hcju/isabelleps huggingface dataset:
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>t1 t2. P (Eq t1 t2)"   and "\<And>x y. P (Neg (Eq (Const x) (Const y)))"   and "\<And>x y. P (Neg (Eq (Var x) (Var y)))"   and "\<And>e ts. P (Pred e ts)"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>; \<And>x3. \<psi> = Neg x3 \<Longrightarrow> P x3\<rbrakk> \<Longrightarrow> P (Neg (Or (Neg \<phi>) \<psi>))"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (Or \<phi> \<psi>)"   and "\<And>\<phi>. P \<phi> \<Longrightarrow> P (Exists \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (Prev I \<phi>)"   and "\<And>I \<phi>. P \<phi> \<Longrightarrow> P (Next I \<phi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>P \<phi>; \<And>x3. \<phi> = Neg x3 \<Longrightarrow> P x3; P \<psi>\<rbrakk> \<Longrightarrow> P (Since \<phi> I \<psi>)"   and "\<And>\<phi> I \<psi>. \<lbrakk>P \<phi>; \<And>x3. \<phi> = Neg x3 \<Longrightarrow> P x3; P \<psi>\<rbrakk> \<Longrightarrow> P (Until \<phi> I \<psi>)"   and "\<And>va vb. P (Neg (Pred va vb))"   and "\<And>vc v. P (Neg (Eq (Var vc) (Const v)))"   and "\<And>v vc. P (Neg (Eq (Const v) (Var vc)))"   and "\<And>va. P (Neg (Neg va))"   and "\<And>v vc vb. P (Neg (Or (Pred v vc) vb))"   and "\<And>v vc vb. P (Neg (Or (Eq v vc) vb))"   and "\<And>v vc vb. P (Neg (Or (Or v vc) vb))"   and "\<And>v vb. P (Neg (Or (Exists v) vb))"   and "\<And>v vc vb. P (Neg (Or (Prev v vc) vb))"   and "\<And>v vc vb. P (Neg (Or (Next v vc) vb))"   and "\<And>v vc vd vb. P (Neg (Or (Since v vc vd) vb))"   and "\<And>v vc vd vb. P (Neg (Or (Until v vc vd) vb))"   and "\<And>va. P (Neg (Exists va))"   and "\<And>va vb. P (Neg (Prev va vb))"   and "\<And>va vb. P (Neg (Next va vb))"   and "\<And>va vb vc. P (Neg (Since va vb vc))"   and "\<And>va vb vc. P (Neg (Until va vb vc))" shows "P a0"
circ_plus_same: fixes x :: "'a" shows "x\<^sup>\<circ> * x = x * x\<^sup>\<circ>"
Y_dim: shows "Y \<subseteq> carrier_vec (n + 1)"
is_full_pmdlI_lt_finite: fixes B :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "finite B"   and "\<And>k. k \<in> component_of_term ` Keys B \<Longrightarrow> \<exists>b\<in>B. b \<noteq> 0 \<and> component_of_term (lt b) = k \<and> lp b = (0::'a)" shows "is_full_pmdl B"
set_bset_eqvt: fixes p :: "perm"   and B :: "'b set['c]" shows "p \<bullet> set_bset B = set_bset (p \<bullet> B)"
inner_loop_def: fixes dst :: "'c"   and succ :: "'c set"   and u :: "'d"   and PRED :: "'c \<Rightarrow> 'd option"   and N :: "'c set" shows "inner_loop dst succ u PRED N \<equiv> FOREACH\<^sub>C\<^bsup>\<lambda>it (f, PRED', N'). PRED' = map_mmupd PRED (succ - it - dom PRED) u \<and> N' = N \<union> (succ - it - dom PRED) \<and> f = (dst \<in> succ - it - dom PRED)\<^esup> succ (\<lambda>(f, PRED, N). \<not> f) (\<lambda>v (f, PRED, N). if v \<in> dom PRED then return (f, PRED, N) else let PRED = PRED(v \<mapsto> u) in assert (v \<notin> N) \<bind> (\<lambda>_. let N = insert v N in return (v = dst, PRED, N))) (False, PRED, N)"
ex_3SPRAY: assumes "\<E> \<noteq> {}" shows "Bex \<E> three_SPRAY"
fresh_heap_expr': fixes a :: "atom"   and \<Gamma> :: "('a \<times> 'b) list"   and e :: "'b" assumes "a \<sharp> \<Gamma>"   and "e \<in> snd ` set \<Gamma>" shows "a \<sharp> e"
