hcju/isabelleps huggingface dataset:
case_23: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x23 :: "v" shows "(case C x23 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f23 x23"
Class_self: fixes a :: "'a" assumes "a \<in> S" shows "a \<in> orbit.Class a"
fps_sin_fps_exp_ii: fixes c :: "complex" shows "fps_sin c = (fps_exp (\<i> * c) - fps_exp (- \<i> * c)) / fps_const (2 * \<i>)"
igFreshIGAbs1STR_imp_igFreshIGAbs1: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igFreshIGAbs1STR MOD" shows "igFreshIGAbs1 MOD"
fine_Nil_iff: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real" shows "fine \<delta> (a, b) [] = (a = b)"
length_zero_is_nil: fixes xs :: "'b list" assumes "length xs = 0" shows "xs = []"
sublist_CONS1_E: fixes h :: "'c"   and l1 :: "'c list"   and l2 :: "'c list" assumes "subseq (h # l1) l2" shows "subseq l1 l2"
I_lift_dnfeq_qe: fixes qe :: "'a list \<Rightarrow> 'a fm"   and \<phi> :: "'a fm"   and xs :: "'b list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> \<not> solvable\<^sub>0 a \<Longrightarrow> is_dnf_qe qe as" shows "I (lift_dnfeq_qe qe \<phi>) xs = I \<phi> xs"
mod_ring_rel_integer_def: fixes x :: "integer"   and y :: "'a mod_ring" shows "mod_ring_rel_integer x y = (\<exists>z. urel_integer x z \<and> mod_ring_rel z y)"
simps_2: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.wt0 wt_sym (Sym f) = PNum (wt_sym f)"
deriv_Q: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
append_Tf: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" shows "Tf t1 \<sqinter> Tf t2 <:: Tf (t1 @ t2)"
step: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set"   and l'' :: "'d"   and Z'' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l'', Z''\<rangle>"
IH: fixes c :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "\<And>w. (\<Sum>i\<le>n. c i * w ^ i) = (0::'a)"   and "k \<le> n" shows "c k = (0::'a)"
periodic: fixes x :: "real" shows "g (x + 2 * pi) = g x"
properfactor_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" shows "properfactor G a b = (a divides\<^bsub>G\<^esub> b \<and> \<not> b divides\<^bsub>G\<^esub> a)"
swapVal_preserves_sWls1: fixes SEM :: "('g, 'h, 'i, 'sort, 'j, 'k, 'l) semDom_scheme"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'k"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "sWlsVal SEM val" shows "sWlsVal SEM (val ^[z1 \<and> z2]_zs)"
step_thread_Some_SomeD: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'a" assumes "\<alpha>.deterministic I"   and "\<alpha>.step_thread update_state s t' = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "s \<in> I" shows "\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> \<alpha>.r_syntax t x (shr s) ta x' m' \<and> \<alpha>.actions_ok s t ta \<and> \<sigma>' = update_state ta"
Proj_range_closed: fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "is_Proj P" shows "closed (range ((*\<^sub>V) P))"
not_prefix_cases: fixes ps :: "'c list"   and ls :: "'c list" assumes "\<not> prefix ps ls" obtains "ps \<noteq> []" and "ls = []"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = a__ # as__" and "ls = x__ # xs__" and "x__ = a__" and "\<not> prefix as__ xs__"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = a # as" and "ls = x # xs" and "x \<noteq> a"
data_dependency_consistent_instrs_issue_expr: fixes T :: "nat set"   and t :: "nat"   and e :: "expr" shows "data_dependency_consistent_instrs T (issue_expr t e)"
subst_heap_delete: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and z :: "Vars.var" shows "(delete x \<Gamma>)[y::h=z] = delete x \<Gamma>[y::h=z]"
meval_MPred': fixes n :: "nat"   and t :: "nat"   and db :: "(char list \<times> 'b list) set"   and e :: "char list"   and ts :: "'b MFOTL.trm list" shows "meval n t db (MPred e ts) = ([Option.these (\<Union>(e', x)\<in>db. if e = e' then {map_option (\<lambda>f. tabulate f 0 n) (match ts x)} else {})], MPred e ts)"
u_map_map: fixes f :: "'f \<rightarrow> 'e"   and g :: "'g \<rightarrow> 'f"   and p :: "'g\<^sub>\<bottom>" shows "u_map\<cdot>f\<cdot>(u_map\<cdot>g\<cdot>p) = u_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>p"
abs_inner: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'b"   and x :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (abs x) i = \<bar>inner x i\<bar>"
