hcju/isabelleps huggingface dataset:
Cd_XD_smallest: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "stable_on ds X"   and "d \<in> ds" shows "Cd d (XD_smallest ds X \<inter> Field (Pd d)) = Cd d (X \<inter> Field (Pd d))"
eval_lowerFm: fixes F :: "atom fm" assumes "freeIn 0 F" shows "\<forall>xs. eval (lowerFm 0 1 F) xs = eval (ExQ F) xs"
intP_def: fixes p :: "'psym"   and al :: "univ list" shows "intP p al \<equiv> eintP p (map2 ntsem (parOf p) al)"
add_lessD1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i + j < k" shows "i < k"
constr_rep_3: shows "Rep_action (\<tau>) = Tau_Rep"
IH': fixes j :: "'sz" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (x2a j). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)" shows "sterm_sem (local.adjoint I \<sigma> \<nu>) (x2a j) = sterm_sem (local.adjoint I \<sigma> \<omega>) (x2a j)"
iFROM_def: fixes n :: "nat" shows "[n\<dots>] \<equiv> {n..}"
num_reachable_def: fixes G :: "'a graph"   and v :: "'a" shows "num_reachable G v = card (succ_tran G v)"
adopt_node_child_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr" assumes "h \<turnstile> ok adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M document_ptr child" shows "child |\<in>| node_ptr_kinds h"
\<Phi>_def: shows "\<Phi> = (\<forall>\<^sub>1 \<exists>\<^sub>1 x\<^sub>1 < x\<^sub>0)"
ide_is_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso a"
equivalent_eq_exists_def: fixes t :: "'a trm"   and s :: "'a trm"   and C :: "'a literal set"   and I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and \<sigma> :: "('a \<times> 'a trm) list"   and L1 :: "'a literal" shows "equivalent_eq_exists t s C I \<sigma> L1 = (\<exists>L\<in>C - {L1}. \<exists>u v. orient_lit_inst L u v pos \<sigma> \<and> t \<lhd> \<sigma> = u \<lhd> \<sigma> \<and> I (s \<lhd> \<sigma>) (v \<lhd> \<sigma>))"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha> set['e]" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = abs_Tree\<^sub>\<alpha> (tConj (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
distinct_1: fixes x2 :: "xname" shows "Object' \<noteq> SXcpt' x2"
Minkowski_inequality_2: fixes p :: "real"   and f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "1 \<le> p"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"   and "integrable M (\<lambda>x. \<bar>g x\<bar> powr p)" shows "(LINT x|M. \<bar>f x + g x\<bar> powr p) powr (1 / p) \<le> (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p) + (LINT x|M. \<bar>g x\<bar> powr p) powr (1 / p)"
bdd_above_Iic: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq (ord.atMost less_eq b)"
is_standard_effect'_def: shows "is_standard_effect' \<equiv> \<lambda>(pre, uu_, vpre, uu_). pre = [] \<and> vpre = None"
simps: fixes C :: "'b literal set" shows "ground_clause C = (vars_of_cl C = {})"
takefill_append: fixes fill :: "'b"   and m :: "nat"   and xs :: "'b list"   and w :: "'b list" shows "takefill fill (m + length xs) (xs @ w) = xs @ takefill fill m w"
validCI_not_act: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'cmp"   and n :: "nat" assumes "\<nexists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) n" shows "eval cid t t' n \<gamma>"
bis_prefix: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "Bisim.mono Retr" shows "bis Retr \<subseteq> Retr (bis Retr)"
set_rw: shows "[0..<min (nrows A) (ncols A) - 1] = [0..<min (nrows A) (ncols A) - 2] @ [min (nrows A) (ncols A) - 2]"
even_mult_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (times a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "a \<cong>\<^sub>C a'" obtains f :: "'a" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f"