type_of_hext_type_of: fixes h :: "'heap"   and w :: "'addr val"   and T :: "ty"   and h' :: "'heap" assumes "typeof\<^bsub>h\<^esub> w = \<lfloor>T\<rfloor>"   and "h \<unlhd> h'" shows "typeof\<^bsub>h'\<^esub> w = \<lfloor>T\<rfloor>"
induct: fixes P :: "'a spmf \<Rightarrow> ('b + 'c \<times> 'd + 'e \<times> 'f) list \<Rightarrow> bool"   and core :: "('a, 'b, 'c, 'e, 'd, 'f) core"   and a0 :: "'a spmf"   and a1 :: "('b + 'c \<times> 'd + 'e \<times> 'f) list" assumes "\<And>p. P p []"   and "\<And>p e tr. P (mk_lossless (p \<bind> (\<lambda>s. cpoke core s e))) tr \<Longrightarrow> P p (Inl e # tr)"   and "\<And>p ia oa tr. P (cond_spmf_fst (p \<bind> (\<lambda>s. cfunc_adv core s ia)) oa) tr \<Longrightarrow> P p (Inr (Inl (ia, oa)) # tr)"   and "\<And>p iu ou tr. P (cond_spmf_fst (p \<bind> (\<lambda>s. cfunc_usr core s iu)) ou) tr \<Longrightarrow> P p (Inr (Inr (iu, ou)) # tr)" shows "P a0 a1"
alignUp_idem: fixes a :: "'b word"   and n :: "nat" assumes "is_aligned a n"   and "n < LENGTH('b)" shows "alignUp a n = a"
ext_method_simp: fixes m :: "qtname \<times> 'a" shows "(declclass m, mthd m) = m"
worst_chain_pred: fixes x :: "'a"   and i :: "nat" assumes "local.bad x" shows "p (local.worst_chain i) (local.worst_chain (Suc i))"
inject_2: fixes x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption"   and y21 :: "'g"   and y22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "(Resumption.resumption.Pause x21 x22 = Resumption.resumption.Pause y21 y22) = (x21 = y21 \<and> x22 = y22)"
p: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) Call p (Q' Z),(A' Z)"
Units_def: shows "transformation.symmetric.Units = {u \<in> transformation.Sym. transformation.symmetric.invertible u}"
StuckPropBlock: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" shows "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Stuck) \<rightarrow> (nrms, css, Stuck)"
one_cong: fixes m :: "int" assumes "residues m" shows "\<one>\<^bsub>residue_ring m\<^esub> = 1 mod m"
max_add_distrib_left: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "subset_mset.max x y + z = subset_mset.max (x + z) (y + z)"
word_diff_ls'_2: fixes y :: "'b word"   and w :: "'b word"   and x :: "'b word" assumes "y < w + x"   and "x \<le> y" shows "y - x < w"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "CCC.seq g f = (C.seq (fst g) (fst f) \<and> CC.seq (snd g) (snd f))"
set_child_nodes_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs" shows "\<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
wordinterval_CIDR_split1_innard_helper: fixes r :: "'b wordinterval"   and a :: "'b word" assumes "wordinterval_lowest_element r = Some a" shows "largest_contained_prefix a r \<noteq> None"
map_le_refl: fixes f :: "'b \<Rightarrow> 'c option" shows "f \<subseteq>\<^sub>m f"
not_empty_eq_Iic_eq_empty: fixes h :: "'b multiset" shows "{} \<noteq> subset_mset.atMost h"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow> g\<guillemotright>" shows "trg \<mu> = trg f"
iop_closed: shows "iop G \<in> carrier G \<rightarrow> carrier G"
simps_1: fixes queue :: "'a list"   and t :: "'a"   and x :: "'b"   and m :: "'c" shows "Round_Robin.enqueue_new_thread queue (NewThread t x m) = queue @ [t]"