intro: fixes G :: "('c, 'd) graph" assumes "valid_graph G"   and "connected_graph_axioms G" shows "connected_graph G"
IH: assumes "spec_ops op_list deps"   and "(oid, op1) \<in> set op_list"   and "(oid, op2) \<in> set op_list" shows "op1 = op2"
lookup_raw_update_by_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k1 :: "'a"   and v :: "'c"   and k2 :: "'a" assumes "oalist_inv xs" shows "lookup_raw (update_by_raw (k1, v) xs) k2 = (if k1 = k2 then v else lookup_raw xs k2)"
eq_fold: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "Max (insert x A) = Finite_Set.fold max x A"
ai_def: shows "ai = f2 / (f2 - f2 / e)"
inject_3: fixes x3 :: "'d"   and y3 :: "'d" shows "(In3 x3 = In3 y3) = (x3 = y3)"
pred_intros_logic_2: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "Measurable.pred M P" shows "Measurable.pred M (\<lambda>x. \<not> P x)"
get_add_neq: fixes empty :: "'c"   and get :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and add :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and to_list :: "'c \<Rightarrow> ('d \<times> 'e) list"   and x :: "'d"   and y :: "'d"   and e :: "'c"   and v :: "'e" assumes "env empty get add to_list"   and "x \<noteq> y" shows "get (add e x v) y = get e y"
maxsimp_vertices: fixes x :: "'a set" assumes "folding_g.chamber x" shows "order.greater_eq (\<Union> X) x"
in_TER_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g"   and "x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "x \<in> TER\<^bsub>quotient_web \<Gamma> f\<^esub> g" shows "x \<in> TER\<^bsub>\<Gamma>\<^esub> (plus_current f g)"
assms_4: shows "\<forall>x. (f \<circ> g) x \<le> x"
HSem_subst_expr_below: fixes e1 :: "'exp"   and x :: "'var"   and e2 :: "'exp"   and \<Gamma> :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" assumes "\<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub>" shows "\<lbrace> (x, e1) # \<Gamma> \<rbrace>\<rho> \<sqsubseteq> \<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>"
min_lvar: shows "min_lvar_not_in_bounds s' = Some x\<^sub>i"
f': shows "set_borel_measurable M (A \<union> B) f"
spec\<^sub>6_def: fixes p :: "prog" shows "spec\<^sub>6 p \<equiv> para p = [''x'', ''y''] \<and> body p = Add (Var ''x'') (Double (Add (Var ''x'') (Var ''y'')))"
x\<beta>: shows "x \<in> elts (\<alpha> \<up> \<beta>)"
simps_1: fixes f :: "'d \<Rightarrow> 'b"   and t :: "'a"   and x :: "'d"   and m :: "'c" shows "convert_new_thread_action f (NewThread t x m) = NewThread t (f x) m"
mset_subset_eq_mono_add_right_cancel: fixes A :: "'c multiset"   and C :: "'c multiset"   and B :: "'c multiset" shows "(A + C \<subseteq># B + C) = (A \<subseteq># B)"
finite_nonempty_carrier_has_maximum: assumes "finite (\<P> outcomes)"   and "\<P> outcomes \<noteq> {}" shows "\<exists>e\<in>\<P> outcomes. \<forall>m\<in>\<P> outcomes. e \<succeq> m"
Times: fixes x :: "'a"   and A :: "'b set"   and B :: "'c set" assumes "x \<in> space M"   and "A \<in> sets N"   and "B \<in> sets L" shows "emeasure (f x \<Otimes>\<^sub>M g x) (A \<times> B) = emeasure (f x) A * emeasure (g x) B"
simps_2: fixes x :: "'b"   and t :: "'b tree" shows "exec (Splay x) [t] = splay x t"
\<Delta>\<Phi>_insert: fixes hp :: "'b tree"   and x :: "'b" assumes "is_root hp" shows "\<Phi> (Pairing_Heap_Tree.insert x hp) - \<Phi> hp \<le> log 2 (real (size hp + 1))"
tendsto_max: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "'b \<Rightarrow> 'a"   and x :: "'a"   and net :: "'b filter"   and Y :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "filterlim X (topological_space.nhds open x) net"   and "filterlim Y (topological_space.nhds open y) net" shows "LIM x net. ord.max less_eq (X x) (Y x) :> topological_space.nhds open (ord.max less_eq x y)"
arcsin_arccos_sqrt_pos: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "arcsin x = arccos (sqrt (1 - x\<^sup>2))"
hyps_1: shows "liftAll (\<lambda>a. qSkel (a #[[x \<and> y]]_zs) = qSkel a) inp"
similar_mat_witD_1: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "P * Q = 1\<^sub>m n"
EmptyMap_OneAncestor: fixes SA :: "('d, 'e, 'f) seqauto" shows "OneAncestor {SA} (EmptyMap (States SA))"
