hcju/isabelleps huggingface dataset:
children_eq_h3: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "h3 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
invert_subcls: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "ws_prog G" shows "(D, C) \<notin> (subcls1 G)\<^sup>*"
abs_minus_cancel: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (uminus a) = abs a"
lens_inv_def: fixes x :: "'c \<Longrightarrow> 'b" shows "inv\<^sub>L x = \<lparr>lens_get = create\<^bsub>x\<^esub>, lens_put = \<lambda>\<sigma>. get\<^bsub>x\<^esub>\<rparr>"
pmdl_closed_monom_mult: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and t :: "'a" assumes "p \<in> pmdl B" shows "monom_mult c t p \<in> pmdl B"
pair_prob_space_def: fixes M1 :: "'c measure"   and M2 :: "'d measure" shows "pair_prob_space M1 M2 \<equiv> pair_sigma_finite M1 M2 \<and> prob_space M1 \<and> prob_space M2"
abc_comp_null_left: fixes A :: "abc_inst list" shows "[] [+] A = A"
\<psi>_in_hom: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<guillemotleft>\<psi> x g : F y \<rightarrow>\<^sub>C x\<guillemotright>"
rep_eq: fixes x :: "rat" shows "Rep_mini_alg_unique (mau_of_rat x) = ma_of_rat x"
sublatticeI: fixes S :: "'a set" assumes "S \<subseteq> A"   and "\<lparr>pset = S, order = induced S r\<rparr> \<in> CompleteLattice" shows "S <<= cl"
not_Some_eq: fixes x :: "'k option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
resultant_smult_right: fixes c :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "c \<noteq> (0::'a)" shows "resultant f (Polynomial.smult c g) = c ^ degree f * resultant f g"
step_p: shows "lnth Ns p \<leadsto>LGC lnth Ns (Suc p)"
F_2: shows "F \<subseteq> {n. (T ^^ n) x \<in> A}"
int_of_real_def: fixes x :: "real" shows "int_of_real x = (SOME y. real_of_int y = x)"
acquire_all_preserves_lock_thread_ok: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option"   and t :: "'b"   and x :: "'c"   and ln :: "'a \<Rightarrow>f nat"   and xw :: "'c \<times> 'a \<Rightarrow>f nat" assumes "lock_thread_ok ls ts"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "lock_thread_ok (acquire_all ls t ln) (ts(t \<mapsto> xw))"
Int_Diff: fixes A :: "'h set"   and B :: "'h set"   and C :: "'h set" shows "A \<inter> B - C = A \<inter> (B - C)"
le_approx_lemma_F: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q" shows "F Q \<subseteq> F P"
frac_def: fixes x :: "'b" shows "frac x \<equiv> x - of_int \<lfloor>x\<rfloor>"
that: assumes "inside (path_image c1 \<union> path_image c) \<inter> inside (path_image c2 \<union> path_image c) = {}"   and "inside (path_image c1 \<union> path_image c) \<union> inside (path_image c2 \<union> path_image c) \<union> (path_image c - {a, b}) = inside (path_image c1 \<union> path_image c2)" shows "thesis"
sinks_aux_subset: fixes U :: "'e set"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "U \<subseteq> sinks_aux I D U xs"
ccw'_mirror_point3: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real"   and w :: "real \<times> real" shows "ccw' (mirror_point x y) (mirror_point x z) (mirror_point x w) = ccw' y z w"
exec_exec_n: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "P \<turnstile> c \<rightarrow>* c'" shows "\<exists>n. P \<turnstile> c \<rightarrow>^n c'"
that: fixes xs :: "(accent \<times> 'b) list" assumes "snd ` set xs \<subseteq> M"   and "(u, v) \<in> local.lconv xs" shows "thesis"
U3: shows "(\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n < 0}. set (case (p, n, s) of (p, n, s) \<Rightarrow> if n = 0 then [(p, 0, Floor s)] else map (case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) (if 0 < n then [0..n] else [n..0]))) = (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n < 0}. set (map (case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) [n..0]))"
