hcju/isabelleps huggingface dataset:
inorder_upd: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c"   and y :: "'d" assumes "t \<in> T h"   and "sorted1 (inorder t)" shows "inorder (upd x y t) = upd_list x y (inorder t)"
dist_remove_def: fixes p :: "'a \<Rightarrow> real"   and x :: "'a" shows "dist_remove p x = (\<lambda>y. if y = x then 0 else p y / (1 - p x))"
simps_2: fixes n :: "nat" shows "index (Suc n) = (LEAST k. p k \<and> index n < k)"
is_nth_power_upto_unit_base: fixes n :: "nat"   and x :: "'a" assumes "is_nth_power n x" shows "is_nth_power_upto_unit n x"
has_pullbacks_def: shows "has_pullbacks = (\<forall>f0 f1. cospan f0 f1 \<longrightarrow> (\<exists>p0 p1. has_as_pullback f0 f1 p0 p1))"
infinite_rangeIt: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "inj f"   and "\<forall>y. x \<noteq> f y" shows "infinite (rangeIt x f)"
convB: fixes A :: "'a set"   and B :: "'a set" assumes "infsetsum'_converges f A"   and "\<And>x. x \<in> A - B \<union> (B - A) \<Longrightarrow> f x = (0::'b)" shows "infsetsum'_converges f B"
wfp_on_imp_minimal: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q)"
M_def: fixes x :: "int" shows "pl.M x = x mod pl"
thetaWhileW_def: shows "thetaWhileW \<equiv> {(while tst do c, while tst do d) |tst c d. compatTst tst \<and> c \<approx>wT d} \<union> {(c1 ;; while tst do c, d1 ;; while tst do d) |tst c1 d1 c d. compatTst tst \<and> c1 \<approx>wT d1 \<and> c \<approx>wT d}"
of_real_fact: fixes n :: "nat" shows "of_real (fact n) = fact n"
game_0_def: fixes \<A> :: "(bool, bool list, bool list) gpv" shows "PRF.game_0 \<A> = prf_key_gen \<bind> (\<lambda>key. exec_gpv (PRF.prf_oracle key) \<A> () \<bind> (\<lambda>(b, uu_). return_spmf b))"
mult_less_multiset\<^sub>D\<^sub>M: fixes M :: "'a multiset"   and N :: "'a multiset" shows "((M, N) \<in> mult {(x, y). x < y}) = less_multiset\<^sub>D\<^sub>M M N"
Diff_insert: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "(A::'b set) - ((a::'b) \<triangleright> (B::'b set)) = A - B - {a}"
quasi_transD: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a"   and z :: "'a" assumes "x \<^bsub>r\<^esub>\<prec> y"   and "y \<^bsub>r\<^esub>\<prec> z"   and "quasi_trans r" shows "x \<^bsub>r\<^esub>\<prec> z"
SFAssRedsThrow: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>"
comp_proj_assoc_1: fixes a0 :: "'a"   and a1 :: "'a"   and a2 :: "'a" assumes "ide a0"   and "ide a1"   and "ide a2" shows "\<pp>\<^sub>1[a0, a1 \<otimes> a2] \<cdot> \<a>[a0, a1, a2] = \<pp>\<^sub>1[a0, a1] \<cdot> \<pp>\<^sub>1[a0 \<otimes> a1, a2]"
B: shows "LIM x inf F (principal {z. f z \<noteq> 0}). - \<bar>Im (g x)\<bar> * pi + - (Re (g x) * ln (cmod (f x))) :> at_top"
2: shows "Suc (card X - 2) = card X - Suc 0"
a_normal: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "abelian_subgroup H G" shows "H \<lhd> add_monoid G"
ext_inject: fixes mor2ZF :: "'d \<Rightarrow> ZF"   and more :: "'e"   and mor2ZF' :: "'d \<Rightarrow> ZF"   and more' :: "'e" shows "(\<lparr>mor2ZF = mor2ZF, \<dots> = more\<rparr> = \<lparr>mor2ZF = mor2ZF', \<dots> = more'\<rparr>) = (mor2ZF = mor2ZF' \<and> more = more')"
face_cycle_succ_inI: fixes x :: "'b"   and y :: "'b" assumes "x \<in> GM.face_cycle_set y" shows "GM.face_cycle_succ x \<in> GM.face_cycle_set y"