sequence_of_def: fixes A :: "'c set"   and enum :: "nat \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'b" shows "sequence_of A enum f = map (\<lambda>n. f (enum n)) [0..<card A]"
fin_S: shows "finite {(R, f). \<exists>j. fst (R, f) \<in> Rs \<and> graph_homomorphism (fst (fst (R, f))) (graph_of (X (j + i))) (snd (R, f)) \<and> nextMax (Range (snd (R, f))) \<le> N}"
ins_rule: fixes t :: "'a btree"   and k :: "nat"   and ti :: "'a btnode ref option"   and x :: "'a" assumes "sorted_less (inorder t)" shows "<btree_assn k t ti> ins k x ti <btupi_assn k (abs_split.ins k x t)>\<^sub>t"
A: fixes n :: "trm"   and u :: "trm"   and v :: "trm"   and l :: "stack" assumes "x \<sharp> (z, n)"   and "t = u to x in v"   and "k = [z]n\<ggreater>l"   and "r = (u to x in v to z in n) \<star> l" shows "P"
0: shows "S'.set (\<Phi> a) = \<phi> ` S.set a"
collapse: fixes le_constraint :: "'a le_constraint" shows "Le_Constraint (lec_rel le_constraint) (lec_poly le_constraint) (lec_const le_constraint) = le_constraint"
simps_2: fixes t :: "'b Trace"   and s :: "'b" shows "tLast (t \<leadsto> s) = s"
dim_ty_sem_Mass_def: fixes uu :: "M itself" shows "dim_ty_sem uu = \<^bold>M"
simps_12: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and x3 :: "'c" shows "(case Wit x3 of Oldf x \<Rightarrow> f1 x | Tag x \<Rightarrow> f2 x | Wit x \<Rightarrow> f3 x) = f3 x3"
ring_r_one: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> 1\<^sub>r\<^bsub>R\<^esub> = x"
has_derivative_inner_left: fixes g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and b :: "'b" assumes "(g has_derivative g') F" shows "((\<lambda>x. g x \<bullet> b) has_derivative (\<lambda>x. g' x \<bullet> b)) F"
fold_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and t :: "('d, 'e) rbt" shows "RBT_Impl.fold f t = fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
coeff_int_monom_mult: fixes a :: "'b"   and d :: "nat"   and f :: "'b poly"   and i :: "int" shows "coeff_int (monom a d * f) i = a * coeff_int f (i - int d)"
fold_fun_left_comm: fixes A :: "name set"   and x :: "name"   and z :: "tm" assumes "finite A" shows "subst x (F x) (Finite_Set.fold (\<lambda>i. subst i (F i)) z A) = Finite_Set.fold (\<lambda>i. subst i (F i)) (subst x (F x) z) A"
ide_implies_can: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.can a"
h_le_g: fixes e :: "'v \<times> 'v" shows "h e \<le> g e"
tvs_mk_all': fixes ty :: "typ"   and idn :: "variable"   and B :: "Core.term" shows "tvs (mk_all idn ty B) = tvs B \<union> tvsT ty"
sublist_split_trans: fixes l1 :: "'a list"   and h :: "'a"   and l1' :: "'a list"   and l2 :: "'a list"   and l2a :: "'a list"   and l2b :: "'a list" assumes "l1 = h # l1'"   and "l2 = l2a @ [h] @ l2b"   and "subseq l1 l2"   and "\<not> ListMem h l2a" shows "subseq l1' l2b"
expands_to_basic: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "basis_wf (b # basis)"   and "length basis = expansion_level TYPE('a)" shows "(b expands_to MS (MSLCons (const_expansion 1, 1) MSLNil) b) (b # basis)"
inner_Basis: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and u :: "'d"   and v :: "'d" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
mon_of_entry: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and p :: "'g" assumes "flowgraph fg" shows "mon_n fg (entry fg p) = mon fg p"