simps: fixes w :: "'a list"   and I :: "(nat + nat set) list" shows "enc (w, I) = {x. \<exists>n. x = cut_same (any, replicate (length I) False) (stream_enc (w, I)) @ replicate n (any, replicate (length I) False)}"
Q_hit_exists: obtains Q_hit :: "'a list" and Q_hit_pre :: "'a list" and Q_hit_post :: "'a list" where "Q_hit \<in> Q" and "y \<in> set Q_hit" and "Q_hit = Q_hit_pre @ y # Q_hit_post"
env_delete_fun_upd2: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b"   and v :: "'b" shows "(env_delete x m)(x := v) = m(x := v)"
star_unfoldl_eq: fixes x :: "('a \<times> 'a) set" shows "Id \<union> x ; x\<^sup>* = x\<^sup>*"
in_\<R>n: shows "((cms\<^sub>A ! n, mem\<^sub>A), cms\<^sub>C ! n, mem\<^sub>C) \<in> \<R>n"
sup_bot_right: fixes x :: "'k multiset" shows "x \<union># {#} = x"
one_le_inverse: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less_eq a one" shows "less_eq one (inverse a)"
elims_2: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)"   and xb :: "nat set" assumes "x \<Turnstile>\<^sub>b xa \<parallel> xb" obtains v :: "nat \<Rightarrow> 'a" and lb :: "nat \<Rightarrow> 'a option" and ub :: "nat \<Rightarrow> 'a option" and S :: "nat set" where "x = v" and "xa = (lb, ub)" and "xb = S" and "\<forall>x\<in>S. in_bounds x v (lb, ub)"
RPCFail_vis: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "\<turnstile> RPCFail send rcv rst p \<longrightarrow> <RPCNext send rcv rst p>_(rst!p, rtrner send!p, caller rcv!p)"
of_nat_mask_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "semiring_1.of_nat one plus zero (mask n) = mask n"
bind_def: fixes s :: "('c, 'e) state"   and k :: "'e \<Rightarrow> ('c, 'd) state" shows "s \<bind> k = State (\<lambda>M. let (a, M') = runState s M in runState (k a) M')"
simps_2: fixes h :: "real poly"   and T :: "real poly list" shows "prod_list_var (h # T) = (if h = 0 then prod_list_var T else h * prod_list_var T)"
triv_finprod_iff_compl_fam_PiE: fixes I :: "'c set"   and S :: "'c \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G" shows "compl_fam S I = (\<forall>f\<in>Pi\<^sub>E I S. finprod G f I = \<one> \<longrightarrow> (\<forall>i\<in>I. f i = \<one>))"
almost_full_on_lists: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "almost_full_on P A" shows "almost_full_on (list_emb P) (lists A)"
rel_gpv''_colossless_gpvD1: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I> :: "('i, 'k) \<I>"   and \<I>' :: "('j, 'l) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "colossless_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "colossless_gpv \<I>' gpv'"
cnj_0: shows "cnj_class.cnj (0::'a) = (0::'a)"
less_imp_le: fixes x :: "'e word"   and y :: "'e word" assumes "x <s y" shows "x \<le>s y"
simps: fixes f :: "int vec list \<times> int vec list"   and uu :: "int iarray iarray \<times> int iarray" shows "small_fs_state (f, uu) = fst f"
add_nonneg_nonneg: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
connected_intermediate_closure: fixes s :: "'c set"   and t :: "'c set" assumes "connected s"   and "s \<subseteq> t"   and "t \<subseteq> closure s" shows "connected t"
n: shows "nrows_iarray (matrix_to_iarray A) - Suc 0 < nrows (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n))"
upd_col_col: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d" shows "upd_col M i (col M i) = M"
rbt_comp_lookup: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_lookup c = ord.rbt_lookup (lt_of_comp c)"
deriv: fixes x :: "real" assumes "x \<in> {a..} - Y" shows "(g has_real_derivative g' x) (at x)"
minus_minus_float: fixes f :: "('a, 'b) IEEE.float" shows "- (- f) = f"
in_real_intervalI: fixes X :: "float interval"   and x :: "real" assumes "real_of_float (lower X) \<le> x"   and "x \<le> real_of_float (upper X)" shows "x \<in>\<^sub>r X"
HAInitStates_def: fixes HA :: "('d, 'e, 'f) hierauto" shows "HAInitStates HA \<equiv> \<Union>A\<in>SAs HA. {InitState A}"