components_are_equivalences: fixes a :: "'a" assumes "B.obj a" shows "C.equivalence_map (\<tau>\<^sub>0 a)"
IA1: shows "Iagree I J {Inl x |x. x \<in> SIGT t1}"
SetPC_sgl: fixes p :: "'a \<Rightarrow> real"   and x :: "'a"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "supp p = {x}" shows "SetPC a (\<lambda>_. p) = (\<lambda>ab P s. p x * a x ab P s)"
nat_helper1: fixes w :: "bit list"   and x :: "bit" assumes "nat_to_bv (bv_to_nat w) = norm_unsigned w" shows "nat_to_bv (2 * bv_to_nat w + bitval x) = norm_unsigned (w @ [x])"
and_mask_eq_iff_shiftr_0: fixes w :: "'b word"   and n :: "nat" shows "(w AND mask n = w) = (w >> n = 0)"
simps_2: fixes l :: "'b tree234"   and a :: "'b"   and r :: "'b tree234" shows "tree\<^sub>i (Up\<^sub>i l a r) = \<langle>l, a, r\<rangle>"
igConsIPresIGWlsSTR_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igConsIPresIGWlsSTR Delta MOD \<equiv> FixSyn.igVarIPresIGWls Delta MOD \<and> FixSyn.igAbsIPresIGWlsSTR Delta MOD \<and> FixSyn.igOpIPresIGWlsSTR Delta MOD"
lcm: fixes m :: "nat"   and n :: "nat" assumes "m \<in> M"   and "n \<in> M" shows "lcm m n \<in> M"
finalRefE: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e : T"   and "is_refT T"   and "final e" obtains "e = null"   | r :: "nat \<times> char list list" where "e = ref r__"   | r :: "nat \<times> char list list" where "e = Throw r"
abs_eq: fixes x :: "'c" shows "ket x = Abs_ell2 (\<lambda>y. if x = y then 1 else 0)"
I_subst_peps: fixes f :: "atom fm"   and r :: "real"   and cs :: "real list"   and xs :: "real list" assumes "nqfree f"   and "R.I (subst\<^sub>+ f (r, cs)) xs" shows "\<exists>leps>r + \<langle>cs,xs\<rangle>. \<forall>x. r + \<langle>cs,xs\<rangle> < x \<and> x \<le> leps \<longrightarrow> R.I f (x # xs)"
assms: shows "n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m'"
add_le_less_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less_eq a b"   and "less c d" shows "less (plus a c) (plus b d)"
ESem_simps_as_defined_4: fixes b :: "bool"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Bool b \<rbrakk>\<^bsub>\<rho>\<^esub> = B\<cdot>(Discr b)"
scaleR_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR 1 x = x"
assms_3: shows "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"
int_and_not_bitANDN_num: fixes n :: "num"   and m :: "num" shows "numeral n AND NOT (numeral m) = (case bitANDN_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> numeral n')"
sinh_add: fixes y :: "'a" shows "sinh (x + y) = sinh x * cosh y + cosh x * sinh y"
list_set_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
split_block_def: fixes A :: "'c mat"   and sr :: "nat"   and sc :: "nat" shows "split_block A sr sc = (let nr = dim_row A; nc = dim_col A; nr2 = nr - sr; nc2 = nc - sc; A1 = mat sr sc (($$) A); A2 = mat sr nc2 (\<lambda>(i, j). A $$ (i, j + sc)); A3 = mat nr2 sc (\<lambda>(i, j). A $$ (i + sr, j)); A4 = mat nr2 nc2 (\<lambda>(i, j). A $$ (i + sr, j + sc)) in (A1, A2, A3, A4))"
in_set_remove1: fixes a :: "'g"   and b :: "'g"   and xs :: "'g list" assumes "a \<noteq> b" shows "(a \<in> set (remove1 b xs)) = (a \<in> set xs)"
exec_noSpec_no_Stuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "noSpec c"   and "\<forall>p\<in>dom \<Gamma>. noSpec (the (\<Gamma> p))"   and "procs c \<subseteq> dom \<Gamma>"   and "\<forall>p\<in>dom \<Gamma>. procs (the (\<Gamma> p)) \<subseteq> dom \<Gamma>"   and "s \<noteq> Stuck" shows "t \<noteq> Stuck"