c: shows "c = (\<lambda>input. callee' input \<bind> (\<lambda>(x, s). inline callee (rpv' x) s))"
\<iota>_triangle_2: shows "MC.tensor \<iota> MC.unity \<cdot> MC.assoc' MC.unity MC.unity MC.unity = MC.tensor MC.unity \<iota>"
simps_1: fixes ze :: "'c"   and on :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "'c list list" shows "mat_powI ze on pl ti n m 0 = mat1I ze on n"
function_of_in_extensional_funcset: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and xs :: "'d list"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "set xs \<subseteq> B"   and "length xs = card A" shows "function_of A enum xs \<in> A \<rightarrow>\<^sub>E B"
A_gets_good_key: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs)"   and "Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
simps_16: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (a ^\<^sub>e 0) vs = True"
Inf_aform'_def: fixes p :: "nat"   and X :: "'b \<times> 'b pdevs" shows "Inf_aform' p X = eucl_truncate_down p (fst X - Radius' p X)"
eq: shows "key_compare (key_order_of_ord ord) = comp_of_ord ord"
intro: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" assumes "finite_comm_group G"   and "subgroup H G"   and "finite_comm_group_adjoin_axioms G H a" shows "finite_comm_group_adjoin G H a"
P: fixes n :: "nat" shows "{x \<in> space M. P n x} \<in> events"
InvariantConsistentAfterApplyBackjump: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantUniqC (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state' = applyBackjump state in InvariantConsistent (getM state')"
triangle_num_Suc: fixes n :: "nat" shows "triangle_num (Suc n) = triangle_num n + Suc n"
uniform_limit_integral: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and g :: "'c \<Rightarrow> 'd"   and F :: "'b filter" assumes "uniform_limit {a..b} f g F"   and "\<And>n. continuous_on {a..b} (f n)"   and "F \<noteq> bot" obtains I :: "'b \<Rightarrow> 'd" and J :: "'d" where "\<And>n. (f n has_integral I n) {a..b}" and "(g has_integral J) {a..b}" and "(I \<longlongrightarrow> J) F"
sym_order_trans: fixes x :: "'a set"   and y :: "'a set"   and z :: "'a set" assumes "Q.sym x y"   and "z \<sqsubseteq>\<^sup>s y"   and "x \<in> Q"   and "y \<in> Q"   and "z \<in> Q" shows "z \<sqsubseteq>\<^sup>s x"
Minimal_SolutionsI: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b"   and "nonzero x"   and "\<not> (\<exists>(u, v)\<in>Linear_Diophantine_Equations.Solutions a b. nonzero u \<and> u @ v <\<^sub>v x @ y)" shows "(x, y) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b"
merge_status_eq_iff_6: fixes a :: "status" shows "(merge_status a SUCCESS = FAILED) = (a = FAILED)"
integral: fixes a :: "'a list"   and b :: "'a list" assumes "a \<otimes>\<^bsub>poly_ring R\<^esub> b = \<zero>\<^bsub>poly_ring R\<^esub>"   and "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)" shows "a = \<zero>\<^bsub>poly_ring R\<^esub> \<or> b = \<zero>\<^bsub>poly_ring R\<^esub>"
not_in_kD_case: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and en :: "'a"   and es :: "nat \<times> k \<times> f \<times> nat \<times> nat \<Rightarrow> 'a" assumes "dip \<notin> kD rt" shows "(case rt dip of None \<Rightarrow> en | Some r \<Rightarrow> es r) = en"
f: shows "continuous_map (subtopology X (X closure_of {x. P x})) Y f"
h_minus_le_h_plus: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_minus i e \<le> h_plus i e"
assms: shows "get_S (cpu_reg_val PSR s1) = 0 \<and> good_context s1 \<and> get_delayed_pool s1 = [] \<and> get_trap_set s1 = {} \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> get_delayed_pool s2 = [] \<and> get_trap_set s2 = {} \<and> good_context s2 \<and> low_equal s1 s2"