simps_4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.fw plus less_eq m n k i (Suc j) = linordered_ab_monoid_add.fw_upd plus less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i j) k i (Suc j)"
return_contT_def: shows "Monad_Overloading.return = return_cont"
Arden_preserves_finite: fixes rhs :: "'b trm set"   and X :: "'b list set" assumes "finite rhs" shows "finite (Arden X rhs)"
wp_loop_step_mono: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "healthy (wp body)"   and "le_trans t u"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "\<And>P. sound P \<Longrightarrow> sound (u P)" shows "le_trans (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
finalVertex_def: fixes g :: "graph"   and v :: "nat" shows "finalVertex g v \<equiv> \<forall>f\<in>set (facesAt g v). final f"
rGamma_series_Weierstrass_def: fixes z :: "'a"   and n :: "nat" shows "rGamma_series_Weierstrass z n = exp (euler_mascheroni * z) * z * (\<Prod>k = 1..n. ((1::'a) + z / of_nat k) * exp (- z / of_nat k))"
simps_2: fixes n :: "nat" shows "mk_tree (Suc n) = (let t = mk_tree n in t \<triangle> t)"
tm_exhaust_apps_sel: fixes s :: "('c, 'd) tm" obtains "s = apps (Hd (head s)) (args s)"
prems: shows "distinct (map fst (a # xs) @ map fst ys)"
polynomial_pf: fixes rs :: "nat list"   and y :: "nat"   and w :: "nat \<Rightarrow> real"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y" shows "insertion w (deep_model_correct_params_y.polynomial_p shared_weights rs y) = deep_model_correct_params_y.polynomial_f shared_weights rs y w"
lipschitz_onD: fixes L :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "L-lipschitz_on X f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) \<le> L * dist x y"
int_sum: fixes f :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "int (sum f A) = (\<Sum>x\<in>A. int (f x))"
dist_less_eq_nth_equal: fixes f :: "'a fps"   and g :: "'a fps"   and i :: "nat" shows "(dist f g < inverse (2 ^ i)) = (\<forall>j\<le>i. fps_nth f j = fps_nth g j)"
strategies_ex: fixes v :: "'a" assumes "v \<in> S" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v S W"
iT_Mult_Min: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "iMin (I \<otimes> k) = iMin I * k"
new1_subset_frmls: fixes \<phi> :: "'d ltlr"   and \<psi> :: "'d ltlr" assumes "\<phi> \<in> new1 \<psi>" shows "\<phi> \<in> subfrmlsr \<psi>"
growth_rate_euclidean_above: fixes D :: "real"   and F :: "'b set"   and R :: "real" assumes "0 < D"   and "F \<subseteq> cball (0::'b) R"   and "0 \<le> R"   and "\<And>x y. \<lbrakk>x \<in> F; y \<in> F; x \<noteq> y\<rbrakk> \<Longrightarrow> D \<le> dist x y" shows "finite F \<and> real (card F) \<le> 1 + (6 / D) ^ DIM('b) * R ^ DIM('b)"
or_lem: fixes a :: "bool"   and c :: "bool"   and b :: "bool"   and d :: "bool" assumes "a = c"   and "b = d" shows "(a \<or> b) = (c \<or> d)"
adm_P: shows "adm (\<lambda>x. P (fst x) (snd x))"
length_rk2_fas_err: fixes rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith" shows "length (rk2_fas_err rkp x0 h cx s2) = length x0"
eq_\<I>_converter_trans: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter"   and conv' :: "('y, 'z, 'aa, 'ab) converter"   and conv'' :: "('y, 'z, 'aa, 'ab) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv''" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv''"
countable_nat_product_event_const: fixes a :: "'c"   and F :: "'c set" assumes "a \<in> F"   and "countable F" shows "countable {x. (\<forall>i. x i \<in> F) \<and> finite {i. x i \<noteq> a}}"
card_UNIV_option: shows "CARD('b option) = (if CARD('b) = 0 then 0 else CARD('b) + 1)"
add_assoc: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> bool" shows "((a \<and>* b) \<and>* c) = (a \<and>* b \<and>* c)"
