hcju/isabelleps huggingface dataset:
keys_monom_multI: fixes v :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and t :: "'b" assumes "v \<in> keys p"   and "c \<noteq> (0::'c)" shows "MPoly_Type_Class.punit.splus t v \<in> keys (MPoly_Type_Class.punit.monom_mult c t p)"
p3: shows "{OrdP (Var k), VarP v} \<turnstile> (SyntaxN.All k' (SeqStTermP v a t u s (Var k) IMP SeqStTermP v a t (Var j') s' (Var k') IMP Var j' EQ u))(j'::=u')"
solve_system_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "solve_system_iarrays A b = (let A' = Gauss_Jordan_iarrays_PA A in (snd A', fst A' *iv b))"
ba1_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
assms_3: shows "c = (m' + a') mod q \<and> y1 = (a' * x1 + b') mod q"
Hausdorff_ac: fixes C :: "'a set" assumes "C \<in> carrier fTo D" shows "\<exists>M\<in>carrier fTo D. C \<subseteq> M \<and> maximal\<^bsub>fTo D\<^esub> M"
add_carrier_mat': fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + B \<in> carrier_mat nr nc"
s1_P2: fixes x :: "nat"   and y :: "nat"   and b :: "nat"   and e2 :: "nat"   and r :: "nat"   and s1 :: "nat" assumes "x < q"   and "y < q"   and "b < q"   and "e2 < q"   and "r < q"   and "s1 < q" shows "((y, b, (e2 * b + q - (x * y + q - r) mod q) mod q, e2), r, (x * y + q - r) mod q) = ((y, b, (e2 * b + q - (x * y + q - r) mod q) mod q, e2), (x * y + q - (x * y + q - r) mod q) mod q, (x * y + q - r) mod q)"
poly_degree_def: fixes p :: "('c monom \<times> 'd) list" shows "poly_degree p \<equiv> max_list (map (\<lambda>(m, c). monom_degree m) p)"
f7: shows "fst (fst (get_curr_win () s1)) = fst (fst (get_curr_win () s2))"
IH: shows "H \<otimes>\<^bsup>n\<^esup> * |Deutsch.zero\<rangle> \<otimes>\<^bsup>n\<^esup> = Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n))"
simps_1: fixes x :: "'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'd"   and g :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'd) list"   and h :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'd) list"   and xs :: "('a \<times> 'e) list" shows "map2_val_pair x f g h xs [] = g xs"
modular_1': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z = x ; (y \<cdot> x\<^sup>\<smile> ; z) \<cdot> z"
simps: fixes a1 :: "'a fset"   and a2 :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)" shows "association_ends_not_unique' a1 a2 = (\<exists>classes associations \<C> from role end\<^sub>1 end\<^sub>2. a1 = classes \<and> a2 = associations \<and> association_ends' classes associations \<C> from role end\<^sub>1 \<and> association_ends' classes associations \<C> from role end\<^sub>2 \<and> end\<^sub>1 \<noteq> end\<^sub>2)"
almost_all_eventually_provable_def: fixes \<phi> :: "'b ltl"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" shows "\<PP>\<^sub>\<infinity> \<phi> \<G> w \<equiv> \<forall>\<^sub>\<infinity>i. \<PP> \<phi> \<G> w i"
hyps_2: fixes vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option" shows "ta_seq_consist P vs (lappend obs obs') = (ta_seq_consist P vs obs \<and> ta_seq_consist P (mrw_values P vs (list_of obs)) obs')"
prems: shows "input2state (xs @ [x]) $ x1 = Some a"
VertauschungA: fixes \<Delta> :: "'b formula list"   and \<DD> :: "'b formula"   and \<EE> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Delta> @ \<DD> # \<EE> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<Delta> @ \<EE> # \<DD> # \<Gamma> \<Rightarrow> \<Theta>"
ex_def: fixes P :: "'b \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.ex P \<equiv> \<lambda>t n. \<exists>y. P y t n"
cdf_nondecreasing: fixes M :: "real measure"   and x :: "real"   and y :: "real" assumes "finite_borel_measure M"   and "x \<le> y" shows "cdf M x \<le> cdf M y"
is_zero_row_upt_k_def: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" shows "is_zero_row_upt_k i k A = (\<forall>j. mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'b))"
