def: fixes F :: "'c downset \<Rightarrow> 'd downset" shows "map_SSup F = SSup \<circ> F \<circ> ds"
subst_compose: fixes \<sigma> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<tau> :: "'f \<Rightarrow> ('c, 'd) Term.term"   and x :: "'e" shows "(\<sigma> \<circ>\<^sub>s \<tau>) x = \<sigma> x \<cdot> \<tau>"
deterministic_def: fixes P :: "'a process" shows "deterministic P \<equiv> \<forall>xs\<in>traces P. \<forall>X. (X \<in> refusals P xs) = (X \<inter> next_events P xs = {})"
partCompat_Un: fixes P :: "'a set set"   and Q :: "'a set set"   and theta :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "partCompat (P \<union> Q) theta f = (partCompat P theta f \<and> partCompat Q theta f)"
Sup_pres_ladj: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_pres f" shows "\<exists>g. f \<stileturn> g"
shiftl1_bl: fixes w :: "'b word" shows "shiftl1 w = of_bl (to_bl w @ [False])"
set_intros_1: fixes x1 :: "'a"   and x2 :: "'a stream" shows "x1 \<in> i.R (x1 ## x2)"
prems_5: shows "getWatch1 stateA = getWatch1 stateB"
devExt_devop: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n\<^sub>s :: "nat"   and n' :: "nat" shows "prefix (devExt t n nid n\<^sub>s (Suc n')) (devExt t n nid n\<^sub>s n') \<or> (\<exists>b. devExt t n nid n\<^sub>s (Suc n') = devExt t n nid n\<^sub>s n' @ [b]) \<and> \<not> Option.is_none (devBC t n nid (n\<^sub>s + Suc n')) \<and> \<parallel>the (devBC t n nid (n\<^sub>s + Suc n'))\<parallel>\<^bsub>t (n\<^sub>s + Suc n')\<^esub> \<and> n\<^sub>s + Suc n' \<le> n \<and> mining (\<sigma>\<^bsub>the (devBC t n nid (n\<^sub>s + Suc n'))\<^esub>t (n\<^sub>s + Suc n'))"
characteristic_iso_2: fixes f :: "'a" assumes "Right.ide f" shows "Right.in_hom (f \<star> \<i>[src f]) (f \<star> src f \<star> src f) (f \<star> src f)"
ProP: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and j :: "'c"   and x :: "'d" assumes "Hintikka A H"   and "Nom j at i in' H"   and "Pro x at j in' H" shows "\<not> (\<^bold>\<not> Pro x) at i in' H"
simps: fixes a1 :: "nat \<Rightarrow> 'n"   and a2 :: "nat" shows "((a1, a2) \<in> scp) = ((\<exists>h \<sigma> n. a1 = path \<sigma> \<and> a2 = n \<and> h \<in> local.hvars \<and> h \<in> local.reads (path \<sigma> n) \<and> (\<forall>k<n. h \<notin> local.writes (path \<sigma> k))) \<or> (\<exists>\<pi> m n. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> scp \<and> n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m) \<or> (\<exists>\<pi> m n v. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> scp \<and> n dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m) \<or> (\<exists>\<pi> m \<pi>' m' n v. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> scp \<and> (\<pi>', m') \<in> scp \<and> n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m'))"
simps: fixes A :: "'a set" shows "random_bst A = (if infinite A \<or> A = {} then return_pmf \<langle>\<rangle> else pmf_of_set A \<bind> (\<lambda>x. random_bst {y \<in> A. y < x} \<bind> (\<lambda>l. random_bst {y \<in> A. x < y} \<bind> (\<lambda>r. return_pmf \<langle>l, x, r\<rangle>))))"
R: fixes a :: "'a" shows "Restr (relations M a) T = Restr (relations M' a) T"
ab1: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "a - x * a + x * b \<le> b"
closed_injective_imp_proper_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "closed_map X Y f"   and "inj_on f (topspace X)" shows "proper_map X Y f"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>" shows "hom'.arr f"
rtrancl_beta_AppL: fixes s :: "lam"   and s' :: "lam"   and t :: "lam" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t"