cexp_mono1: fixes p1 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p1 \<le>o r1"   and "Card_order q" shows "p1 ^c q \<le>o r1 ^c q"
induct: fixes P1 :: "state \<Rightarrow> stmt \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool"   and P2 :: "state \<Rightarrow> expr \<Rightarrow> val \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool"   and x1 :: "state"   and x2 :: "stmt"   and x3 :: "nat"   and x4 :: "state"   and x5 :: "state"   and x6 :: "expr"   and x7 :: "val"   and x8 :: "nat"   and x9 :: "state" assumes "\<And>s n. P1 s Skip n s"   and "\<And>s0 c1 n s1 c2 s2. \<lbrakk>s0 -c1-n\<rightarrow> s1; P1 s0 c1 n s1; s1 -c2-n\<rightarrow> s2; P1 s1 c2 n s2\<rbrakk> \<Longrightarrow> P1 s0 (c1;; c2) n s2"   and "\<And>s0 e v n s1 c1 c2 s2. \<lbrakk>s0 -e\<succ>v-n\<rightarrow> s1; P2 s0 e v n s1; s1 -(if v \<noteq> Null then c1 else c2)-n\<rightarrow> s2; P1 s1 (if v \<noteq> Null then c1 else c2) n s2\<rbrakk> \<Longrightarrow> P1 s0 (If (e) c1 Else c2) n s2"   and "\<And>s0 x c n. s0<x> = Null \<Longrightarrow> P1 s0 (While (x) c) n s0"   and "\<And>s0 x c n s1 s2. \<lbrakk>s0<x> \<noteq> Null; s0 -c-n\<rightarrow> s1; P1 s0 c n s1; s1 -While (x) c-n\<rightarrow> s2; P1 s1 (While (x) c) n s2\<rbrakk> \<Longrightarrow> P1 s0 (While (x) c) n s2"   and "\<And>s x n. P2 s (LAcc x) (s<x>) n s"   and "\<And>s e v n s' x. \<lbrakk>s -e\<succ>v-n\<rightarrow> s'; P2 s e v n s'\<rbrakk> \<Longrightarrow> P1 s (x :== e) n (lupd(x\<mapsto>v) s')"   and "\<And>s e a n s' f. \<lbrakk>s -e\<succ>Addr a-n\<rightarrow> s'; P2 s e (Addr a) n s'\<rbrakk> \<Longrightarrow> P2 s (e..f) (get_field s' a f) n s'"   and "\<And>s0 e1 a n s1 e2 v s2 f. \<lbrakk>s0 -e1\<succ>Addr a-n\<rightarrow> s1; P2 s0 e1 (Addr a) n s1; s1 -e2\<succ>v-n\<rightarrow> s2; P2 s1 e2 v n s2\<rbrakk> \<Longrightarrow> P1 s0 (e1..f:==e2) n (upd_obj a f v s2)"   and "\<And>s a C n. new_Addr s = Addr a \<Longrightarrow> P2 s (new C) (Addr a) n (new_obj a C s)"   and "\<And>s e v n s' C. \<lbrakk>s -e\<succ>v-n\<rightarrow> s'; P2 s e v n s'; case v of Null \<Rightarrow> True | Addr a \<Rightarrow> obj_class s' a \<preceq>C C\<rbrakk> \<Longrightarrow> P2 s (Cast C e) v n s'"   and "\<And>s0 e1 a n s1 e2 p s2 C m s3. \<lbrakk>s0 -e1\<succ>a-n\<rightarrow> s1; P2 s0 e1 a n s1; s1 -e2\<succ>p-n\<rightarrow> s2; P2 s1 e2 p n s2; lupd(This\<mapsto>a) (lupd(Par\<mapsto>p) (del_locs s2)) -Meth (C, m)-n\<rightarrow> s3; P1 (lupd(This\<mapsto>a) (lupd(Par\<mapsto>p) (del_locs s2))) (Meth (C, m)) n s3\<rbrakk> \<Longrightarrow> P2 s0 ({C}e1..m(e2)) (s3<Res>) n (set_locs s2 s3)"   and "\<And>s a D C m n s'. \<lbrakk>s<This> = Addr a; D = obj_class s a; D \<preceq>C C; init_locs D m s -Impl (D, m)-n\<rightarrow> s'; P1 (init_locs D m s) (Impl (D, m)) n s'\<rbrakk> \<Longrightarrow> P1 s (Meth (C, m)) n s'"   and "\<And>s Cm n s'. \<lbrakk>s -body Cm-n\<rightarrow> s'; P1 s (body Cm) n s'\<rbrakk> \<Longrightarrow> P1 s (Impl Cm) (Suc n) s'" shows "(x1 -x2-x3\<rightarrow> x4 \<longrightarrow> P1 x1 x2 x3 x4) \<and> (x5 -x6\<succ>x7-x8\<rightarrow> x9 \<longrightarrow> P2 x5 x6 x7 x8 x9)"
a_get_root_node_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_root_node ptr = Heap_Error_Monad.bind (local.a_get_ancestors ptr) (\<lambda>ancestors. return (last ancestors))"
twiddleHeap_def: fixes \<beta> :: "(Location \<times> Location) set"   and h1 :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and h2 :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" shows "twiddleHeap \<beta> h1 h2 = (\<beta> \<in> Pbij \<and> Pbij_Dom \<beta> \<subseteq> Dom h1 \<and> Pbij_Rng \<beta> \<subseteq> Dom h2 \<and> (\<forall>l ll v w. (l, ll) \<in> \<beta> \<longrightarrow> lookup h1 l = Some v \<longrightarrow> lookup h2 ll = Some w \<longrightarrow> twiddleObj \<beta> v w))"
