hcju/isabelleps huggingface dataset:
bit_drop_bit_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (drop_bit n a) = bit a \<circ> (+) n"
add: fixes x :: "'a"   and y :: "'a" shows "(x + y) *s x = x *s x + y *s x"
indep_events_def_alt: fixes A :: "'c \<Rightarrow> 'a set"   and I :: "'c set" shows "indep_events A I = indep_sets (\<lambda>i. {A i}) I"
simps_2: fixes \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('c, 'd) Term.term"   and pair_fun :: "'a"   and X :: "'b list"   and F :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and F' :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "comp_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<Gamma> pair_fun \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = (F' = [] \<and> (\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. is_Var (\<Gamma> (Var x))) \<or> (\<forall>u\<in>subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> pair' pair_fun ` set F'). is_Fun u \<longrightarrow> args u = [] \<or> (\<exists>s\<in>set (args u). s \<notin> Var ` set X)))"
F_def: shows "F \<equiv> \<lambda>n. if n = 0 then 0::'a else inverse (of_int n) * a $$ (n - 1)"
prog: shows "{X. F \<squnion> G \<in> stable X} \<in> progress_set F UNIV B"
cases: fixes G :: "('a, 'b) graph_scheme"   and a :: "'a list" assumes "cycle G a" obtains v :: "'a" and p :: "'a list" where "a = p" and "path G v p v" and "p \<noteq> []" and "distinct p"
exp_double: fixes z :: "'b" shows "exp ((2::'b) * z) = (exp z)\<^sup>2"
fin_make_untimed_inf_truncate_Nonempty_all0: fixes z :: "nat \<Rightarrow> 'a list" assumes "z 0 \<noteq> []" shows "\<forall>i. fin_make_untimed (inf_truncate z i) \<noteq> []"
Tr: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg" assumes "eq a b"   and "eq b c" shows "eq a c"
IH: fixes YS :: "(real \<times> real pdevs) list"   and YSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) (YS # YSS)" shows "prove_nonneg prnt i p (slp_of_fas [fa]) YSS"
genus_impl_correct: fixes G :: "'b list \<times> ('b \<times> 'b) list"   and M :: "('b \<times> 'b) list list" assumes "distinct (fst G)"   and "distinct (snd G)"   and "lists_digraph_map G M" shows "pre_digraph_map.euler_genus (with_proj (list_digraph G)) (to_map' (snd G) M) = genus_impl G M"
focus_plus: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X (p + q) = focus X p + focus X q"
Refl: fixes Ax :: "('a, 'b) Axioms"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" assumes "Axioms Ax"   and "Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e : B" shows "Vx : A \<turnstile> e\<equiv>e : B \<in> Axioms.Theory Ax"
real_cond_exp_intA: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real"   and A :: "'b set" assumes "sigma_finite_subalgebra M F"   and "integrable M f"   and "A \<in> sets F" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A (real_cond_exp M F f)"
set: fixes x1 :: "'b form multiset"   and x2 :: "'b form multiset" shows "set_sequent ( x1 \<Rightarrow>* x2) = \<Union> (set_form ` set_mset x1) \<union> \<Union> (set_form ` set_mset x2)"
closed_sum_zero_right: fixes A :: "'b set" shows "A +\<^sub>M {0::'b} = closure A"
sums_mult_iff: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b"   and d :: "'b" assumes "c \<noteq> (0::'b)" shows "(\<lambda>n. c * f n) sums (c * d) = f sums d"
contains_pred_def: fixes A :: "'a set"   and x :: "'a" shows "contains_pred A x = (if x \<in> A then Predicate.single () else \<bottom>)"
match_pre_def: shows "Solve_SASP.match_pre \<equiv> \<lambda>(x, v) s. s x = Some v"
reflect_poly_reflect_poly: fixes p :: "'a poly" assumes "constant_term p \<noteq> (0::'a)" shows "reflect_poly (reflect_poly p) = p"
exec_step_ind_Init_Error: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = \<lfloor>(sfs, Error)\<rfloor>" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Throwing Cs (addr_of_sys_xcpt NoClassDefFoundError)) # frs, sh)"
append: fixes xs :: "'b list"   and ys :: "'b list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