boundsl_def: fixes s :: "('d, 'c) state" shows "\<B>\<^sub>l s = map_option snd \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>l s)"
leg_simps_4: shows "C.dom Dom\<mu>_Dom\<nu>.leg1 = Dom\<mu>_Dom\<nu>.apex"
mem_0_Ord: fixes k :: "V" assumes "Ord k"   and "k \<noteq> 0" shows "0 \<in>\<^sub>\<circ> k"
sup_least: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e"   and z :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x"   and "less_eq z x" shows "less_eq (sup y z) x"
homeomorphic_map_maps: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "homeomorphic_map X Y f = (\<exists>g. homeomorphic_maps X Y f g)"
lcm_gcd_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm" shows "lcm x (gcd y z) = gcd (lcm x y) (lcm x z)"
subset_path_image_join: fixes g1 :: "real \<Rightarrow> 'c"   and s :: "'c set"   and g2 :: "real \<Rightarrow> 'c" assumes "path_image g1 \<subseteq> s"   and "path_image g2 \<subseteq> s" shows "path_image (g1 +++ g2) \<subseteq> s"
assms_3: shows "\<pi> =\<^bsub>n\<^esub> \<pi>'"
self_financing_def: fixes Mkt :: "('b, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real" shows "self_financing Mkt p = (\<forall>n. val_process Mkt p (Suc n) = cls_val_process Mkt p (Suc n))"
has_vector_derivative_at_within: fixes f :: "real \<Rightarrow> 'c"   and f' :: "'c"   and x :: "real"   and S :: "real set" assumes "(f has_vector_derivative f') (at x)" shows "(f has_vector_derivative f') (at x within S)"
pc_def: shows "pc = p \<circ>\<^sub>p [:st, tt - st:]"
position_in_def: fixes p :: "indices list"   and s :: "'b trm" shows "position_in p s = (\<exists>t. subterm s p t)"
HEndPhase2_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv5_inner s q"   and "p \<noteq> q" shows "HInv5_inner s' q"
simps: fixes v1 :: "'form \<times> nat list"   and p1 :: "('form, 'var) out_port"   and v2 :: "'form \<times> nat list"   and p2 :: "('form, 'var) antecedent" shows "edge_begin ((v1, p1), v2, p2) = v1"
rep_eq: fixes x :: "'b" shows "blinfun_apply (blinfun_inner_right x) = (\<bullet>) x"
reds_length: fixes E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>" shows "length es = length es'"
coeff_int_def: fixes p :: "'a poly"   and i :: "int" shows "coeff_int p i = (if i < 0 then 0::'a else poly.coeff p (nat i))"
\<T>_equals_\<Z>_induct_step: fixes k :: "nat"   and u :: "nat" assumes "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u))"   and "\<T> k u = \<Z> k u" shows "\<T> k (Suc u) = \<Z> k (Suc u)"
filtermap_tan_at_left_inf: fixes a :: "real" assumes "cos a = 0" shows "filtermap tan (at_left a) = at_top"
s_mul_ext_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "s_mul_ext ns s \<equiv> (mult2_alt_s (ns\<inverse>) (s\<inverse>))\<inverse>"
combine_pairs'_prop: fixes \<P> :: "((('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'a set \<times> ('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'a set \<times> ('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set) set"   and \<delta> :: "('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<Rightarrow> 'a set \<Rightarrow> 'a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option"   and q\<^sub>0 :: "'a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option"   and w :: "nat \<Rightarrow> 'a set" shows "(\<forall>P\<in>\<P>. accepting_pair\<^sub>R \<delta> q\<^sub>0 P w) = accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (local.combine_pairs' \<P>) w"
gcd_cong_l: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x \<sim> x'"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "y \<in> carrier G" shows "somegcd G x y \<sim> somegcd G x' y"
