hcju/isabelleps huggingface dataset:
disjnt_def: fixes A :: "'d set"   and B :: "'d set" shows "disjnt A B = (A \<inter> B = {})"
I_def: shows "I = {k. k < n \<and> (i, k) \<in> (graph_of_mat A)\<^sup>*}"
less_le_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
enc_satisfies_binary_pred_iff_indRelR_satisfies_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>S a. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) a) = (\<forall>(P, Q)\<in>indRelR. \<forall>a. Pred (P, Q) a)"
mismatch_pair_lcp: fixes u :: "'b list"   and v :: "'b list" shows "mismatch_pair u v = (u ! \<^bold>|u \<and>\<^sub>p v\<^bold>|, v ! \<^bold>|u \<and>\<^sub>p v\<^bold>|)"
isAbr_simps_3: fixes f :: "'e" shows "isAbr (Fault f) = False"
assms_5: shows "h \<turnstile> to_tree_order ptr'' \<rightarrow>\<^sub>r to''"
IVT_two_functions_real_ivl: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "continuous_on {a--b} f"   and "continuous_on {a--b} g"   and "f a < g a"   and "g b < f b" shows "\<exists>x\<in>{a--b}. f x = g x"
val_process_def: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "val_process Mkt p = (if \<not> portfolio p then \<lambda>n w. 0 else (\<lambda>n w. \<Sum>x\<in>support_set p. prices Mkt x n w * p x (Suc n) w))"
intro: fixes \<phi> :: "char list \<Rightarrow> nat" assumes "inj \<phi>" shows "poly_embed \<phi>"
Lxy_def: fixes xs :: "'a list"   and S :: "'a set" shows "Lxy xs S = filter (\<lambda>z. z \<in> S) xs"
is_terminal_nonterminal: fixes x :: "symbol" assumes "is_terminal x"   and "is_nonterminal x" shows "False"
subset_eq_diff_conv_zmset: fixes A :: "'b zmultiset"   and C :: "'b zmultiset"   and B :: "'b zmultiset" shows "(A - C \<subseteq>#\<^sub>z B) = (A \<subseteq>#\<^sub>z B + C)"
Osum_Refl: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Refl r"   and "Refl r'" shows "Refl (r \<union>o r')"
guarantees_imp: fixes Y :: "'b program set" assumes "Y = UNIV guarantees Y" shows "ex_prop Y"
fps_divide_compose: fixes c :: "'b fps"   and b :: "'b fps"   and a :: "'b fps" assumes "c $ 0 = (0::'b)"   and "b $ 0 \<noteq> (0::'b)" shows "a / b oo c = (a oo c) / (b oo c)"
poincare_between_x_axis_0uv: fixes x :: "real"   and y :: "real" assumes "- 1 < x"   and "x < 1"   and "x \<noteq> 0"   and "- 1 < y"   and "y < 1"   and "y \<noteq> 0" shows "poincare_between 0\<^sub>h (of_complex (cor x)) (of_complex (cor y)) = (x < 0 \<and> y < 0 \<and> y \<le> x \<or> 0 < x \<and> 0 < y \<and> x \<le> y)"
Process_eq_spec: fixes P :: "'a process"   and Q :: "'a process" shows "(P = Q) = (F P = F Q \<and> D P = D Q)"
simps_10: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "list_all2 R [] []"
cring_simprules_19: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = y"
A2: shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> False"
foldr_cong: fixes a :: "'b"   and b :: "'b"   and l :: "'c list"   and k :: "'c list"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a" shows "foldr f l a = foldr g k b"
applyBackjumpEffect: 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 "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 "InvariantUniqC (getC state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "let l = getCl state; bClause = getC state; bLiteral = opposite l; level = getBackjumpLevel state; prefix = prefixToLevel level (getM state); state'' = applyBackjump state in (formulaEntailsClause F0 bClause \<and> isUnitClause bClause bLiteral (elements prefix) \<and> getM state'' = prefix @ [(bLiteral, False)]) \<and> getF state'' = getF state"
submonoid_invertible: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> N" shows "invertible u"
conway_monoid_split: fixes x :: "('b, 'c) boffa_mon" assumes "conway_assm2 x" shows "\<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV} = (1::'c) + \<Sum> {x\<^bsub>i\<^esub> |i. i \<in> UNIV}"