lub_least: fixes A :: "'c llist set"   and z :: "'c llist" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z" shows "lSup A \<sqsubseteq> z"
wf_Un: fixes r :: "('d \<times> 'd) set"   and s :: "('d \<times> 'd) set" assumes "wf r"   and "wf s"   and "Domain r \<inter> Range s = {}" shows "wf (r \<union> s)"
sub_instr_sub1_privilege: fixes s' :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
inpt_str_comp_b: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and r' :: "'b \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a \<times> 'b"   and x :: "nat \<Rightarrow> 'c" shows "prec_st (\<lambda>u u' x. inpt_st r (fst u) (fst u') x \<and> (\<forall>y. r (fst u) (fst u') x y \<longrightarrow> inpt_st r' (snd u) (snd u') y)) (\<lambda>(u, v) (u', v'). r u u' OO r' v v') u x = (\<box> lft_pred_st (inpt_st (rel_st r r'))) u x"
flow_tendsto: fixes t :: "real"   and x0 :: "'a"   and ts :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "t \<in> existence_ivl0 x0"   and "(ts \<longlongrightarrow> t) F" shows "((\<lambda>s. flow0 x0 (ts s)) \<longlongrightarrow> flow0 x0 t) F"
lunit_commutes_with_L: fixes f :: "'a" assumes "Left.ide f" shows "Left.lunit (Left.L f) = Left.L (Left.lunit f)"
axioms_1: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "Group.group G"
one_in_ring_n_pd_one_K: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "1\<^sub>r\<^bsub>O\<^bsub>K P n\<^esub>\<^esub> = 1\<^sub>r"
d_def: fixes X :: "'a set" shows "d X = Sigma_Algebra.measure M X - Sigma_Algebra.measure N X"
proj_fm: fixes J :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'd"   and x :: "'d"   and m :: "'d \<Rightarrow> 'f" assumes "function_to_finmap J f f'"   and "x \<in> J" shows "(function_to_finmap.fm J f f' m)\<^sub>F (f x) = m x"
c: shows "\<Gamma> \<Rightarrow> \<Delta> \<down> cut_bound n n F"
un_Node1_def: fixes tree :: "'a tree" shows "un_Node1 tree = (case tree of \<langle>x21, x22, x23\<rangle> \<Rightarrow> x21)"
rr: fixes x :: "real" assumes "poly p x = 0" shows "- rr < x \<and> x < rr"
encL: fixes S :: "'procS" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk> SourceTerm S"
dsj_Semicong_1: fixes H :: "fmla set"   and A :: "fmla"   and C :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A OR C"   and "H \<turnstile> A IMP B" shows "H \<turnstile> B OR C"
hensel_binary: fixes D :: "int poly"   and H :: "int poly"   and C :: "int poly"   and D' :: "int poly"   and H' :: "int poly" assumes "coprime_m D H"   and "C =m D * H"   and "Mp D = D"   and "Mp H = H"   and "monic D"   and "hensel_binary C D H = (D', H')" shows "pn.eq_m C (D' * H') \<and> monic D' \<and> D =m D' \<and> H =m H' \<and> pn.Mp D' = D' \<and> pn.Mp H' = H'"
match_ifaceAny: fixes i :: "char list" shows "match_iface ifaceAny i"
Pxz: shows "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"
crdt_ops_no_future_ref: fixes xs :: "('a \<times> 'b) list"   and oid :: "'a"   and oper :: "'b"   and ys :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set"   and ref :: "'a" assumes "crdt_ops (xs @ [(oid, oper)] @ ys) deps"   and "ref \<in> deps oper" shows "ref \<notin> fst ` set ys"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "(*\<^sub>V) (x*) = (*\<^sub>V) x\<^sup>\<dagger>"
bal_sub_height: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ls :: "('c btree \<times> 'c) list"   and a :: "'c btree \<times> 'c"   and rs :: "('c btree \<times> 'c) list"   and t :: "'c btree" assumes "split split"   and "bal (Node (ls @ a # rs) t)" shows "case rs of [] \<Rightarrow> True | (sub, sep) # x \<Rightarrow> height sub = height t"
upper_bounds_def: fixes D :: "('c, 'd) Order_scheme"   and H :: "'c set" shows "upper_bounds D H = {x. ub\<^bsub>D\<^esub> H x}"
oghoare_while_valid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and i :: "'e set"   and A :: "'e set"   and b :: "'e set"   and Q :: "'e set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c i, A"   and "i \<inter> b \<subseteq> P"   and "i \<inter> - b \<subseteq> Q" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> i While b c Q, A"