proper_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf" assumes "Reachability_Problem K S S1 S2" shows "Reachability_Problem.proper K S S1 S2 ct = (ct \<in> Pi\<^sub>E S K \<and> (\<forall>s\<in>Reachability_Problem.S\<^sub>r K S S1 S2. 0 < Reachability_Problem.v K S S1 S2 (Finite_Markov_Decision_Process.simple K S ct s)))"
simps: fixes s :: "'s"   and f :: "'s \<Rightarrow> 'm" shows "local.put_state s (StateT f) = StateT (\<lambda>_. f s)"
ltl_def: fixes llist :: "'c llist" shows "ltl llist = (case llist of LNil \<Rightarrow> LNil | LCons x21 x22 \<Rightarrow> x22)"
lebesgue_measurable_vimage_borel: fixes f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "f \<in> borel_measurable lebesgue"   and "T \<in> sets borel" shows "{x. f x \<in> T} \<in> sets lebesgue"
Collect_case_prod_Sigma: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
comp_def: fixes g :: "'c"   and f :: "'c" shows "g \<cdot>\<^sup>o\<^sup>p f \<equiv> f \<cdot>\<^sub>C g"
assms: shows "set \<Phi>\<^sub>2 \<subseteq> set \<Phi>\<^sub>1"
t5: shows "typ_of (subst_bv (Fv x \<tau>') s) = typ_of (Abs \<tau>' s $ Fv x \<tau>')"
D_mono: fixes A :: "'b set option"   and A' :: "'b set option"   and e :: "('b, 'c, 'd) exp" assumes "A \<sqsubseteq> A'"   and "\<D> e A" shows "\<D> e A'"
norm_triangle_ineq: fixes x :: "'a"   and y :: "'a" shows "norm (x + y) \<le> norm x + norm y"
distinct_conv_nth: fixes xs :: "'c list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
assms_4: shows "pd_kleene_allegory_class.acyclic w"
s4: fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" shows "rel_ar (y O rel_ar x) O rel_ar x = rel_ar y O rel_ar x"
convergent_norm: fixes f :: "nat \<Rightarrow> 'a" assumes "convergent f" shows "convergent (\<lambda>n. norm (f n))"
higher_differentiable_on_divide: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<noteq> (0::'c)"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x / g x) n"
reduced_bdt_checkers_axioms: shows "reduced_bdt_checkers Boolean_Expression_Example.ifex_of bval"
star_trans_eq: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
select_convs_1: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cLengthF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cLengthF"
gcsrp_map_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "gcsrp_map G H \<equiv> \<lambda>X\<in>set_rcs G H. SOME x. x \<in> X"
Beta: fixes r :: "lam"   and x :: "name"   and s :: "lam"   and ss :: "lam list"   and t :: "lam" assumes "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>l t" shows "(Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>l t"
f_accessible_def: fixes \<delta> :: "('a, 'b) ta_rule set"   and Q0 :: "'a set" shows "f_accessible \<delta> Q0 \<equiv> (f_succ \<delta>)\<^sup>* `` Q0"
span_superset: fixes S :: "'c set" shows "S \<subseteq> m2.span S"
foldl_cons_def: fixes g :: "('f, 'g) generator"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and z :: "'e"   and s :: "'g" shows "foldl_cons g f z s = foldl f z (unstream g s)"
S_def: shows "S \<equiv> \<Union>t\<in>T. {t - {LNil}}"
P_\<psi>: fixes s :: "'s" assumes "P s" shows "s \<notin> \<psi>"
nth_bin_to_bl_aux: fixes n :: "nat"   and m :: "nat"   and bl :: "bool list"   and w :: "int" assumes "n < m + length bl" shows "bin_to_bl_aux m w bl ! n = (if n < m then bit w (m - 1 - n) else bl ! (n - m))"