sum_list_inc_eq_sum_list_Suc: fixes u :: "nat list"   and y :: "nat list"   and i :: "nat" assumes "length u = length y"   and "i < length y"   and "\<exists>j\<ge>i. j < length y \<and> u ! j < y ! j" shows "sum_list (inc y i u) = Suc (sum_list u)"
n_dist_var1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(n x \<oplus> n y) \<cdot> (t x \<oplus> n z) = t x \<cdot> n y \<oplus> n x \<cdot> n z"
measurable_bounded_by_integrable_imp_integrable: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "S \<in> sets lebesgue" shows "f integrable_on S"
Un_subset_iff: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
0: shows "FvarsT \<chi> = FvarsT r - snd ` set txs \<union> \<Union> {if x \<in> FvarsT r then {u} else {} |u x. u \<in> var \<and> (Var u, x) \<in> set (zip (map Var us) (map snd txs))}"
rep_eq: fixes x :: "'b list" shows "\<langle>finset x\<rangle>\<^sub>f = set x"
rel_FGl_def: fixes L1 :: "'au \<Rightarrow> 'bh \<Rightarrow> bool"   and L2 :: "'av \<Rightarrow> 'bi \<Rightarrow> bool"   and L3 :: "'bb \<Rightarrow> 'bn \<Rightarrow> bool"   and Co1 :: "'aw \<Rightarrow> 'bj \<Rightarrow> bool"   and Co2 :: "'ax \<Rightarrow> 'bk \<Rightarrow> bool"   and Co3 :: "'bc \<Rightarrow> 'bo \<Rightarrow> bool"   and Co4 :: "'bd \<Rightarrow> 'bp \<Rightarrow> bool"   and Contra1 :: "'ay \<Rightarrow> 'bl \<Rightarrow> bool"   and Contra2 :: "'az \<Rightarrow> 'bm \<Rightarrow> bool"   and Contra3 :: "'be \<Rightarrow> 'bq \<Rightarrow> bool"   and Contra4 :: "'bf \<Rightarrow> 'br \<Rightarrow> bool" shows "rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 = rel_F (rel_G L1 L2 Co1 Co2 Contra1 Contra2) L1 L3 Co1 Co3 Co4 Contra1 Contra3 Contra4"
distinct_11: fixes x21 :: "tm"   and x22 :: "tm"   and x4 :: "fm" shows "x21 EQ x22 \<noteq> Neg x4"
sdiv_word_def: fixes v :: "'b word"   and w :: "'b word" shows "v sdiv w = word_of_int (sint v sdiv sint w)"
height_balance: fixes t :: "'a tree"   and h :: "nat" assumes "\<not> bal_i (size t) h" shows "height (balance_tree t) < h"
eq4: shows "int (card X) - int (poly_deg f) - int j gchoose (card X - j) = (- 1) ^ (card X - j) * (int (poly_deg f) - 1 gchoose (card X - j))"
fps_diff_code: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "fps_of_ratfps x - fps_of_ratfps y = fps_of_ratfps (x - y)"
card_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "card (f ` A) = card A"
share_shared_eq: fixes \<S>' :: "nat \<Rightarrow> bool option"   and a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list" assumes "\<S>' a = \<S> a" shows "share sb \<S>' a = share sb \<S> a"
that: fixes as' :: "'edge list"   and a' :: "'edge" assumes "as = as' @ [a']"   and "n -as'\<rightarrow>\<^sub>\<surd>* sourcenode a'"   and "valid_edge a'"   and "n' = targetnode a'" shows "thesis"
check_poly_weak_anti_mono_smart_def: fixes discrete :: "bool" shows "check_poly_weak_anti_mono_smart discrete \<equiv> if discrete then check_poly_weak_anti_mono_discrete else check_poly_weak_anti_mono"
assms_3: shows "ord_Zp y < ord_Zp x"
a_get_mode_locs_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "get_mode_locs shadow_root_ptr \<equiv> {preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode)}"
singletonI: fixes A :: "'c set"   and y :: "'c" assumes "\<And>x. x \<in> A \<Longrightarrow> x = y"   and "y \<in> A" shows "A = {y}"
lhd_LCons: fixes x21 :: "'c"   and x22 :: "'c llist" shows "lhd (LCons x21 x22) = x21"
preallocated_def: fixes h :: "'heap" shows "preallocated h \<equiv> \<forall>C\<in>sys_xcpts. typeof_addr h (addr_of_sys_xcpt C) = \<lfloor>Class_type C\<rfloor>"
priK_keys_of: fixes A :: "agent" shows "priK A \<in> keys_of A"
SN_rel_on_alt_def: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_rel_on_alt R S T = (\<forall>f. (\<forall>i. (f i, f (Suc i)) \<in> R \<union> S) \<and> f 0 \<in> T \<longrightarrow> \<not> (\<exists>\<^sub>\<infinity>j. (f j, f (Suc j)) \<in> R))"
gcd_dvd_lcm: fixes a :: "'a"   and b :: "'a" shows "gcd a b dvd lcm a b"
invariant_iff_compl_invariant: fixes M :: "'a set" shows "invariant M = invariant (X - M)"