simps_9: fixes v1 :: "'a"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (V v1) (Minus v va) = True"
in_set_zip: fixes p :: "'a \<times> 'b"   and xs :: "'a list"   and ys :: "'b list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
op_ls_empty_def: shows "op_ls_empty = op_set_empty"
FAccThrow: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and F :: "char list"   and Cs :: "char list 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" shows "P,E \<turnstile> \<langle>Throw r\<bullet>F{Cs},s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
distinct_25: shows "IndexOfOp \<noteq> string_binop.GreaterEqOp"
oth_class_taut_5_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> \<phi> \<^bold>& \<^bold>\<not>\<psi> in v]"
ipresIGVar_def: fixes h :: "'x \<Rightarrow> 'y"   and MOD :: "('z, 'aa, 'ab, 'ac, 'ad, 'ae, 'x, 'af, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ab, 'aj, 'ak, 'ae, 'y, 'al, 'am) model_scheme" shows "ipresIGVar h MOD MOD' \<equiv> \<forall>xs x. h (igVar MOD xs x) = igVar MOD' xs x"
mrNoneMT: fixes r :: "'b"   and p :: "'b list"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'c" assumes "r \<in> set p"   and "applied_rule_rev C x p = \<bottom>" shows "x \<notin> dom (C r)"
Cauchy_converges_subseq: fixes u :: "nat \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> nat"   and l :: "'b" assumes "Cauchy u"   and "strict_mono r"   and "(u \<circ> r) \<longlonglongrightarrow> l" shows "u \<longlonglongrightarrow> l"
stricts_2: shows "PRJ('e)\<cdot>\<bottom> = \<bottom>"
subseq_filter_left: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "subseq (filter P xs) xs"
cs_sorted_list_of_cd': fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" shows "cs\<^bsup>\<pi>\<^esup> k = map \<pi> (sorted_list_of_set {i. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i}) @ [\<pi> k]"
l1_inv2_def: shows "l1_inv2 \<equiv> {s. \<forall>KE A B Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> progress s Rb = Some {xpkE, xsk} \<longrightarrow> guessed_frame Rb xpkE = Some KE \<longrightarrow> can_signal s A B \<longrightarrow> 0 < l1_state.signals s (Running A B \<langle>KE, NonceF (Rb $ sk)\<rangle>)}"
start_atom: fixes a :: "'b" shows "start (atom a) = [True]"
circ_sup_one_right_unfold: fixes x :: "'a" assumes "reflexive x" shows "x\<^sup>\<star> * x = x\<^sup>\<star>"
HIGH_ASSIGN: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and e :: "Expr" assumes "CONTEXT x = high" shows "G \<rhd> Assign x e : twiddle"
right: fixes x :: "'a"   and t :: "'a tree" assumes "x \<in> set_tree (right t)"   and "P x (right t)" shows "P x t"
free_tv_comp_subst: fixes s1 :: "nat \<Rightarrow> typ"   and s2 :: "nat \<Rightarrow> typ" shows "free_tv (\<lambda>u. $ s1 (s2 u)) \<subseteq> free_tv s1 \<union> free_tv s2"
DIAndSubst342_def: shows "DIAndSubst342 = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Pc (Inl pid3) && Pc (Inl pid4)) else if C = pid2 then Some (Pc (Inl pid4)) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
finite_fv\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('d, 'c) strand_step" shows "finite (fv\<^sub>s\<^sub>t\<^sub>p x)"
bind_map_option: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e option"   and g :: "'d \<Rightarrow> 'c option" shows "map_option f x \<bind> g = x \<bind> g \<circ> f"
measurable_marg_dens2': fixes x :: "nat"   and y :: "nat" assumes "x \<in> V"   and "y \<in> V" shows "(\<lambda>(x, y). marg_dens2 \<Y> x y x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"
WTVal: fixes v :: "val"   and T :: "ty"   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 :: "char list \<Rightarrow> ty option" assumes "typeof v = \<lfloor>T\<rfloor>" shows "P,E \<turnstile> Val v :: T"
l2_lkr_others_def: fixes A :: "agent" shows "l2_lkr_others A \<equiv> {(s, s'). A \<noteq> test_owner \<and> A \<noteq> test_partner \<and> s' = s\<lparr>l2_state.bad := {A} \<union> l2_state.bad s\<rparr>}"
grd_def: fixes S :: "'e \<Rightarrow> 'd" shows "grd S = - S \<bottom>"
simps_1: fixes x :: "name"   and a :: "coname"   and d :: "coname"   and z :: "name"   and P :: "trm" shows "Ax x a{d:=z.P} = (if d = a then Cut <a>.Ax x a z.P else Ax x a)"
ereal_uminus_uminus: fixes a :: "ereal" shows "- (- a) = a"