cs_join: fixes a :: "'a"   and b :: "'a" shows "(a \<down>\<^sup>* b) = ((a, b) \<in> {(x, y). x \<rightarrow> y}\<^sup>\<down>)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F F F"
420: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> - - (x \<sqinter> y) = - - (x \<sqinter> y)"
VPow_mono: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "VPow A \<subseteq>\<^sub>\<circ> VPow B"
continuous_map_of_fst: fixes X :: "'b topology"   and Y :: "'c topology"   and Z :: "'d topology"   and f :: "'b \<Rightarrow> 'd" shows "continuous_map (prod_topology X Y) Z (f \<circ> fst) = (topspace Y = {} \<or> continuous_map X Z f)"
concl: shows "concl_of \<iota> \<in> Red_F N"
HomCod_def: shows "HomCod = 3\<^sub>\<nat>"
hd_prog_append_Prog\<^sub>s\<^sub>b: fixes x :: "'a"   and sb :: "'a memref list"   and p :: "'a"   and p' :: "'a"   and mis :: "instr list" shows "hd_prog x (sb @ [Prog\<^sub>s\<^sub>b p p' mis]) = hd_prog p sb"
LAss: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and T :: "ty"   and v' :: "val"   and l' :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"   and "E V = \<lfloor>T\<rfloor>"   and "P \<turnstile> T casts v to v' "   and "l' = l(V \<mapsto> v')" shows "P,E \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v',(h, l')\<rangle>"
pull_up_senior_rank: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "senior x n = senior y n" shows "rank x n = rank y n"
asIMOD_def: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "asIMOD SEM \<equiv> \<lparr>igWls = \<lambda>s X. \<forall>val. (sWlsVal SEM val \<or> X val = undefined) \<and> (sWlsVal SEM val \<longrightarrow> sWls SEM s (X val)), igWlsAbs = \<lambda>(xs, s) A. \<forall>val. (sWlsVal SEM val \<or> A val = undefined) \<and> (sWlsVal SEM val \<longrightarrow> sWlsAbs SEM (xs, s) (A val)), igVar = \<lambda>ys y val. if sWlsVal SEM val then val ys y else undefined, igAbs = \<lambda>xs x X val. if sWlsVal SEM val then sAbs xs (\<lambda>sX. if sWls SEM (asSort xs) sX then X (val (x := sX)_xs) else sDummy SEM (SOME s. sWls SEM s (X val))) else undefined, igOp = \<lambda>delta inp binp val. if sWlsVal SEM val then sOp SEM delta (lift (\<lambda>X. X val) inp) (lift (\<lambda>A. A val) binp) else undefined, igFresh = \<lambda>ys y X. \<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> X val = X val', igFreshAbs = \<lambda>ys y A. \<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> A val = A val', igSwap = \<lambda>zs z1 z2 X val. if sWlsVal SEM val then X (val ^[z1 \<and> z2]_zs) else undefined, igSwapAbs = \<lambda>zs z1 z2 A val. if sWlsVal SEM val then A (val ^[z1 \<and> z2]_zs) else undefined, igSubst = \<lambda>ys Y y X val. if sWlsVal SEM val then X (val (y := Y val)_ys) else undefined, igSubstAbs = \<lambda>ys Y y A val. if sWlsVal SEM val then A (val (y := Y val)_ys) else undefined\<rparr>"
col_obtuse__bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.Obtuse Bet Cong A B C" shows "Bet A B C"
span_superset: fixes S :: "'a set" shows "S \<subseteq> ideal S"
mult_ge1_powers_1: fixes a :: "'b" assumes "(1::'b) \<le> a" shows "(1::'b) \<le> a * a"