cps23_3: fixes x :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "x \<doteq> y \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then if x \<tau> = null \<or> y \<tau> = null then \<lfloor>\<lfloor>x \<tau> = null \<and> y \<tau> = null\<rfloor>\<rfloor> else \<lfloor>\<lfloor>oid_of (x \<tau>) = oid_of (y \<tau>)\<rfloor>\<rfloor> else invalid \<tau>"
finite_atLeastAtMost: fixes l :: "nat"   and u :: "nat" shows "finite {l..u}"
simps_12: fixes b :: "char list exp"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "fv (if (b) e\<^sub>1 else e\<^sub>2) = fv b \<union> fv e\<^sub>1 \<union> fv e\<^sub>2"
dickson_grading_dgrad_dummy: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.graded_dickson_powerprod plus minus zero lcs" shows "plus.dickson_grading plus (graded_dickson_powerprod.dgrad_dummy plus)"
OUTfromV_VARto_holds: shows "OUTfromV_VARto"
dgrad_p_set_closed_red_rtrancl: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "p \<in> dgrad_p_set d m"   and "(red F)\<^sup>*\<^sup>* p q" shows "q \<in> dgrad_p_set d m"
divide_fps_const: fixes f :: "'b fps"   and c :: "'b" shows "f / fps_const c = fps_const (inverse c) * f"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Const x2 = Const y2) = (x2 = y2)"
real_cond_exp_diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M Invariants (\<lambda>x. f x - g x) x = real_cond_exp M Invariants f x - real_cond_exp M Invariants g x"
h1_def: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c" shows "h1 f s = f s"
mkv_lemma: fixes ODE :: "('sf, 'sz) ODE"   and x :: "'sz"   and y :: "'sz"   and I :: "('sf, 'a, 'sz) interp"   and a :: "(real, 'sz) vec"   and b :: "(real, 'sz) vec"   and c :: "(real, 'sz) vec" assumes "ORadmit ODE" shows "Radj x y (mk_v I (OUrename x y ODE) (a, b) c) = mk_v I ODE (RSadj x y a, RSadj x y b) (RSadj x y c)"
simps: fixes a :: "fmla" shows "(a \<in> special_axioms) = (\<exists>A i x. a = A(i::=x) IMP exi i A)"
pi_half_ge_zero: shows "0 \<le> pi / 2"
assms: shows "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = Some l"
join_ir_nth: fixes i :: "nat"   and "is" :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "i < length is" shows "join_ir is r (I i) = Some (is ! i)"
prems_3: shows "no \<in> set_of (Node lt a rt)"
prj_simps_1: shows "C.seq fg.prj\<^sub>1 \<p>\<^sub>1[g.leg0 \<cdot> fg.prj\<^sub>0, f.leg1]"
normalized_src_ports_def2: fixes ms :: "'b common_primitive match_expr" shows "normalized_src_ports ms = normalized_n_primitive (is_Src_Ports, src_ports_sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) ms"
MP: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "\<turnstile> p \<^bold>\<longrightarrow> q"   and "\<turnstile> p" shows "\<turnstile> q"
change_Lam_Variable: fixes y' :: "Vars.var"   and y :: "Vars.var"   and e :: "exp" assumes "y' \<noteq> y \<Longrightarrow> atom y' \<sharp> (e, y)" shows "Lam [y]. e = Lam [y']. ((y \<leftrightarrow> y') \<bullet> e)"
ipset_from_cidr_lowest: fixes a :: "'b word"   and n :: "nat" shows "a \<in> ipset_from_cidr a n"
nfoldli_rule: fixes I :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> bool"   and l0 :: "'a list"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b nres"   and P :: "'b \<Rightarrow> bool" assumes "I [] l0 \<sigma>0"   and "\<And>x l1 l2 \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I l1 (x # l2) \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (l1 @ [x]) l2)"   and "\<And>l1 l2 \<sigma>. \<lbrakk>l0 = l1 @ l2; I l1 l2 \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma>. \<lbrakk>I l0 [] \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "nfoldli l0 c f \<sigma>0 \<le> SPEC P"
allDecisionsThenExistsBackjumpLevel: fixes M :: "(Literal \<times> bool) list"   and c :: "Literal list"   and l :: "Literal" assumes "uniq (elements M)"   and "\<forall>l'. l' el c \<longrightarrow> opposite l' el decisions M"   and "isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M)" shows "\<exists>level. isBackjumpLevel level l c M"
