hcju/isabelleps huggingface dataset:
tagged_division_split_left_inj_content: fixes \<D> :: "('a \<times> 'a set) set"   and S :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set"   and k :: "'a"   and c :: "real" assumes "\<D> tagged_division_of S"   and "(x1, K1) \<in> \<D>"   and "(x2, K2) \<in> \<D>"   and "K1 \<noteq> K2"   and "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"   and "k \<in> Basis" shows "content (K1 \<inter> {x. x \<bullet> k \<le> c}) = 0"
rel_frontier_affine_hull: fixes S :: "'b set" shows "rel_frontier S \<subseteq> affine hull S"
gen_keys_simps_1: shows "RBT_Impl.gen_keys [] rbt.Empty = []"
simps_3: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list" shows "separate (AllowPortFromTo v va vb # y # z) = (if first_bothNet (AllowPortFromTo v va vb) = first_bothNet y then separate (AllowPortFromTo v va vb \<oplus> y # z) else AllowPortFromTo v va vb # separate (y # z))"
sel: fixes x2 :: "'b" shows "the (Some x2) = x2"
real_le_lsqrt: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y"   and "x \<le> y\<^sup>2" shows "sqrt x \<le> y"
pos: shows "(0::'a) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
chop_Nil: fixes n :: "nat" shows "chop n [] = []"
Int_Diff: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" shows "A \<inter> B - C = A \<inter> (B - C)"
RInitThrow: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh' :: "sheap"   and a :: "nat"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, i)\<rfloor>"   and "sh' = sh(C \<mapsto> (sfs, Error))" shows "P \<turnstile> \<langle>RI (C,Throw a) ; [] \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>Throw a,(h, l, sh'),b\<rangle>"
simps: fixes xs :: "'a list" shows "sort_and_count_inversions xs = (if length xs \<le> 1 then (xs, 0) else let (xs1, xs2) = split_list xs; (xs1', m) = sort_and_count_inversions xs1; (xs2', n) = sort_and_count_inversions xs2 in (merge_lists xs1' xs2', m + n + inversion_number_between_sorted xs1' xs2'))"
valid_split: fixes l :: "color list" shows "valid l = (l = [] \<or> l ! 0 = B \<and> valid (tl l) \<or> 3 \<le> length l \<and> (\<forall>i<length l. l ! i = R) \<or> (\<exists>j<length l. 3 \<le> j \<and> (\<forall>i<j. l ! i = R) \<and> l ! j = B \<and> valid (drop (j + 1) l)))"
natural_transformation_axioms_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "natural_transformation_axioms A B F G \<tau> \<equiv> ((\<forall>f. \<not> partial_magma.arr A f \<longrightarrow> \<tau> f = partial_magma.null B) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> partial_magma.dom B (\<tau> f) = F (partial_magma.dom A f))) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> partial_magma.cod B (\<tau> f) = G (partial_magma.cod A f)) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f)"
ennreal_lt_0: fixes x :: "real" assumes "x < 0" shows "ennreal x = 0"
coinduct_strong: fixes R :: "('d, 'e, 'f) resumption \<Rightarrow> ('d, 'e, 'f) resumption \<Rightarrow> bool"   and resumption :: "('d, 'e, 'f) resumption"   and resumption' :: "('d, 'e, 'f) resumption" assumes "R resumption resumption'"   and "\<And>resumption resumption'. R resumption resumption' \<Longrightarrow> is_Done resumption = is_Done resumption' \<and> (is_Done resumption \<longrightarrow> is_Done resumption' \<longrightarrow> result resumption = result resumption') \<and> (\<not> is_Done resumption \<longrightarrow> \<not> is_Done resumption' \<longrightarrow> output resumption = output resumption' \<and> ((=) ===> (\<lambda>resumption resumption'. R resumption resumption' \<or> resumption = resumption')) (resume resumption) (resume resumption'))" shows "resumption = resumption'"