fun_range_witness: fixes f :: "ZF"   and y :: "ZF" assumes "isFun f"   and "y |\<in>| HOLZF.Range f" shows "\<exists>x. x |\<in>| HOLZF.Domain f \<and> f \<acute> x = y"
apos_natpos: fixes a :: "ant" assumes "a \<noteq> \<infinity>"   and "0 \<le> a" shows "0 \<le> na a"
prj0_def: fixes f :: "'a"   and g :: "'a" shows "prj0 f g \<equiv> if cospan f g then snd (SOME x. limit_cone cospan_shape.comp (\<cdot>) (cospan_diagram.map (\<cdot>) f g) (local.dom (fst x)) (cospan_diagram.mkCone (\<cdot>) f g (fst x) (snd x))) else null"
rbt_lookup_rbt_less: fixes k :: "'a"   and t :: "('a, 'c) rbt" assumes "t |\<guillemotleft> k" shows "rbt_lookup t k = None"
to_complex_cvec_def: fixes z :: "complex \<times> complex" shows "to_complex_cvec z = (let (x, y) = z in x / y)"
elims: fixes x :: "ternaryvalue"   and y :: "bool option" assumes "ternary_to_bool x = y" obtains "x = TernaryTrue" and "y = Some True"   | "x = TernaryFalse" and "y = Some False"   | "x = TernaryUnknown" and "y = None"
b_e_type_call_indirect_2: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Call_indirect i" shows "\<exists>ts'' tf1 tf2. ts = ts'' @ tf1 @ [T_i32] \<and> ts' = ts'' @ tf2 \<and> types_t \<C> ! i = tf1 _> tf2"
map_cong: fixes x :: "'a list"   and ya :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set ya \<Longrightarrow> f z = g z" shows "map f x = map g ya"
x_fv_ex: shows "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"
assms_2: shows "execute (g x) h' = Some (y, h'')"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (add_monoid G)"   and "a \<in> H" shows "\<ominus> a \<in> H"
expand: fixes dlts :: "'c dlts"   and dlts' :: "'c dlts" assumes "un_DLTS dlts = un_DLTS dlts'" shows "dlts = dlts'"
\<Delta>\<Phi>_pass2: fixes hs :: "'b hp list" assumes "hs \<noteq> []" shows "\<Phi> (pass\<^sub>2 hs) - \<Phi>_hps hs \<le> log 2 (real (size_hps hs))"
map_raw_id: fixes xs :: "('a \<times> 'e) list \<times> 'o" assumes "oalist_inv xs" shows "map_raw id xs = xs"
inner_mult_left': fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'b"   and m :: "real"   and b :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(a * of_real m) \<bullet> b = m * (a \<bullet> b)"
LPDi_Times: fixes i :: "nat"   and r :: "mregex"   and s :: "mregex" shows "LPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesR (\<Union>j\<le>i. LPDi j r) s \<union> (\<Union>j\<le>i. LPDi j s)"
hyps_3: shows "w \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {wa. w \<rightarrow>\<^sup>* wa}\<^esub> v"
rep_eq: fixes x :: "'a regular" shows "Rep_regular (x\<^sup>\<star>) = Rep_regular x\<^sup>\<star>"
mpoly_map_vars_sum: fixes f :: "nat \<Rightarrow> nat"   and g :: "'c \<Rightarrow> 'b mpoly"   and A :: "'c set" shows "mpoly_map_vars f (sum g A) = (\<Sum>x\<in>A. mpoly_map_vars f (g x))"
a': shows "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Compound F Fs, n) \<in> derivable R*"
space_measure_of: fixes A :: "'c set set"   and \<Omega> :: "'c set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
Liminf_upto_llist_infinity: fixes Xs :: "'b set llist" shows "Liminf_upto_llist Xs \<infinity> = Liminf_llist Xs"
dgrad_p_set_le_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set"   and m :: "nat" assumes "dgrad_p_set_le d F G"   and "G \<subseteq> dgrad_p_set d m" shows "F \<subseteq> dgrad_p_set d m"
small_upair: fixes x :: "'a"   and y :: "'a" shows "small {x, y}"
vars_t: shows "vars_mset s \<subseteq># vars_mset t"
fag_single_free1: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "fags_cond f z i a"   and "n\<Odot>a\<^bsub>f,i,z\<^esub> = m\<Odot>a\<^bsub>f,i,z\<^esub>" shows "n = m"
lvars_def: fixes t :: "(nat \<times> linear_poly) list" shows "lvars t = set (map lhs t)"
set_2: fixes x21 :: "'e"   and x22 :: "'e list" shows "set (Cons x21 x22) = insert x21 (set x22)"