\<omega>_dec_of_Cantor_aux: fixes \<alpha> :: "V"   and \<alpha>s :: "V list"   and ms :: "nat list"   and m :: "nat" assumes "sorted_wrt (\<lambda>x y. y < x) (\<alpha> # \<alpha>s)"   and "length \<alpha>s = length ms" shows "sorted_wrt (\<lambda>x y. y \<le> x) (\<omega>_of_Cantor (\<alpha> # \<alpha>s) (m # ms))"
hyps: shows "\<forall>S\<in>M. S \<subseteq> V \<and> (\<forall>x. x \<in> S \<longrightarrow> (\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> x S W))"
induct: fixes P4 :: "(cell \<times> interface) list \<Rightarrow> bool"   and P1 :: "sdiagram \<Rightarrow> bool"   and P2 :: "cell \<Rightarrow> bool"   and P3 :: "cell \<times> interface \<Rightarrow> bool"   and cell_interface_prod :: "cell \<times> interface" assumes "\<And>x. P4 x \<Longrightarrow> P1 (SDiagram x)"   and "\<And>x. P2 (Filler x)"   and "\<And>x1a x2 x3. P2 (Basic x1a x2 x3)"   and "\<And>x1a x2. P1 x2 \<Longrightarrow> P2 (Exists_sdia x1a x2)"   and "\<And>x1a x2 x3 x4. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P2 (Choose_sdia x1a x2 x3 x4)"   and "\<And>x1a x2 x3. P1 x2 \<Longrightarrow> P2 (Loop_sdia x1a x2 x3)"   and "\<And>x1a x2. P2 x1a \<Longrightarrow> P3 (x1a, x2)"   and "P4 []"   and "\<And>x1a x2. \<lbrakk>P3 x1a; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1a # x2)" shows "P3 cell_interface_prod"
assms_1: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "degree (A $$ (i, j)) \<le> k"
inj: shows "inj_on ((\<circ>) g) {\<pi>. \<pi> permutes {0..<n}}"
replicate_empty: fixes n :: "nat"   and x :: "'g" shows "(replicate n x = []) = (n = 0)"
raWrite_def: fixes arr :: "nat \<times> (nat \<Rightarrow> 'a)"   and i :: "nat"   and "value" :: "'a"   and fillValue :: "'a" shows "raWrite arr i value fillValue \<equiv> let len = fst arr; f = snd arr in if i < len then (len, f(i := value)) else (i + 1, fillAndUpdate len f i value fillValue)"
fresh_fun_app: fixes a :: "atom"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "a \<sharp> f"   and "a \<sharp> x" shows "a \<sharp> f x"
last_is_acc_code_def: fixes G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "last_is_acc_code G_impl \<equiv> \<lambda>x. let y' = x in case y' of (xa, xb) \<Rightarrow> RETURN (List.all_interval_nat (\<lambda>xc. bs_mem xc (as_top xa)) 0 (igbgi_num_acc G_impl))"
foldSetD_closed: fixes A :: "'e set"   and z :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "(A, z) \<in> foldSetD D f e" shows "z \<in> D"
simps: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state"   and a3 :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>a1, a2\<rangle> \<Rightarrow> a3 = ((\<exists>t. a1 = [] \<and> a2 = t \<and> a3 = t) \<or> (\<exists>m. a1 = [Rule m action.Accept] \<and> a2 = Undecided \<and> a3 = Decision FinalAllow \<and> Semantics.matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m action.Drop] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> Semantics.matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Reject] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> Semantics.matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Log] \<and> a2 = Undecided \<and> a3 = Undecided \<and> Semantics.matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Empty] \<and> a2 = Undecided \<and> a3 = Undecided \<and> Semantics.matches \<gamma> m p) \<or> (\<exists>m a. a1 = [Rule m a] \<and> a2 = Undecided \<and> a3 = Undecided \<and> \<not> Semantics.matches \<gamma> m p) \<or> (\<exists>rs X. a1 = rs \<and> a2 = Decision X \<and> a3 = Decision X) \<or> (\<exists>rs\<^sub>1 t rs\<^sub>2 t'. a1 = rs\<^sub>1 @ rs\<^sub>2 \<and> a2 = Undecided \<and> a3 = t' \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t') \<or> (\<exists>m chain rs\<^sub>1 m' rs\<^sub>2. a1 = [Rule m (Call chain)] \<and> a2 = Undecided \<and> a3 = Undecided \<and> Semantics.matches \<gamma> m p \<and> \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2) \<and> Semantics.matches \<gamma> m' p \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided) \<or> (\<exists>m chain rs t. a1 = [Rule m (Call chain)] \<and> a2 = Undecided \<and> a3 = t \<and> Semantics.matches \<gamma> m p \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t))"