uncurry_def: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
beta: fixes l :: "Label"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and a :: "sterm" assumes "l \<in> dom f"   and "lc (Obj f T)"   and "lc a" shows "Call (Obj f T) l a -> (the (f l)\<^bsup>[Obj f T,a]\<^esup>)"
l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs \<and> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
f1: fixes r :: "real"   and ra :: "real"   and rb :: "real"   and rc :: "real" shows "r + ra * rb - ra * rc = r - ra * (rc - rb)"
L3_5v_lemma4: fixes b :: "'a set"   and i :: "'a set"   and c :: "'a com"   and s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a"   and n :: "nat" assumes "(While b i c, s) -Pn\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "\<exists>k. (fwhile b c k, s) -P*\<rightarrow> (Parallel Ts, t)"
subst_lit_is_neg: fixes L :: "'a literal"   and \<sigma> :: "'s" shows "is_neg (L \<cdot>l \<sigma>) = is_neg L"
pr1_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>hfsetcat.pr1 a b : hfsetcat.prod a b \<rightarrow> a\<guillemotright>"
classical_P: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "let PP = \<lambda>t. subst P t xx in prv (B.neg (B.neg (PP \<langle>\<phi>\<rangle>)))" shows "prv (PP \<langle>\<phi>\<rangle>)"
simps: fixes a1 :: "nat list list"   and a2 :: "nat list list"   and a3 :: "nat list list"   and a4 :: "nat list list" shows "merge a1 a2 a3 a4 = (a1 = [] \<and> a2 = [] \<and> a3 = [] \<and> a4 = [] \<or> (\<exists>as. a1 = as \<and> a2 = [] \<and> a3 = [concat as] \<and> a4 = [] \<and> as \<noteq> []) \<or> (\<exists>as1 bs1 as2 bs2 as bs. a1 = as1 @ as2 \<and> a2 = bs1 @ bs2 \<and> a3 = concat as1 # as \<and> a4 = concat bs1 # bs \<and> as1 \<noteq> [] \<and> bs1 \<noteq> [] \<and> concat as1 < concat bs1 \<and> concat bs1 < concat as2 \<and> merge as2 bs2 as bs))"
simps_12: fixes C :: "char list"   and M :: "char list"   and es :: "'c exp list" shows "ss_exp (C\<bullet>\<^sub>sM(es)) = (case map_vals_of es of None \<Rightarrow> the (ss_exps es) | \<lfloor>t\<rfloor> \<Rightarrow> C\<bullet>\<^sub>sM(es))"
hyps_3: shows "sum f F_ = \<Sum> (f ` F_)"
normloat_id: fixes x :: "float" shows "normfloat x = x"
walk_take: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and n :: "nat" assumes "Digraph G"   and "Digraph.walk G xs" shows "Digraph.walk G (take n xs)"
sepDom: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "dom (Cp (list2FWpolicy p)) = dom (Cp (list2FWpolicy (separate p)))"
simps_2: fixes x :: "'a"   and p :: "nat"   and m :: "('a \<times> nat) list"   and n :: "('a \<times> nat) list" shows "monom_mult_list ((x, p) # m) n = (case n of [] \<Rightarrow> (x, p) # m | (y, q) # n' \<Rightarrow> if x = y then (x, p + q) # monom_mult_list m n' else if x < y then (x, p) # monom_mult_list m n else (y, q) # monom_mult_list ((x, p) # m) n')"
map_relation_def: fixes f :: "'c \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "map_relation f R = (\<lambda>x y. R (f x) (f y))"
th: shows "\<forall>j\<in>set [1..c * d]. bound0 (simpfm (\<sigma> p c (Add e (C j))))"
restrict_spmf_eq_return_pmf_None: fixes p :: "'c spmf"   and A :: "'c set" shows "(p \<upharpoonleft> A = return_pmf None) = (set_spmf p \<inter> A = {})"