ok_def: fixes F :: "'c program"   and G :: "'c program" shows "F ok G \<equiv> Acts F \<subseteq> AllowedActs G \<and> Acts G \<subseteq> AllowedActs F"
simple_restrict_some: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and i :: "nat"   and X :: "'c option list"   and A :: "nat set" assumes "getIJ getIJ"   and "i < length X"   and "i \<in> A" shows "restrict A X ! i = X ! i"
split_part: fixes P :: "bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
in_subnet: fixes a :: "'a"   and e :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(a, e) \<sqsubset> {{(x1, y). P x1 y}} = P a e"
mat_base_case: shows "M_mat [[1], [- 1]] [[], [0]] = mat_of_rows_list 2 [[1, 1], [1, - 1]]"
dom_MkArr: fixes A :: "'e set"   and B :: "'e set"   and F :: "'e \<Rightarrow> 'e" assumes "arr (MkArr A B F)" shows "local.dom (MkArr A B F) = MkIde A"
assms_2: shows "snd (cs c cid) \<noteq> NotStarted"
prems_1: shows "dom f' = dom (f(l \<mapsto> t))"
zip_dist: shows "zip (linearize tss @ [nt]) ([tp - len..<tp] @ [tp]) = zip (linearize tss) [tp - len..<tp] @ [(nt, tp)]"
swap_preserves_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "wls s X" shows "wls s (X #[z1 \<and> z2]_zs)"
eq2: shows "(\<Prod>i = 0..<Suc b. monom 1 1) mod f = monom 1 1 mod f \<otimes>\<^bsub>R\<^esub> ((\<Prod>i = 0..<b. monom 1 1) mod f)"
compute_numberarith_25: shows "Num.BitM num.One = num.One"
intro: fixes x :: "complex"   and y :: "complex"   and z :: "complex" assumes "x + y + z = 1"   and "x\<^sup>2 + y\<^sup>2 + z\<^sup>2 = 2"   and "x ^ 3 + y ^ 3 + z ^ 3 = 3" shows "power_sum_puzzle_example x y z"
degree_reciprocal: fixes P :: "'a poly"   and p :: "nat" assumes "degree P \<le> p" shows "degree (reciprocal_poly p P) \<le> p"
tauStepChainBang: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<parallel> !P \<Longrightarrow>\<^sub>\<tau> P'"   and "guarded P" shows "\<Psi> \<rhd> !P \<Longrightarrow>\<^sub>\<tau> P'"
card_ran_le_dom: fixes m :: "'c \<Rightarrow> 'd option" assumes "finite (dom m)" shows "card (ran m) \<le> card (dom m)"
simps_5: fixes b1 :: "bool"   and v :: "'b gexp"   and va :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux (Bc b1) (Nor v va) = True"
echelon_form_of_column_k_def: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A' :: "(('d, 'e) vec, 'f) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k bezout A' k = (let (A, i) = A' in if (\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'d)) \<or> i = nrows A then (A, i) else if \<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'d) then (A, i + 1) else let n = LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d) \<and> mod_type_class.from_nat i \<le> n; interchange_A = interchange_rows A (mod_type_class.from_nat i) n in (bezout_iterate interchange_A (nrows A - 1) (mod_type_class.from_nat i) (mod_type_class.from_nat k) bezout, i + 1))"
set_integrable_cong: fixes M :: "'b measure"   and M' :: "'b measure"   and A :: "'b set"   and A' :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c" assumes "M = M'"   and "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "set_integrable M A f = set_integrable M' A' f'"
assms_3: shows "ball a r \<inter> T \<subseteq> S"
supp_perm_perm_eq: fixes x :: "'a"   and p :: "perm"   and q :: "perm" assumes "\<forall>a\<in>supp x. p \<bullet> a = q \<bullet> a" shows "p \<bullet> x = q \<bullet> x"
simps_3: fixes val :: "('a, 'b, 'c) val"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(ApplyR\<^sub>\<E> val \<E>) [e] = Apply (VE val) (\<E> [e])"
alluopairs_set1: fixes xs :: "'a list" shows "set (alluopairs xs) \<subseteq> {(x, y). x \<in> set xs \<and> y \<in> set xs}"
T8: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<circle>(F \<and> G) = (\<circle>F \<and> \<circle>G)"