assms_5: shows "prv (imp \<chi> (imp \<psi> \<phi>))"
hyps: assumes "p \<circ>\<^sub>p q \<noteq> 0" shows "order x (p \<circ>\<^sub>p q) = order x (q - [:poly q x:]) * order (poly q x) p"
antimono_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
1: shows "(xs, ys) \<in> listrel1 (r\<^sup>*)"
hom_mult: fixes x :: "'pq mod_ring poly"   and y :: "'pq mod_ring poly" shows "#(x * y) = #x * #y"
filter_paths_conv_free_restr: fixes S :: "'a set"   and t :: "'a ttree" shows "filter (\<lambda>x'. x' \<in> S) ` paths t = paths (ttree_restr S t)"
ASSUME_leI: fixes \<Phi> :: "bool"   and M :: "'ma"   and M' :: "'ma" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "bind (ASSUME \<Phi>) (\<lambda>_. M) \<le> M'"
add_replicate: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "foldl (+) k (replicate m n) = k + m * n"
eval_wp_Seq: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" shows "wp (a ;; b) P = (wp a \<circ> wp b) P"
contains_propagates_fex: fixes n :: "nat"   and m :: "nat"   and g :: "form" assumes "infinite (deriv s)"   and "contains f n (m, FEx g)" shows "\<exists>y. contains f (n + y) (0, finst g m) \<and> contains f (n + y) (Suc m, FEx g)"
assms: shows "at \<in> set (hOfFax \<sigma>)"
assms_3: fixes x :: "nat" assumes "0 < x"   and "x < c" shows "of_nat x \<noteq> (0::'a)"
simps_5: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_aexp_inputs (Minus v va) = enumerate_aexp_inputs v \<union> enumerate_aexp_inputs va"
nprv_addImp3LemmaI: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<phi>4 :: "'fmla"   and F :: "'fmla set" assumes "prv (imp \<phi>1 (imp \<phi>2 (imp \<phi>3 \<phi>4)))"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "nprv F \<phi>1"   and "nprv F \<phi>2"   and "nprv F \<phi>3" shows "nprv F \<phi>4"
tl_append: fixes xs :: "'b list"   and ys :: "'b list" shows "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z # zs \<Rightarrow> zs @ ys)"
mapWithIndex_def: fixes f :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and xs :: "'e list" shows "mapWithIndex f xs = map (\<lambda>(i, t). f i t) (enumerate 0 xs)"
divide_le_posI_ennreal: fixes x :: "ennreal"   and z :: "ennreal"   and y :: "ennreal" assumes "0 < x"   and "z \<le> x * y" shows "z / x \<le> y"
fin_dim_1: shows "vectorspace.fin_dim class_ring (vs (local.span {vec n f}))"
diagseq_mono: fixes n :: "nat" shows "diagseq n < diagseq (Suc n)"
iMOD_plus: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and k :: "nat" assumes "x \<in> [ r, mod m ]" shows "(x + k \<in> [ r, mod m ]) = (k mod m = 0)"
supchamberD_1: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "folding_g.chamber (supchamber v)"
simps_2: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and i :: "String.literal"   and l :: "tm list" shows "semantics_term e f (Fn i l) = f i (semantics_list e f l)"
subspace_sum_orthogonal_comp: fixes U :: "'a set" assumes "subspace U" shows "U + U\<^sup>\<bottom> = UNIV"
minus_one_power_iff: fixes n :: "nat" shows "(- (1::'b)) ^ n = (if even n then 1::'b else - (1::'b))"
Cauchy_NSCauchy: fixes X :: "nat \<Rightarrow> 'a" assumes "Cauchy X" shows "NSCauchy X"
wf_rcv_append''': fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and t :: "('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
vcpower_of_vempty: fixes n :: "V" assumes "n \<noteq> []\<^sub>\<circ>" shows "[]\<^sub>\<circ> ^\<^sub>\<times> n = []\<^sub>\<circ>"
foldl_list_all2_Cons1: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and ys :: "'b list"   and s :: "'c" shows "foldl_list_all2 f P Q (x # xs) ys s = (\<exists>y ysa. ys = y # ysa \<and> Q x y s \<and> (P x y s \<longrightarrow> foldl_list_all2 f P Q xs ysa (f x y s)))"
substitute_only_empty_both: fixes f :: "'a \<Rightarrow> 'a ttree"   and t' :: "'a ttree"   and T :: "'a set"   and t :: "'a ttree" assumes "const_on f (carrier t') TTree.empty" shows "substitute f T (t \<otimes>\<otimes> t') = substitute f T t \<otimes>\<otimes> t'"