induct_matrix_elementary: fixes P :: "((real, 'a) vec, 'a) vec \<Rightarrow> bool"   and A :: "((real, 'a) vec, 'a) vec" assumes "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (A ** B)"   and "\<And>A i. row i A = 0 \<Longrightarrow> P A"   and "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A $ i $ j = 0) \<Longrightarrow> P A"   and "\<And>m n. m \<noteq> n \<Longrightarrow> P (\<chi>i j. mat 1 $ i $ Fun.swap m n id j)"   and "\<And>m n c. m \<noteq> n \<Longrightarrow> P (\<chi>i j. if i = m \<and> j = n then c else of_bool (i = j))" shows "P A"
rel_transp: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "transp R1"   and "transp R2" shows "transp (tllist_all2 R1 R2)"
bisim2: shows "P,es,h \<turnstile> (es', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"
IH: shows "(x \<oplus> y) [^] n = (\<Oplus>i\<in>{..n}. [(n choose i)] \<cdot> (x [^] i \<otimes> y [^] (n - i)))"
k_exists: shows "\<exists>i. local.F (Suc i) = 0"
bounds: shows "real_of_float (lb_arctan prec (- x)) \<le> arctan (real_of_float (- x)) \<and> arctan (real_of_float (- x)) \<le> real_of_float (ub_arctan prec (- x))"
zero_diff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and a :: "'k" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
Times_same_infinite_bij_betw: fixes A :: "'a set" assumes "infinite A" shows "\<exists>f. bij_betw f (A \<times> A) A"
owner_iii_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "(Owner n, \<lbrace>Num 3, PubKey A\<rbrace>) \<in> s \<longrightarrow> PriKey A \<in> used s"
cod_comp: fixes g :: "J.Arr"   and f :: "J.Arr" assumes "J.seq g f" shows "J.cod (g \<cdot>\<^sub>J f) = J.cod g"
fps_sin_def: fixes c :: "'b" shows "fps_sin c = Abs_fps (\<lambda>n. if even n then 0::'b else (- (1::'b)) ^ ((n - 1) div 2) * c ^ n / of_nat (fact n))"
cross_ratio_swap_12: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" assumes "cross_ratio_correct p q r s"   and "cross_ratio_correct q p r s" shows "cross_ratio q p r s = 1 / cross_ratio p q r s"
WPO_S_subst: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'b) Term.term" assumes "(s, t) \<in> WPO_S" shows "(s \<cdot> \<sigma>, t \<cdot> \<sigma>) \<in> WPO_S"
simps: fixes n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_1_main n i j A = (if n \<le> j then A else if i = 0 then ??.Jordan_Normal_Form_Existence.step_1_main n (j + 1) (j + 1) A else let i' = i - 1; ev_left = A $$ (i', i'); ev_below = A $$ (j, j); aij = A $$ (i', j) in Let (if ev_left \<noteq> ev_below \<and> aij \<noteq> (0::'b) then add_col_sub_row (aij / (ev_below - ev_left)) i' j A else A) (??.Jordan_Normal_Form_Existence.step_1_main n i' j))"
exec_instr_xcpts: fixes \<sigma>' :: "jvm_state"   and i :: "instr"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics' :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and a :: "nat" assumes "\<sigma>' = exec_instr i P h stk loc C M pc ics' frs sh"   and "fst \<sigma>' = \<lfloor>a\<rfloor>" shows "i = Throw \<or> a \<in> {a. \<exists>x\<in>sys_xcpts. a = addr_of_sys_xcpt x}"
thetaSeqWD_Wbis: shows "thetaSeqWD \<subseteq> Wbis"
lemma_3_12_ii_i: fixes H :: "'a set" assumes "H \<in> normalfilters"   and "H \<in> properfilters"   and "\<forall>x a. x \<notin> H \<longrightarrow> (\<exists>n. x ^ n l\<rightarrow> a \<in> H)" shows "H \<in> maximalfilters"