eligibility: shows "ground_resolution_with_selection.eligible S_G As (D + negs (mset As))"
asms_1: fixes a :: "'a"   and b :: "'a" shows "Bf__ a b \<and> Q"
rel_cset_iff: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f cset"   and b :: "'g cset" shows "rel_cset R a b = ((\<forall>t. cin t a \<longrightarrow> (\<exists>u. cin u b \<and> R t u)) \<and> (\<forall>t. cin t b \<longrightarrow> (\<exists>u. cin u a \<and> R u t)))"
simps_1: fixes uu :: "'n \<Rightarrow> 'o fundef option"   and \<Sigma> :: "Unboxed.type option list" shows "sp uu [] \<Sigma> = Ok \<Sigma>"
small_pscope: fixes c :: "com"   and \<pi>' :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and c' :: "com"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option" assumes "c \<noteq> SKIP"   and "small_step \<pi>' (c, s) = Some (c', s')" shows "small_step \<pi> (PScope \<pi>' c, s) = Some (PScope \<pi>' c', s')"
snd_combined_restrict: fixes S :: "Vars.var set"   and p :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "snd (combined_restrict S p) = snd p G|` S"
wf_dickson_less_p: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "dickson_grading d" shows "wfP (gd_term.dickson_less_p pair_of_term ord_term_strict d m)"
simps: fixes P :: "'a prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" shows "(P \<turnstile> lub(U, V) = T) = (P \<turnstile> U \<le> T \<and> P \<turnstile> V \<le> T \<and> (\<forall>x. P \<turnstile> U \<le> x \<longrightarrow> P \<turnstile> V \<le> x \<longrightarrow> P \<turnstile> T \<le> x))"
partn_lst_E: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and \<alpha> :: "V list"   and n :: "nat"   and f :: "'a set \<Rightarrow> nat"   and l :: "nat" assumes "Partitions.partn_lst r B \<alpha> n"   and "f \<in> [B]\<^bsup>n\<^esup> \<rightarrow> {..<l}"   and "l = length \<alpha>" obtains i :: "nat" and H :: "'a set" where "i < l" and "H \<subseteq> B" and "ordertype H r = \<alpha> ! i" and "f ` [H]\<^bsup>n\<^esup> \<subseteq> {i}"
concurrent_store_delete_independent_technical: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and mo :: "'a"   and ir :: "'a" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Store e mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Store e mo i) (ir, Delete is e)"
t\<^sub>p_def: fixes s :: "'c list"   and q :: "'c"   and a :: "nat \<times> nat list" shows "t\<^sub>p s q a = (let (mf, sws) = a in index (swaps sws s) q + length sws)"
LI_k: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "\<exists>k\<in>I. LI K v I = v k"
aux: fixes v :: "'b option"   and x :: "'b" assumes "v = Some x" shows "the v = x"
R\<^sub>C_of_strong_low_bisim_mm: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" assumes "abs.strong_low_bisim_mm \<R>\<^sub>A"   and "secure_refinement \<R>\<^sub>A \<R> P"   and "sym P" shows "conc.strong_low_bisim_mm (R\<^sub>C_of \<R>\<^sub>A \<R> P)"
amd_empty_def: fixes dflt :: "'b" shows "amd_empty dflt \<equiv> dyn_array_new_sz dflt 16"
start_of_Suc3: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and n :: "nat" assumes "ly = layout_of ap"   and "abc_fetch as ap = Some (Goto n)" shows "start_of ly (Suc as) = start_of ly as + 1"
assms_3: shows "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"
simps_6: fixes x1 :: "nat"   and x3 :: "nat" shows "Const x1 \<noteq> Greater x3"
infer_simp_4: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and \<tau> :: "PreSimplyTyped.type"   and A :: "'a trm" shows "infer \<Gamma> (Fn x \<tau> A) = (case infer (\<Gamma>(x \<mapsto> \<tau>)) A of None \<Rightarrow> None | Some \<sigma> \<Rightarrow> Some (TArr \<tau> \<sigma>))"
defect_ge_1: fixes N :: "'b quasinorm" shows "1 \<le> defect N"
cc_d_true: fixes \<sigma> :: "'s" assumes "is_ground_subst \<sigma>" shows "I \<Turnstile>m (mset CAs + {#DA#}) \<cdot>cm \<sigma>"