args_def_or_valid: shows "\<tau> \<Turnstile> \<delta> self \<and> f_\<upsilon> a1 \<tau>"
left_neutral: fixes a :: "'g multiset" shows "{#} \<union># a = a"
assms_2: fixes p :: "'a" assumes "p \<in> prime_factors x" shows "multiplicity p x \<le> multiplicity p y"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#size ta, size sa#} < {#size t, size s#}"   and "wary ta"   and "wary sa"   and "ta >\<^sub>t sa" shows "subst \<rho> ta >\<^sub>t subst \<rho> sa"
commute: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "inf_prod a b = inf_prod b a"
mertens_bound_strong: fixes n :: "nat" assumes "0 < n" shows "\<MM> (real n) - ln (real n) \<in> {- 1 - 9 / pi\<^sup>2<..ln 4}"
eq_alt_conversep: shows "(=) = (BNF_Def.Grp UNIV id)\<inverse>\<inverse>"
aallI: fixes vs :: "state \<Rightarrow> 'a"   and F :: "(state \<Rightarrow> 'b) \<Rightarrow> behavior \<Rightarrow> bool"   and sigma :: "behavior" assumes "basevars vs"   and "\<And>x. basevars (x, vs) \<Longrightarrow> F x sigma" shows "(\<forall>\<forall> x. F x) sigma"
dom_char: shows "Span.dom = (\<lambda>\<mu>. if Span.arr \<mu> then \<lparr>Chn = span_in_category.apex (\<odot>) (Dom \<mu>), Dom = Dom \<mu>, Cod = Dom \<mu>\<rparr> else Span.null)"
init_fin_lift_state_def: fixes s :: "status"   and \<sigma> :: "('c, 'd) locks \<times> (('d \<Rightarrow> ('e \<times> 'c \<Rightarrow>f nat) option) \<times> 'f) \<times> ('d \<Rightarrow> 'g wait_set_status option) \<times> 'd set" shows "init_fin_lift_state s \<sigma> = (locks \<sigma>, (\<lambda>t. map_option (\<lambda>(x, ln). ((s, x), ln)) (thr \<sigma> t), shr \<sigma>), wset \<sigma>, interrupts \<sigma>)"
add_mset_add_mset_same_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(a, A = a, B) = (A = B)"
Re_pos_in_ball: fixes z :: "complex"   and t :: "complex" assumes "0 < Re z"   and "t \<in> ball z (Re z / 2)" shows "0 < Re t"
take_cols_var_def: fixes A :: "'b mat"   and inds :: "nat list" shows "take_cols_var A inds = mat_of_cols (dim_row A) (map ((!) (cols A)) inds)"
lin_independent_length_le_dim: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and us :: "'b list" assumes "FinDimVectorSpace smult V"   and "set us \<subseteq> V"   and "R_scalar_mult.R_lin_independent UNIV smult us" shows "length us \<le> R_scalar_mult.dim_R UNIV smult V"
IH: shows "(S, A) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (S, A @ B)"
cospan_\<mu>\<nu>: shows "C.cospan \<mu>.leg0 \<nu>\<pi>.leg1"
h_3: shows "(flow0 x h, Dflow x h o\<^sub>L d) \<in> X1"
if_false_eval\<^sub>w: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "\<not> ev\<^sub>B mem b" shows "\<langle>Stmt.If b t c', mds, mem\<rangle> \<leadsto> \<langle>c', mds, mem\<rangle>"
moebius_comp_cmat_def: fixes M1 :: "complex \<times> complex \<times> complex \<times> complex"   and M2 :: "complex \<times> complex \<times> complex \<times> complex" shows "moebius_comp_cmat M1 M2 = M1 *\<^sub>m\<^sub>m M2"
set_take_disj_set_drop_if_distinct: fixes vs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct vs"   and "i \<le> j" shows "set (take i vs) \<inter> set (drop j vs) = {}"
LNil_all: fixes A :: "'b set" shows "LNil \<in> A\<^sup>\<infinity>"
thick: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "\<exists>D E. D \<in> X - {C} \<and> z \<lhd> D \<and> E \<in> X - {C, D} \<and> z \<lhd> E"
ih: fixes x :: "hf" assumes "\<And>y. (y, x) \<in> hmem_rel \<Longrightarrow> P y" shows "P x"