oexp_empty: fixes r :: "('d \<times> 'd) set" assumes "Well_order r" shows "r ^o {} = {(\<lambda>x. undefined, \<lambda>x. undefined)}"
pairwise_empty: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "pairwise P {}"
valid_path_split_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and as :: "'a list"   and as' :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path kind get_return_edges (as @ as')" shows "CFG.valid_path kind get_return_edges as"
vec_first_lincomb_list: fixes Xs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set Xs \<subseteq> carrier_vec n"   and "m \<le> n" shows "vec_first (n.lincomb_list c Xs) m = m.lincomb_list c (map (\<lambda>v. vec_first v m) Xs)"
is_node_ptr_kind_def: fixes ptr :: "(_) object_ptr" shows "is_node_ptr_kind ptr = (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \<noteq> None)"
big_power: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> O[F](g)" shows "(\<lambda>x. f x ^ n) \<in> O[F](\<lambda>x. g x ^ n)"
linear_list_comb: fixes f :: "term"   and xs :: "term list" assumes "linear f"   and "linears xs"   and "fdisjnt (frees f) (freess xs)"   and "\<not> is_free f" shows "linear (list_comb f xs)"
assms: shows "(ic, p, q) \<in> set (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)"
a_set_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "a_set_disconnected_nodes_locs document_ptr \<equiv> all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update)"
norm_cblinfun_ceuclidean_le: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "norm a \<le> (\<Sum>x\<in>CBasis. norm (cblinfun_apply a x))"
ns_mul_ext_refl_local: fixes ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and s :: "('a \<times> 'a) set" assumes "locally_refl ns A" shows "(A, A) \<in> ns_mul_ext ns s"
scale_minus_left: fixes a :: "'b"   and x :: "'b" shows "- a * x = - (a * x)"
assms: shows "fv\<^sub>s\<^sub>s\<^sub>t S \<subseteq> subst_domain \<sigma>"
decisive6_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} y x" shows "decisive swf A Is {j} x y"
l_new_shadow_root_get_tag_name_axioms_def: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_tag_name_axioms get_tag_name_locs \<equiv> \<forall>h new_shadow_root_ptr h' r ptr'. h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h'"
PO_m1_inv4_rfresh_trans: shows "{m1_inv4_rfresh \<inter> m1_inv3r_init \<inter> m1_inv2r_serv \<inter> m1_inv1r_cache \<inter> m1_secrecy \<inter> m1_inv1_ifresh} TS.trans m1 {> m1_inv4_rfresh}"
hyps: shows "gval (In v l) s = (if s v = None then invalid else gval (fold gOr (map (\<lambda>x. Eq (V v) (L x)) l) (Bc False)) s)"
finite_consistency: fixes C :: "'b fm set set" assumes "consistency C"   and "subset_closed C" shows "consistency (mk_finite_char C)"
finite_natarray2: fixes m :: "nat"   and n :: "nat" shows "finite {(x, y). x < m \<and> y < n}"
vsubst_def: fixes u :: "nat"   and v :: "nat"   and w :: "nat" shows "vsubst u v w = (if u < w then u else if u = w then v else u - 1)"
chain: shows "Complete_Partial_Order.chain (\<le>) (f ` Y)"
interleave_cons2: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "x # l \<in> a \<otimes> x # b"
equivalence_comp_left_complement: fixes x :: "'a" assumes "equivalence x" shows "x * - x = - x"