assms_5: shows "u \<noteq> \<^bold>\<I>"
rtrancl_last_visit: fixes q :: "'b"   and q' :: "'b"   and R :: "('b \<times> 'b) set"   and S :: "'b set" assumes "(q, q') \<in> R\<^sup>*" obtains "(q, q') \<in> (R - UNIV \<times> S)\<^sup>*"   | qt :: "'b" where "qt \<in> S" and "(q, qt) \<in> R\<^sup>+" and "(qt, q') \<in> (R - UNIV \<times> S)\<^sup>*"
union_monad_def: shows "Set_Impl.union_monad = (\<union>)"
join_compl_top_left1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- x \<squnion> (x \<squnion> y) = \<top>"
unstream_simps_2: fixes g :: "('e, 'f) generator"   and s :: "'f"   and s' :: "'f" assumes "generator g s = Skip s'" shows "unstream g s = unstream g s'"
sump: fixes s :: "'s" shows "sum (p s) (supp (p s)) \<le> 1"
orientable_5_implies_1: fixes x :: "'a" assumes "orientable_5 (x \<squnion> x\<^sup>T)" shows "orientable_1 x"
un: fixes x :: "real" assumes "\<exists>q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0" shows "\<exists>!q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0"
FunctorId: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and C :: "('d, 'f, 'h) Category_scheme"   and D :: "('e, 'g, 'i) Category_scheme"   and X :: "'d" assumes "Ftor F : C \<longrightarrow> D"   and "X \<in> obj\<^bsub>C\<^esub>" shows "F ## id\<^bsub>C\<^esub> X = id\<^bsub>D\<^esub> (F @@ X)"
Lim_def: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'d filter"   and f :: "'d \<Rightarrow> 'b" assumes "class.t2_space open" shows "t2_space.Lim open A f = (THE l. filterlim f (topological_space.nhds open l) A)"
simps_3: fixes x1 :: "'a"   and x2 :: "'a"   and xs :: "'a list" shows "changes (x1 # x2 # xs) = (if x1 * x2 < (0::'a) then 1 + changes (x2 # xs) else if x2 = (0::'a) then changes (x1 # xs) else changes (x2 # xs))"
list_all2_iff: fixes P :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and xs :: "'k list"   and ys :: "'l list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
simps_2: fixes k :: "nat" shows "Stirling 0 (Suc k) = 0"
infinity_simps_3: shows "IEEE.exponent \<infinity> = emax TYPE(('a, 'b) IEEE.float)"
measurable_snd: fixes M1 :: "'a measure"   and M2 :: "'b measure" shows "snd \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M2"
D\<^sub>i\<^sub>i\<^sub>i: fixes x :: "i" shows "((\<^bold>\<forall>x. (dom x) \<cdot> x \<cong> x \<^bold>\<leftarrow> E ((dom x) \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> (dom x) \<cong> x \<^bold>\<leftarrow> E (x \<cdot> (dom x)))) \<^bold>\<and> x \<cdot> (dom x) \<cong> x \<^bold>\<leftarrow> E x"
extremum_uniqueI: fixes a :: "'c" assumes "\<top> \<le> a" shows "a = \<top>"
domConc: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp (list2FWpolicy b))"   and "b \<noteq> []" shows "x \<in> dom (Cp (list2FWpolicy (a @ b)))"
bind_commute_spmf: fixes p :: "'s spmf"   and q :: "'t spmf"   and f :: "'s \<Rightarrow> 't \<Rightarrow> 'r spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
to_nat_from_nat_id: fixes i :: "nat" assumes "i < CARD('d)" shows "Bij_Nat.to_nat (Bij_Nat.from_nat i) = i"
prm_unit_action: fixes a :: "'b"   and b :: "'b" shows "[a \<leftrightarrow> b] $ a = b"
AssertAllState''_atoms_imply_bounds_index: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "set ats \<Turnstile>\<^sub>i \<B>\<I> (assert_bound_loop ats (init t))"
index_add_mat_1: fixes i :: "nat"   and B :: "'c mat"   and j :: "nat"   and A :: "'c mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A + B) $$ (i, j) = A $$ (i, j) + B $$ (i, j)"
assms_5: shows "b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
risk_neutral_iff': fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 \<le> q"   and "q \<le> 1"   and "filt_equiv nat_filtration M N" shows "risk_neutral_prob N = (q = (1 + r - d) / (u - d))"
ltake_lappend1: fixes n :: "enat"   and xs :: "'a llist"   and ys :: "'a llist" assumes "n \<le> llength xs" shows "ltake n (lappend xs ys) = ltake n xs"
add_increasing: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "{#} \<subseteq># a"   and "b \<subseteq># c" shows "b \<subseteq># a + c"