new_element_get_shadow_root_is_l_new_element_get_shadow_root: shows "l_new_element_get_shadow_root ShadowRootClass.type_wf get_shadow_root get_shadow_root_locs"
image_dual: fixes dual :: "'b \<Rightarrow> 'b" assumes "class.dual dual" shows "(`) dual \<circ> (`) dual = id"
lzip_lmap2: fixes xs :: "'c llist"   and f :: "'e \<Rightarrow> 'd"   and ys :: "'e llist" shows "lzip xs (lmap f ys) = lmap (\<lambda>(x, y). (x, f y)) (lzip xs ys)"
i_set_mult_subset_i_set: fixes k :: "nat" shows "i_set_mult k \<subseteq> i_set"
simps_5: fixes k :: "name"   and e :: "(name, hf) finfun"   and A :: "fm" assumes "atom k \<sharp> e" shows "eval_fm e (SyntaxN.Ex k A) = (\<exists>x. eval_fm (finfun_update e k x) A)"
elims_3: fixes x :: "'e \<times> ('f, 'g, 'h, 'i) SLin_state_scheme" assumes "\<not> P18 x" obtains s1 :: "'e" and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (abortVals s2 \<noteq> {} \<longrightarrow> (\<exists>p. status s2 p \<noteq> Sleep))"
rel_def: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "stk1.rel x y \<equiv> (fst x \<in> I \<and> fst y \<in> I) \<and> (snd x \<in> \<FF>1 (fst x) \<and> snd y \<in> \<FF>1 (fst y)) \<and> (\<exists>W. W \<in> I \<and> W \<subseteq> fst x \<inter> fst y \<and> \<rho>1 (fst x) W (snd x) = \<rho>1 (fst y) W (snd y))"
2: fixes ars :: "String.literal set list"   and y :: "String.literal \<Rightarrow> String.literal set list option"   and type :: "String.literal" assumes "exe_osig_conds sa"   and "trans (fst (translate_osig sa))"   and "\<forall>x\<in>set cto. typ_ok_sig (map_of cto, map_of tao, translate_osig sa) (snd x)"   and "ars \<in> ran y"   and "snd (translate_osig sa) type = Some y" shows "the (map_of tao type) = length ars"
obj_def': fixes a :: "'b" shows "B.obj a = (B.arr a \<and> trg\<^sub>B a = a)"
add_pos_nonneg: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "{#} \<subset># a"   and "{#} \<subseteq># b" shows "{#} \<subset># a + b"
qGood_qTerm_templateInduct: fixes rel :: "(('a, 'b, 'varSort, 'var, 'c) qTerm \<times> ('a, 'b, 'varSort, 'var, 'c) qTerm) set"   and phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" assumes "\<And>X Y. \<lbrakk>qGood X; (X, Y) \<in> rel\<rbrakk> \<Longrightarrow> qGood Y \<and> qSkel Y = qSkel X"   and "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; \<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
case_2: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x21 :: "agent"   and x22 :: "msg" shows "(case Gets x21 x22 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f2 x21 x22"
pdevs_apply_coord_pdevs: fixes i :: "nat"   and x :: "nat" shows "pdevs_apply (coord_pdevs i) x = (if x = i then 1 else 0)"
Image_def: fixes r :: "('b \<times> 'a) set"   and s :: "'b set" shows "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
stirling_row_aux_correct: fixes n :: "'a"   and y :: "'a"   and xs :: "'a list" shows "stirling_row_aux n y xs = zip_with_prev (\<lambda>a b. a + n * b) y xs @ [1::'a]"
C_1: shows "C \<in> components (- (T \<union> U))"
inset: shows "m' \<in> set (normalize_ports_generic normalize_pos rewrite_neg m)"
am_d_def: fixes x :: "'a" shows "d x = ad (ad x)"
Quotient_refl2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'b"   and s :: "'b" assumes "Quotient R Abs Rep T"   and "R r s" shows "R s s"
Cons: fixes n :: "nat"   and m :: "nat"   and ms :: "nat list" shows "omega_sum_aux (Suc n) (m # ms) = \<omega> \<up> ord_of_nat n * ord_of_nat m + omega_sum_aux n ms"