aux_alpha_ignoreSecond_qTermLessQSwapped_wf: shows "wf (inv_image qTermQSwappedLess aux_alpha_ignoreSecond)"
llist_all2_lSupI: fixes Y :: "('c llist \<times> 'd llist) set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "Complete_Partial_Order.chain (rel_prod (\<sqsubseteq>) (\<sqsubseteq>)) Y"   and "\<forall>(xs, ys)\<in>Y. llist_all2 P xs ys" shows "llist_all2 P (lSup (fst ` Y)) (lSup (snd ` Y))"
assms_2: fixes a :: "'a" assumes "a \<notin> S" shows "x a \<sqsubseteq> z a"
ex_perp_cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "A \<noteq> B" shows "\<exists>Q. A B Perp Q C \<and> Coplanar A B P Q"
equalityCE: fixes A :: "'g set"   and B :: "'g set"   and c :: "'g" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
ring_distrib1: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<plusminus>\<^bsub>R\<^esub> z) = x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<plusminus>\<^bsub>R\<^esub> x \<cdot>\<^sub>r\<^bsub>R\<^esub> z"
extended_bicategory_axioms: shows "extended_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg"
abelian_group_def: fixes G :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d" shows "abelian_group G composition unit \<equiv> Group_Theory.group G composition unit \<and> commutative_monoid G composition unit"
get_ancestors_remains_not_in_ancestors: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ancestors' :: "(_) object_ptr list"   and node :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h'"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "h' \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors'"   and "\<And>p children children'. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes p \<rightarrow>\<^sub>r children; h' \<turnstile> Shadow_DOM.get_child_nodes p \<rightarrow>\<^sub>r children'\<rbrakk> \<Longrightarrow> set children' \<subseteq> set children"   and "node \<notin> set ancestors"   and "object_ptr_kinds h = object_ptr_kinds h'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.type_wf h'" shows "node \<notin> set ancestors'"
stutter_extend_pred_of_enex_conv: fixes enex :: "('b \<Rightarrow> 'd set) \<times> ('d \<Rightarrow> 'b \<Rightarrow> 'b)" shows "stutter_extend_edges UNIV (rel_of_enex enex) = rel_of_enex (stutter_extend_enex enex)"
sorted_less_rev_set_eq: fixes xs :: "'a list" assumes "Sorted_Less.sorted (rev xs)" shows "sorted_list_of_set (set xs) = rev xs"
\<A>'_1: shows "\<A>' \<in> set (tr \<A> [])"
space_T: fixes s :: "'s \<times> 's option" shows "space (KN.T s) = space (stream_space (count_space UNIV))"
simps_7: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<rho> \<lbrace>t\<rbrace>"
bag_cards_nonempty: assumes "V \<noteq> {}" shows "bag_cards \<noteq> {}"
A: shows "\<guillemotleft>\<r>[f] : f \<star> src f \<Rightarrow> f\<guillemotright>"
accimethds_def: fixes G :: "prog"   and pack :: "pname"   and I :: "qtname" shows "accimethds G pack I = (if G \<turnstile> Iface I accessible_in pack then DeclConcepts.imethds G I else (\<lambda>k. {}))"
simply_connected_def: fixes S :: "'a set" shows "simply_connected S \<equiv> \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and> path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S \<longrightarrow> homotopic_loops S p q"