assms_1: fixes l :: "nat list \<times> nat list"   and i :: "nat" assumes "l \<in> set subs1"   and "i \<in> set (fst l) \<or> i \<in> set (snd l)" shows "i < n1"
add_pow_ldistr: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "a \<in> carrier P"   and "b \<in> carrier P" shows "[k] \<cdot>\<^bsub>P\<^esub> a \<otimes>\<^bsub>P\<^esub> b = [k] \<cdot>\<^bsub>P\<^esub> (a \<otimes>\<^bsub>P\<^esub> b)"
minus_minus: fixes x :: "'a" assumes "x \<in> E" shows "- (- x) = x"
pred_fmap_id: fixes f :: "'d \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "pred_fmap id (fmmap f m) = pred_fmap f m"
sum_nonpos: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> bot" shows "sup_monoid.sum f A \<le> bot"
hyps: shows "{i} \<times> below (2 * n) \<in> tiling domino"
ovalid_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalid P f Q \<equiv> \<forall>s r. P s \<and> f s = Some r \<longrightarrow> Q r s"
plus_wb_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c" assumes "wb_lens x"   and "wb_lens y"   and "x \<bowtie> y" shows "wb_lens (x +\<^sub>L y)"
test_converse: fixes x :: "'a" shows "x \<cdot> 1' = x\<^sup>T \<cdot> 1'"
add_right_cancel: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "cancel_semigroup_add_ow U plus"   and "b \<in> U"   and "a \<in> U"   and "c \<in> U" shows "(plus b a = plus c a) = (b = c)"
ptr_in_heap: shows "ptr |\<in>| object_ptr_kinds h3"
directedE: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "directed A less_eq"   and "x \<in> A"   and "y \<in> A" obtains z :: "'b" where "z \<in> A" and "less_eq x z" and "less_eq y z"
incoming_def: fixes G :: "('a, 'b) graph_scheme"   and y :: "'a" shows "\<^bold>I\<^bold>N\<^bsub>G\<^esub> y = {x. (x, y) \<in> \<^bold>E\<^bsub>G\<^esub>}"
assms_3: shows "snd' \<equiv> op_sem Snd"
subst_mgt_refl: fixes \<theta> :: "'f \<Rightarrow> ('e, 'd) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta>"
declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('d strand_label \<times> ('b, 'e) stateful_strand_step) list"   and \<I> :: "'e \<Rightarrow> ('b, 'c) Term.term" shows "declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<I> \<equiv> {t. \<langle>\<star>, receive\<langle>t\<rangle>\<rangle> \<in> set \<A>} \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
MStar_1: fixes \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex" assumes "ok (length \<phi>s) mr"   and "fv_regex (from_mregex mr \<phi>s) \<subseteq> A"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels"   and "ok_rctxt \<phi>s \<kappa> \<kappa>'"   and "\<forall>ms\<in>\<kappa> ` RPD mr. qtable n A (mem_restr R) (\<lambda>v. Q (map the v) (from_mregex ms \<phi>s)) (lookup rel ms)" shows "qtable n A (mem_restr R) (\<lambda>v. \<exists>a\<in>Regex.rpd\<kappa> \<kappa>' (Formula.sat \<sigma> V (map the v)) j (from_mregex mr \<phi>s). Q (map the v) a) (r\<delta> \<kappa> rel rels mr)"
composition_of_substs_cl: fixes C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" shows "subst_cl (subst_cl C \<sigma>) \<eta> = subst_cl C (\<sigma> \<lozenge> \<eta>)"
fst_guaranteeStripPair: fixes f :: "'d"   and g :: "'e set" shows "fst (guaranteeStripPair f g) = f"
set_intros_1: fixes x31 :: "'e"   and x32 :: "'f" shows "x31 \<in> set1_step (Yield x31 x32)"
seq_extract_singleton: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "{i} \<upharpoonleft>\<^sub>l xs = [xs ! i]"
distribF: shows "dist_over_sup (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
cons_to_append: fixes a :: "'a"   and b :: "'a list" shows "a # b = [a] @ b"
interior_def: fixes S :: "'b set" shows "interior S = \<Union> {T. open T \<and> T \<subseteq> S}"
inject_4: fixes x4 :: "val"   and y4 :: "val" shows "(Val x4 = Val y4) = (x4 = y4)"
config_rand: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and init :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init \<and> distinct (fst x) = distinct init \<and> length (fst x) = length init"