in_atlas_order_le: fixes l :: "enat"   and c :: "('a, 'b) chart" assumes "l \<le> k"   and "c \<in> atlas" shows "c \<in> c_manifold.atlas charts l"
l0f2e: shows "1 / (f2 * (e - 1)) \<le> real l0"
distinct_2: fixes name' :: "name"   and trm' :: "trm"   and name :: "name" shows "Lam [name'].trm' \<noteq> Var name"
cterms_includes_sterms_of_oseq_reachable: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and A :: "((nat \<Rightarrow> 'f) \<times> ('f, 'g, 'e, 'h) seqp, 'g seq_action) automaton"   and i :: "nat"   and S :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> 'g seq_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'f) \<Rightarrow> (nat \<Rightarrow> 'f) \<Rightarrow> bool" assumes "wellformed \<Gamma>"   and "control_within \<Gamma> (init A)"   and "automaton.trans A = oseqp_sos \<Gamma> i" shows "\<Union> (sterms \<Gamma> ` snd ` oreachable A S U) \<subseteq> cterms \<Gamma>"
S5: shows "mod7 (c_fst (c_pair 5 (c_pair n1 n2))) = 5"
assms_3: fixes s :: "'b" assumes "I S s" shows "P s"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "c \<in> A" shows "addA (addA a b) c = addA a (addA b c)"
group_eq_aux: fixes a :: "'a"   and b :: "'a" shows "a + (b - a) = b"
H_assign_var: fixes P :: "(char list \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "\<forall>s. P s \<longrightarrow> Q (s(v := e s))" shows "PRE P (v ::= e) POST Q"
bdd_aboveI2: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and M :: "'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "subset_mset.bdd_above (f ` A)"
homo: shows "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0"
hcomplex_hcnj_hcomplex_of_hypreal: fixes x :: "real star" shows "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
filterof_def: fixes X :: "'a set" shows "filterof X = \<Inter> {F \<in> filters. X \<subseteq> F}"
sup_continuousD: fixes F :: "'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> 'a" assumes "sup_continuous F"   and "incseq M" shows "F (Sup (range M)) = (SUP i. F (M i))"
\<omega>_def: shows "\<omega> \<equiv> ZFC_in_HOL.set (range ord_of_nat)"
Diff_partition: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
alpha_abs_res_abs_set: fixes as :: "atom set"   and x :: "'b"   and p :: "perm"   and as' :: "atom set"   and x' :: "'b" assumes "(as, x) \<approx>res (=) supp p (as', x')" shows "(as \<inter> supp x, x) \<approx>set (=) supp p (as' \<inter> supp x', x')"
before_vs: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs"   and "before vs ram1 ram2" shows "vs = fst (splitAt ram1 vs) @ ram1 # fst (splitAt ram2 (snd (splitAt ram1 vs))) @ ram2 # snd (splitAt ram2 vs)"
union_mulex_on_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and A :: "'b multiset"   and C :: "'b multiset"   and B :: "'b multiset"   and D :: "'b multiset" assumes "mulex_on P F A C"   and "mulex_on P F B D" shows "mulex_on P F (A + B) (C + D)"
"1_1": shows "ns = ns\<^sub>1 @ x # ns'"
trancl_sub_rel_of: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "sub_rel_of g subs" shows "sub_rel_of g (subs\<^sup>+)"
sel: fixes state :: "('b, 'call, 'ret) gpv \<times> 's + ('b, 'call, 'ret) rpv \<times> ('ret \<times> 's, 'call', 'ret') gpv" shows "the_gpv (local.inline_aux state) = (case state of Inl (c, s) \<Rightarrow> map_spmf (\<lambda>result. case result of Inl (x, s) \<Rightarrow> Pure (x, s) | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. local.inline_aux (Inr (rpv, oracle input)))) (local.inline1 c s) | Inr (rpv, c) \<Rightarrow> map_spmf (\<lambda>result. case result of Inl (Inl (x, s)) \<Rightarrow> Pure (x, s) | Inl (Inr (out, oracle, rpv)) \<Rightarrow> IO out (\<lambda>input. local.inline_aux (Inr (rpv, oracle input))) | Inr (out, c) \<Rightarrow> IO out (\<lambda>input. local.inline_aux (Inr (rpv, c input)))) (the_gpv c \<bind> (\<lambda>generat. case generat of Pure (x, s') \<Rightarrow> map_spmf Inl (local.inline1 (rpv x) s') | IO out c \<Rightarrow> return_spmf (Inr (out, c)))))"