Nml_Nmlize_2: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Src \<^bold>\<lfloor>t\<^bold>\<rfloor> = E.Src t"
phi_def: shows "phi \<equiv> init \<and> \<box>[m1 \<or> m2]_(x, y) \<and> Live"
assms_6: shows "slist_1 \<in> state_set (state_list s as)"
cSup_singleton: fixes x :: "'c multiset" shows "Sup {x} = x"
alignUp_div_helper: fixes k :: "nat"   and n :: "nat"   and x :: "'b word"   and a :: "'b word" assumes "k < 2 ^ (LENGTH('b) - n)"   and "x = 2 ^ n * word_of_nat k"   and "a \<le> x"   and "n < LENGTH('b)"   and "a mod 2 ^ n \<noteq> 0" shows "a div 2 ^ n < word_of_nat k"
K2_centre_def: shows "K2_centre = proj2_abs (vector [0, 0, 1])"
cond1_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and bs :: "'b list"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "cond1 f bs y xs ys \<equiv> (\<exists>b. b \<in> set bs \<and> fst (f b y) \<and> snd (mul_ext f (remove1 b xs) (filter (\<lambda>y. \<not> fst (f b y)) ys))) \<or> (\<exists>b. b \<in> set bs \<and> snd (f b y) \<and> fst (mul_ext f (remove1 b xs) ys))"
setmult_subset_G: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"   and "K \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)" shows "H <#>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> K \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"
baEANow: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'c"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'c"   and n :: "nat"   and \<phi> :: "'c \<Rightarrow> bool" assumes "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)"   and "active c (t n)" shows "\<phi> (tCMP c (t n))"
preorder_family_def: fixes dom :: "'a set"   and carrier :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "preorder_family dom carrier R \<equiv> dom \<noteq> {} \<and> (\<forall>i. i \<in> dom \<longrightarrow> Order_Predicates.preorder_on carrier (R i)) \<and> (\<forall>i x y. i \<notin> dom \<longrightarrow> \<not> R i x y)"
hyps_8: shows "[a] @- b\<^sub>1 @- u' =\<^sub>I u"
genideal_zero: shows "Idl\<^bsub>\<Z>\<^esub> {0} = {0}"
signed_word_eqI: fixes v :: "'c word"   and w :: "'c word" assumes "signed v = signed w" shows "v = w"
poly_no_roots_greater: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x>a. poly p x \<noteq> 0) = (p \<noteq> 0 \<and> count_roots_above p a = 0)"
bind_def: fixes x :: "('e, 'g) state"   and f :: "'g \<Rightarrow> ('e, 'f) state" shows "x \<bind> f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
dictator_def: fixes F :: "(indi \<Rightarrow> alt \<Rightarrow> real) \<Rightarrow> alt \<Rightarrow> real"   and i :: "indi" assumes "arrow F" shows "arrow.dictator F i \<equiv> \<forall>a b. arrow.dictates F i a b"
im_uminus: fixes g :: "'g" shows "T (- g) = - T g"
finite': fixes P :: "'a \<Rightarrow> bool" assumes "finite {x. \<not> P x}" shows "eventually P F"
l1_inv2_def: shows "l1_inv2 \<equiv> {s. \<forall>gnx A B Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> in_progressS (progress s Rb) {xny, xgnx, xgny, xsk} \<longrightarrow> guessed_frame Rb xgnx = Some gnx \<longrightarrow> can_signal s A B \<longrightarrow> 0 < signalsInit s (Running A B (Exp gnx (NonceF (Rb $ ny))))}"
adopt_node_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> Shadow_DOM.adopt_node_locs parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
ord_pp_strict_def: fixes cmp_term :: "'b nat_term_order"   and s :: "'b"   and t :: "'b" shows "ord_pp_strict_punit cmp_term s t = lt_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))"
not_psubset_empty: fixes A :: "'a set" shows "\<not> A \<subset> {}"
open_segment_commute: fixes a :: "'b"   and b :: "'b" shows "{a<--<b} = {b<--<a}"