below_halfspace_def: fixes sctn :: "'c sctn" shows "below_halfspace sctn = Collect (le_halfspace sctn)"
I_dnf: fixes \<phi> :: "atom fm"   and xs :: "real list" assumes "nqfree \<phi>" shows "(\<exists>as\<in>set (dnf \<phi>). \<forall>a\<in>set as. I\<^sub>R a xs) = R.I \<phi> xs"
ann_matches_imp_assertionsR: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "ann_matches \<Gamma> \<Theta> a c"   and "\<not> pre_par a" shows "assertionsR \<Gamma> \<Theta> Q A a c (pre a)"
filter_dropWhile: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter (\<lambda>x. \<not> P x) (dropWhile P xs) = filter (\<lambda>x. \<not> P x) xs"
intros_10: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<longrightarrow>\<^sub>! [r]\<^sub>! q)"
run_bind_option: fixes return :: "'b option \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> ('b option \<Rightarrow> 'a) \<Rightarrow> 'a"   and x :: "('b, 'a) optionT"   and f :: "'b \<Rightarrow> ('b, 'a) optionT" shows "run_option (bind_option return bind x f) = bind (run_option x) (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y))"
monomial_0_iff: fixes t :: "'a"   and c :: "'b" shows "(monomial c t = 0) = (c = (0::'b))"
Sup_apply: fixes A :: "('b \<Rightarrow> 'a) set"   and x :: "'b" shows "Sup A x = (SUP f\<in>A. f x)"
finite_prod: shows "finite UNIV = (finite UNIV \<and> finite UNIV)"
lim: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. g n $$ (i, j)) \<longlonglongrightarrow> A $$ (i, j)"
continuous_Ree: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> octo" assumes "continuous F g" shows "continuous F (\<lambda>x. Ree (g x))"
Int_insert_left_if1: fixes a :: "'a"   and C :: "'a set"   and B :: "'a set" assumes "a \<in> C" shows "insert a B \<inter> C = insert a (B \<inter> C)"
pfp_def: fixes f :: "'b \<Rightarrow> 'b" shows "pfp f = while_option (\<lambda>x. \<not> f x \<le> x) f"
SN_rel_on_conv: shows "SN_rel_on = SN_rel_on_alt"
finite_pow2_le: fixes n :: "nat" shows "finite {i. 2 ^ i \<le> n}"
u_adjoint_def: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" shows "((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)\<^sup>#) (x::'b::refinement_lattice) \<equiv> \<Squnion> {y::'b::refinement_lattice. F y \<sqsubseteq> x}"
Infinitesimal_hcmod_iff: fixes z :: "complex star" shows "(z \<in> Infinitesimal) = (hcmod z \<in> Infinitesimal)"
purgeIdle_simps_3: fixes st :: "'a"   and u :: "'b"   and c :: "'c"   and sl :: "('a, 'b, 'c) state list" shows "GM_sec_model.purgeIdle (State st u c # sl) = State st u c # GM_sec_model.purgeIdle sl"
meas: shows "\<forall>j\<in>R. B j \<in> measurable_sets M"
EX_State_SA: fixes SA :: "('a, 'b, 'c) seqauto" shows "\<exists>S. S \<in> States SA"
shadow_root_ptr_kinds_eq2_h2: shows "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h' |\<union>| {|shadow_root_ptr|}"
T_Var: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows " \<Gamma> \<turnstile> Var x : T "
word_bw_comms_2: fixes x :: "'b word"   and y :: "'b word" shows "x OR y = y OR x"
marg_dens2_cexpr_def: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and x :: "nat"   and y :: "nat"   and e :: "cexpr" shows "marg_dens2_cexpr \<Gamma> vs x y e = cexpr_comp_aux (Suc x) (fst\<^sub>c (CVar 0)) (cexpr_comp_aux (Suc y) (snd\<^sub>c (CVar 0)) (map_vars Suc (integrate_vars \<Gamma> (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs) e)))"