set_mset_comp_mset: shows "set_mset \<circ> mset = set"
fresh_3: fixes a :: "atom"   and var :: "Syntax.var"   and "term" :: "Syntax.term" shows "a \<sharp> Lam var term = (a \<sharp> term \<or> a \<in> set [atom var])"
monic_gcd_dvd: fixes f :: "'b poly"   and g :: "'b poly"   and h :: "'b poly" assumes "f dvd g"   and "monic f"   and "gcd g h \<in> {1, g}" shows "gcd f h \<in> {1, f}"
smooth_on_def: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "k-smooth_on S f = (\<forall>n. enat n \<le> k \<longrightarrow> higher_differentiable_on S f n)"
route2match_correct: fixes a :: "('b, 'c) routing_rule_scheme"   and p :: "('b, 'd) simple_packet_scheme" assumes "valid_prefix (routing_match a)" shows "prefix_match_semantics (routing_match a) (p_dst p) = simple_matches (route2match a) p"
tllist_all2_flip: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" shows "tllist_all2 (flip P) (flip Q) xs ys = tllist_all2 P Q ys xs"
stop_\<I>_full: shows "stop_\<I> \<I>_full = \<I>_full"
right_gpv_Inr_transfer: fixes gpv :: "('a, 'c, 'e) gpv" shows "rel_gpv'' (=) (\<lambda>l r. l = Inr r) (\<lambda>l r. l = Inr r) (right_gpv gpv) gpv"
abc_sym: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "[[c b a]]"
prime: fixes x :: "nat" assumes "x \<in> A2" shows "prime x"
lex_two_compat'2: fixes ns1 :: "('b \<times> 'b) set"   and s1 :: "('b \<times> 'b) set"   and s2 :: "('c \<times> 'c) set"   and ns2 :: "('c \<times> 'c) set" assumes "ns1 O s1 \<subseteq> s1"   and "s1 O ns1 \<subseteq> s1"   and "s1 O s1 \<subseteq> s1"   and "ns1 O ns1 \<subseteq> ns1"   and "s2 O ns2 \<subseteq> s2" shows "lex_two s1 ns1 s2 O lex_two s1 ns1 ns2 \<subseteq> lex_two s1 ns1 s2"
guaranteesD: fixes F :: "'d program"   and X :: "'d program set"   and Y :: "'d program set"   and G :: "'d program" assumes "F \<in> X guarantees Y"   and "F ok G"   and "F \<squnion> G \<in> X" shows "F \<squnion> G \<in> Y"
summatory_totient_asymptotics'': shows "sum_upto (\<lambda>n. real (totient n)) \<sim>[at_top] (\<lambda>x. 3 / pi\<^sup>2 * x\<^sup>2)"
is_mset_setD: fixes X :: "'d multiset"   and x :: "'d" assumes "is_mset_set X"   and "x \<in># X" shows "count X x = 1"
n\<phi>: shows "n = cmod \<langle>\<phi>, \<psi>\<rangle> / norm \<phi>"
divide_poly_main_hom: fixes lc :: "'a"   and f :: "'a poly"   and g :: "'a poly"   and h :: "'a poly"   and i :: "nat"   and j :: "nat" shows "map_poly hom (divide_poly_main lc f g h i j) = divide_poly_main (hom lc) (map_poly hom f) (map_poly hom g) (map_poly hom h) i j"
SOME_Basis: shows "(SOME i. i \<in> Basis) \<in> Basis"
length_nfresh: fixes S :: "'a set"   and n :: "nat" shows "length (nfresh S n) = n"
indexed_Ints_vec_def: fixes I :: "nat set" shows "indexed_Ints_vec I = {x. \<forall>i<dim_vec x. i \<in> I \<longrightarrow> x $ i \<in> \<int>}"
ww'_2: shows "w' \<le> - 1 \<and> w' * exp w' = x"
set_bind_pmf: fixes M :: "'d pmf"   and N :: "'d \<Rightarrow> 'c pmf" shows "set_pmf (M \<bind> N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"