pi_divide_HNatInfinite_not_zero: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "hypreal_of_real pi / hypreal_of_hypnat N \<noteq> 0"
error_free_Jump: fixes j :: "jump"   and s :: "st" shows "error_free (Some (Jump j), s)"
chain_cons: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and x :: "'b" assumes "chain R xs"   and "R x (lhd xs)" shows "chain R (LCons x xs)"
valid_shift_end_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and nt :: "nat" assumes "valid_mmsaux args cur aux auxlist"   and "cur \<le> nt" shows "valid_mmsaux args cur (shift_end args nt aux) (filter (\<lambda>(t, rel). enat (nt - t) \<le> right (args_ivl args)) auxlist)"
qps_after_measure_is_qp: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and P :: "nat \<Rightarrow> complex mat" assumes "measurement d n M"   and "\<And>k. k < n \<Longrightarrow> is_quantum_predicate (P k)" shows "is_quantum_predicate (matrix_sum d (\<lambda>k. adjoint (M k) * P k * M k) n)"
containFalsum: fixes \<Gamma> :: "'a form multiset"   and R :: "('a sequent list \<times> 'a sequent) set"   and \<Delta> :: "'a form multiset" assumes "ff \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable R*"
var_rename_is_renaming_2: fixes n :: "nat" shows "subst_range (var_rename_inv n) \<subseteq> range Var"
one_neq_neg_one: shows "(1::'a) \<noteq> - (1::'a)"
simps: fixes c :: "CTypeId"   and a :: "nat"   and f :: "AttId" shows "objV c a..f = objLoc (catt c f) a"
coeff_def: fixes p :: "'q mpoly" shows "MPoly_Type.coeff p = lookup (mapping_of p)"
negligible_def: fixes s :: "'c set" shows "negligible s = (\<forall>a b. (indicat_real s has_integral 0) (cbox a b))"
winding_number_valid_path: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "winding_number \<gamma> z = 1 / (complex_of_real (2 * pi) * \<i>) * contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
mset_set_set_mset_msubset: fixes A :: "'d multiset" shows "mset_set (set_mset A) \<subseteq># A"
Int_Union: fixes A :: "'f set"   and B :: "'f set set" shows "A \<inter> \<Union> B = \<Union> ((\<inter>) A ` B)"
a_neigh_neq_a': shows "a_neigh \<noteq> a'"
tensor_mat_mult: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat"   and m4 :: "'b mat" assumes "m1 \<in> carrier_mat st.d1 st.d1"   and "m2 \<in> carrier_mat st.d1 st.d1"   and "m3 \<in> carrier_mat st.d2 st.d2"   and "m4 \<in> carrier_mat st.d2 st.d2" shows "st.tensor_mat (m1 * m2) (m3 * m4) = st.tensor_mat m1 m3 * st.tensor_mat m2 m4"
obtain_v1_v2_3: shows "disjnt vs (\<Union> S)"
def_hashmap_size_list_def: shows "def_hashmap_size = (\<lambda>_. 2 * def_hashmap_size TYPE('a))"
T_insert_def: fixes a :: "'b"   and t :: "'b tree" shows "T_insert a t = int (T_merge \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>\<rangle> t + 1)"
pairwise_trans_eq_elapsed: fixes xs :: "('c \<Rightarrow> 'a) stream" assumes "stream_trans xs"   and "pred_stream (\<lambda>u. u \<in> V) xs" shows "PTA_Reachability.pairwise eq_elapsed xs"
equicont: fixes x :: "'a"   and e :: "real" assumes "x \<in> S"   and "0 < e" shows "\<exists>d>0. \<forall>n y. y \<in> S \<and> norm (x - y) < d \<longrightarrow> norm (\<F> n x - \<F> n y) < e"
NI_return_def: shows "NI_return \<equiv> nondetM_base.return_nondet return_id csingle"
sum_component: fixes f :: "'d \<Rightarrow> ('b, 'c) vec"   and S :: "'d set"   and i :: "'c" shows "sum f S $ i = (\<Sum>x\<in>S. f x $ i)"
mod_minus_cong: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and a' :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a b = modulo a' b" shows "modulo (uminus a) b = modulo (uminus a') b"
that: fixes x' :: "'b" assumes "x = Some x'"   and "f x' = Some y" shows "thesis"