mult_1s_2: fixes a :: "'e" shows "a * Numeral1 = a"
linear_iff: fixes f :: "'b \<Rightarrow> 'c" shows "linear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x))"
cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes document :: "('a, 'b, 'c, 'h option) RDocument_scheme" shows "cast document = RObject.extend (RObject.truncate document) (Inr (Inl (RObject.more document)))"
awalk_ConsI: fixes v :: "'a"   and es :: "'b list"   and w :: "'a"   and e :: "'b"   and u :: "'a" assumes "awalk v es w"   and "e \<in> arcs G"   and "arc_to_ends G e = (u, v)" shows "awalk u (e # es) w"
take_rows_from_matrix_def: fixes matr :: "'b mat"   and indices_to_keep :: "nat list" shows "take_rows_from_matrix matr indices_to_keep = mat_of_rows (dim_col matr) (take_indices (rows matr) indices_to_keep)"
modRule2Characterise: fixes Ps :: "'a list"   and C :: "('b, 'c) sequent" assumes "(Ps, C) \<in> modRules2" shows "Ps \<noteq> [] \<and> (\<exists>F Fs. C = ( \<Empt> \<Rightarrow>* \<LM> Modal F Fs  \<RM>) \<or> C = ( \<LM> Modal F Fs  \<RM> \<Rightarrow>* \<Empt>))"
AxSlopedLineInVerticalPlane: fixes e :: "'a Point"   and f :: "'a Point"   and g :: "'a Point" assumes "onAxisT e"   and "onAxisT f"   and "e \<noteq> f"   and "\<not> onAxisT g" shows "\<forall>s. \<exists>p. collinear e g p \<and> space2 p f = s * s * time2 p f"
mr_cos_oneTr: fixes H :: "'a set" assumes "submodule R M H" shows "H = \<zero> \<uplus>\<^bsub>M\<^esub> H"
keys_lower: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "keys (punit.lower p v) = {u \<in> keys p. u \<prec> v}"
gt_antisym: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary s"   and "wary t"   and "t >\<^sub>t s" shows "\<not> s >\<^sub>t t"
lm014: fixes X :: "'b set"   and x :: "'c"   and Y :: "'b set"   and y :: "'c" shows "runiq (X \<times> {x} +< Y \<times> {y})"
filter_sup: fixes F :: "'a set"   and G :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G" shows "ord_class.filter (filter_sup F G)"
stereographic_sphere_circle_set': fixes H :: "circline" shows "stereographic ` sphere_circle_set (inv_stereographic_circline H) = circline_set H"
r_s_div_s: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "(r * s - Suc 0) div s = r - Suc 0"
vlrestriction_vdomain: shows "F \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> F = F"
capacity_pos: fixes e :: "'v \<times> 'v" assumes "e \<in> \<^bold>E\<^bsub>\<Delta>\<^esub>" shows "0 < capacity \<Delta> e"
assms_2: shows "has_as_product J1 D1 a1"
ifex_of_ltl_reduced_bdt_checker: shows "reduced_bdt_checkers ifex_of_ltl (\<lambda>y s. {x. s x} \<Turnstile>\<^sub>P y)"
inf0: fixes a :: "'b::bounded_semilattice_inf_top" shows "a::'b::bounded_semilattice_inf_top \<equiv> a \<sqinter> \<top>"
assms: fixes x :: "'a"   and y :: "'a" assumes "f x = f y" shows "x = y"
elims: fixes x :: "'a list"   and xa :: "nat"   and y :: "nat" assumes "\<ff> x xa = y" obtains s :: "'a list" where "x = s__" and "xa = 0" and "y = 0"   | s :: "'a list" and v :: "nat" where "x = s" and "xa = Suc v" and "y = length (intrinsic_border (take (Suc v) s)) + 1"
induct: fixes P :: "C \<Rightarrow> bool"   and x :: "C" assumes "adm P"   and "P \<bottom>"   and "\<And>Ca. P Ca \<Longrightarrow> P (C\<cdot>Ca)" shows "P x"