sel_12: fixes x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "the_set_term \<langle>x61: x62 \<in> x63\<rangle> = x63"
invalid_assn_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and x :: "'c"   and y :: "'d" shows "invalid_assn R x y \<equiv> \<up> (\<exists>h. h \<Turnstile> R x y) * true"
assms_2: shows "coreflexive (H * d[[H * d]])"
countable_basis: fixes p :: "'a set \<Rightarrow> bool"   and B :: "'a set set" assumes "countable_basis p B" shows "countable B"
def_ass_allUses_aux: fixes g :: "'g"   and ns :: "'node list"   and n :: "'node"   and v :: "'var \<times> 'node \<times> Def" assumes "g \<turnstile> Entry g-ns\<rightarrow>n" shows "lookupDefNode g n (var g v) \<in> set ns"
insert_def: shows "local.insert \<equiv> \<lambda>k x ti. ins k x ti \<bind> (\<lambda>ti'. case ti' of T\<^sub>i sub \<Rightarrow> return sub | Up\<^sub>i l a r \<Rightarrow> pfa_init (2 * k) (l, a) 1 \<bind> (\<lambda>kvs. ref (Btnode kvs r) \<bind> (\<lambda>t'. return (Some t'))))"
reachable_equiv_traces: fixes F :: "'c program" shows "reachable F = {s. \<exists>evs. (s, evs) \<in> traces (Init F) (Acts F)}"
1: fixes x :: "'a" assumes "\<forall>i. \<exists>n\<ge>i. P n x" shows "Min {i. P i x} = the None"
dp_p_fun1: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_p_fun x"
sup_inf_le_distr_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion> b \<sqinter> c \<le> (a \<squnion> b) \<sqinter> (a \<squnion> c)"
enum_UNIV: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "set enum = UNIV"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "sup_class.sup x (inf_class.inf x y) = x"
RedLAss: 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 V :: "String.literal"   and v :: "'addr val"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option" shows "extTA,P,t \<turnstile> \<langle>V:=Val v,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>unit,(h, l(V \<mapsto> v))\<rangle>"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. n < 4 \<Longrightarrow> P n"   and "\<And>n. \<lbrakk>4 \<le> n; P (nat \<lfloor>real n / 4\<rfloor>); P (nat \<lfloor>real n / 2\<rfloor>)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
nat_inc_closed: fixes n :: "nat" shows "[n] \<cdot> \<one> \<in> carrier R"
sup_lexord1: fixes k :: "'c \<Rightarrow> 'b"   and A :: "'c"   and B :: "'c"   and s :: "'c"   and c :: "'c" assumes "k A = k B" shows "sup_lexord A B k s c = c"
S5n_common_knowledge_fixed_point: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and a :: "'e"   and as :: "'e list"   and \<phi> :: "('e, 'f) Kform" assumes "S5n M"   and "w \<in> worlds M"   and "a \<in> set as" shows "M, w \<Turnstile> \<^bold>C\<^bsub>as\<^esub> \<phi> = M, w \<Turnstile> Kand \<^bold>K\<^sub>a \<phi> \<^bold>K\<^sub>a \<^bold>C\<^bsub>as\<^esub> \<phi>"
tlofOPT2: fixes Q :: "'a"   and x :: "'a"   and y :: "'a"   and QS :: "'a list"   and R :: "'a list" assumes "Q \<in> {x, y}"   and "set QS \<subseteq> {x, y}"   and "R \<in> {[x, y], [y, x]}" shows "tl (OPT2 ((Q # QS) @ [x, x]) R) = OPT2 (QS @ [x, x]) (step R Q (hd (OPT2 ((Q # QS) @ [x, x]) R)))"
weak_get_put: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
semiattractive_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "semiattractive A r = semiattractive A r'"
rep_eq: fixes x :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and xa :: "('g, 'e) uexpr"   and xb :: "('h, 'e) uexpr" shows "\<lbrakk>bop x xa xb\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b))"
oVeblen_0: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "oVeblen F 0 = F"
If_cong: fixes b :: "tr"   and b' :: "tr"   and t :: "'r"   and t' :: "'r"   and e :: "'r"   and e' :: "'r" assumes "b = b'"   and "b = TT \<Longrightarrow> t = t'"   and "b = FF \<Longrightarrow> e = e'" shows "If b then t else e = If b' then t' else e'"
prefixI: fixes ys :: "'d list"   and xs :: "'d list"   and zs :: "'d list" assumes "ys = xs @ zs" shows "prefix xs ys"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f)) local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f))"