image_Pow_surj: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A = B" shows "(`) f ` Pow A = Pow B"
dom_sl_def: fixes G :: "nat graph_rec" shows "dom_sl G \<equiv> cfg_doms G"
nlex_sum: fixes X :: "'a set"   and f :: "'a \<Rightarrow> real \<times> real" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> Counterclockwise_2D_Arbitrary.lex (f x) 0" shows "Counterclockwise_2D_Arbitrary.lex (sum f X) 0"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R\<setminus>I"   and "b \<in> R\<setminus>I" shows "a \<cdot> b \<in> R\<setminus>I"
VSigma_iff: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" shows "(\<langle>a, b\<rangle> \<in> elts (VSigma A B)) = (a \<in> elts A \<and> b \<in> elts (B a))"
ParamUses_call_source_length: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length (ParamUses (sourcenode a)) = length ins"
step: fixes x :: "'a" assumes "\<forall>i<k. P (iterate i\<cdot>f\<cdot>x)" shows "P (iterate k\<cdot>f\<cdot>x)"
neg_inf_eq_sup: fixes 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 inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "uminus (inf a b) = sup (uminus a) (uminus b)"
sees_field_i_i_i_o_iI: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xc :: "ty"   and xd :: "char list" assumes "x \<turnstile> xa sees xb:xc in xd" shows "pred.eval (sees_field_i_i_i_o_i x xa xb xd) xc"
simps_12: fixes f1 :: "'a"   and f2 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "BitList \<Rightarrow> 'a \<Rightarrow> 'a" shows "rec_BitList f1 f2 f3 BitListNil = f1"
Ass: fixes C :: "'b literal set"   and S :: "'b literal set set" assumes "C \<in> S" shows "S \<turnstile> C"
complement_intro: fixes z :: "i"   and x :: "i" assumes "CMC"   and "\<forall>w. P w z = (\<not> O w x)" shows "\<midarrow> x = z"
coeff_uminus: fixes lp :: "linear_poly"   and var :: "nat" shows "coeff (- lp) var = - coeff lp var"
perm_empty2: fixes xs :: "'a list" shows "(xs <~~> []) = (xs = [])"
smult_carrier_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and k :: "'c" assumes "A \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m A \<in> carrier_mat nr nc"
dist_degree_factorize_main: fixes v :: "'a mod_ring poly"   and w :: "'a mod_ring poly"   and d :: "nat"   and res :: "(nat \<times> 'a mod_ring poly) list"   and facts :: "(nat \<times> 'a mod_ring poly) list"   and u :: "'a mod_ring poly" assumes "local.dist_degree_factorize_main v w d res = facts"   and "w = monom 1 1 ^ CARD('a) ^ d mod v"   and "square_free u"   and "monic u"   and "u = v * prod_list (map snd res)"   and "\<And>f. \<lbrakk>irreducible f; f dvd v\<rbrakk> \<Longrightarrow> d < degree f"   and "\<And>i f. (i, f) \<in> set res \<Longrightarrow> i \<noteq> 0 \<and> degree f \<noteq> 0 \<and> monic f \<and> (\<forall>g. irreducible g \<longrightarrow> g dvd f \<longrightarrow> degree g = i)" shows "u = prod_list (map snd facts) \<and> (\<forall>i f. (i, f) \<in> set facts \<longrightarrow> factors_of_same_degree i f)"
ord_unique: fixes x :: "'a"   and d :: "nat" assumes "x \<in> carrier (mult_of R)" shows "(G.ord x = d) = (\<forall>n. (x [^] n = \<one>) = (d dvd n))"
solves_ode_domainD: fixes y :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "real set"   and X :: "'d set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
trimmed_neg_def: fixes F :: "'a" shows "trimmed_neg F = (trimmed F \<and> fst (dominant_term F) < 0)"
filterlim_inverse_at_infinity: shows "filterlim inverse at_infinity (at (0::'a))"
aXb': shows "Derives1 \<delta> i r ((a @ [X]) @ b)"
sim_action_is_write_action_eq: fixes obs :: "('a, 'b) obs_event action"   and obs' :: "('a, 'b) obs_event action" assumes "obs \<approx> obs'" shows "is_write_action obs = is_write_action obs'"
simps_6: fixes i :: "'a"   and p :: "'a pfm" shows "reduce (K\<^sub>! i p) = K\<^sub>! i (reduce p)"