one_less_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "0 < n" shows "less one (power.power one times a n)"
nat_to_bv_length_bv_to_nat: fixes xs :: "bit list"   and n :: "nat" assumes "length xs = n"   and "xs \<noteq> []" shows "nat_to_bv_length (bv_to_nat xs) n = xs"
power_down_neg_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_down prec b n < 0) = (b < 0 \<and> odd n)"
welldef_def: shows "welldef \<equiv> {F. Init F \<noteq> {}}"
mult_less_cancel_left_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less (times c a) (times c b) = less a b"
simps_1: fixes a :: "('c, 'd) Node" shows "nlcons a Empty = Single a"
not_None_eq: fixes x :: "'f option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
bind_scale_spmf: fixes r :: "real"   and p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" assumes "r \<le> 1" shows "scale_spmf r p \<bind> f = p \<bind> (\<lambda>x. scale_spmf r (f x))"
img_is_subfield_2: fixes K :: "'a set" assumes "subfield K R"   and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>" shows "subfield (h ` K) S"
BIJ: shows "bij_betw (f(B\<^sub>1 := u)) (P\<^sub>1 \<union> wrap B\<^sub>1) (\<Union> (P\<^sub>2 \<union> wrap (B\<^sub>2 \<union> {u})))"
comp_fun_commute_axioms: shows "comp_fun_commute tcf"
pmf_transition_of_st_mat: fixes A :: "'a st_mat"   and a :: "'a"   and i :: "'a" shows "pmf (transition_of_st_mat A a) i = st_mat A $h i $h a"
simps_1: fixes n :: "nat"   and l :: "real" shows "T Ins (n, l) = (if real n + 1 \<le> f2 * l then 1 else real n + 1)"
geodesic_segment_param_def: fixes G :: "'b set"   and x :: "'b"   and t :: "real" shows "geodesic_segment_param G x t = (if \<exists>w. w \<in> G \<and> dist x w = t then SOME w. w \<in> G \<and> dist x w = t else SOME w. w \<in> G)"
all_pairs_set: fixes xs :: "'a list" shows "set (all_pairs xs) = set xs \<times> set xs"
mult_eq': shows "(\<otimes>\<^bsub>R\<^esub>) = (\<lambda>x y. x * y mod n)"
sep_conjE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" obtains x :: "'a" and y :: "'a" where "P x" and "Q y" and "sep_disj x y" and "h = plus x y"
map_poly_inj_idom_hom_axioms: shows "map_poly_inj_idom_hom of_int"
A30: shows "A3 = 0\<^sub>m n 1"
prob_space_distr: fixes f :: "'s stream \<Rightarrow> 'a"   and s :: "'s"   and M' :: "'a measure" assumes "K.T.random_variable s M' f" shows "prob_space (distr (K.T s) M' f)"
r: shows "\<forall>v. r v \<noteq> (0::'a) \<longrightarrow> v \<in> B"
regular_event_preserves_process_snapshots: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and r :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "regular_event ev" shows "ps c r = ps c' r"
context_unique: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty"   and U :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>"   and "(x, U) \<in> set \<Gamma>" shows "T = U"
low_proc_non0\<^sub>2_def: fixes s\<^sub>L :: "stmt" shows "low_proc_non0\<^sub>2 s\<^sub>L \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> the (\<sigma> ''lowIn'') \<noteq> 0 \<and> s\<^sub>L \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''lowOut'') = the (\<sigma> ''lowIn'') + 1"
simps_2: fixes i :: "nat"   and k :: "nat"   and x :: "'b"   and xs :: "'b list" shows "take_nths i k (x # xs) = (if i = 0 then x # take_nths (2 ^ k - 1) k xs else take_nths (i - 1) k xs)"
succeed_t_def: fixes i :: "nat" shows "succeed_t i = {n. \<exists>q. state_rank q n = Some i \<and> q \<notin> F - {q\<^sub>0} \<and> \<delta> q (w n) \<in> F}"
imod_iless: fixes m :: "enat"   and n :: "enat" assumes "m < n" shows "m mod n = m"
assms_2: shows "lt p \<prec>\<^sub>t v"