indRelLPOp_indRelLPO_eq: fixes Enc :: "'a \<Rightarrow> 'b" shows "encoding.indRelLPOp Enc = (\<lambda>x xa. (x, xa) \<in> encoding.indRelLPO Enc)"
that: fixes c :: "real" assumes "\<And>s. s \<in> complex_cone' \<alpha> - {0} \<Longrightarrow> cmod (stirling_integral n s) \<le> c / cmod s ^ n" shows "thesis"
cases: fixes x :: "dir" obtains "x = left"   | "x = right"
left_treeE: fixes u :: "nat"   and s :: "nat"   and t :: "nat" assumes "u \<in> left_tree s t" shows "u \<in> V \<and> s \<in> set (u \<leadsto> t)"
set_in_cone: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "Vs \<subseteq> cone Vs"
intros_1: fixes lholed :: "Lholed"   and es :: "e list" assumes "lholed = LBase [] []" shows "Lfilled_exact 0 lholed es es"
le_fininf_suffixE: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w :: "'a stream" assumes "traces ind"   and "traces.le_fininf ind w\<^sub>1 w" obtains w\<^sub>2 :: "'a stream" where "traces.eq_inf ind w (w\<^sub>1 @- w\<^sub>2)"
Invariants_sets: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T" shows "sets (qmpt.Invariants M T) = {A \<in> sets M. T -` A \<inter> space M = A}"
P'_def: shows "P' = tensor_P (proj_psi_l R) (proj_k R)"
vars_partitioned_renamings_apart: fixes Cs :: "('a, nat) Term.term literal multiset list" shows "vars_partitioned (subst_cls_lists Cs (renamings_apart Cs))"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "sup (dual xa) (dual x) = dual (inf xa x)"
set_zip_D: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs \<and> y \<in> set ys"
assms: shows "\<forall>f\<in>set fs. is_lit_plus f"
assms_2: shows "s adds\<^sub>p v \<ominus> t"
invariant_def: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat" shows "invariant ts \<S> m \<equiv> valid_ownership \<S> ts \<and> valid_reads m ts \<and> valid_history program_step ts \<and> valid_sharing \<S> ts \<and> tmps_distinct ts \<and> valid_data_dependency ts \<and> valid_sops ts \<and> load_tmps_fresh ts \<and> enough_flushs ts \<and> valid_program_history ts \<and> valid ts"
finite_completion: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto> A' i \<union> C"   and "\<And>i. i \<in> I \<Longrightarrow> F \<in> A' i co A' i \<union> C" shows "F \<in> \<Inter> (A ` I) \<longmapsto> \<Inter> (A' ` I) \<union> C"
contains_def: fixes A :: "'a set"   and x :: "'a" shows "contains A x = (x \<in> A)"
maximal_literal_is_unique_def: fixes t :: "'a trm"   and s :: "'a trm"   and C' :: "'a literal set"   and L' :: "'a literal"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "maximal_literal_is_unique t s C' L' S \<sigma> = (\<forall>s''. eq_occurs_in_cl t s'' (C' - {L'}) \<sigma> \<longrightarrow> (s'', t) \<in> trm_ord \<longrightarrow> (s, t) \<in> trm_ord \<longrightarrow> trm_rep s S \<noteq> trm_rep s'' S)"
distinct_39: fixes x41 :: "trm"   and x42 :: "trm"   and x6 :: "(char list \<times> trm) list" shows "x41 \<bullet> x42 \<noteq> Rcd x6"
return_bind: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and x :: "'d"   and f :: "'d \<Rightarrow> 'e" assumes "monad return bind" shows "bind (return x) f = f x"
reach_PairI: fixes s :: "'state"   and a :: "'act"   and ou :: "'out"   and s' :: "'state" assumes "reach s"   and "step s a = (ou, s')" shows "reach s'"
smult_def: fixes \<alpha> :: "'b"   and A :: "'b tensor" shows "\<alpha> \<cdot> A = tensor_from_vec (dims A) (vec_smult \<alpha> (vec A))"
opp_chamberD2: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>" shows "f ` (THE D. D \<in> \<C> - f \<turnstile> \<C> \<and> f ` D = C) = C"
mbc_prop: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "MBC t n \<in> actHn (t n) \<and> pow_cond t n (length (bc (\<sigma>\<^bsub>MBC t n\<^esub>t n)))"
vpfx: shows "valid_prefix (routing_match a) \<and> valid_prefixes as"
zeta'_def: shows "zeta' = (\<lambda>s. \<Prod>p\<in>P. inverse (1 - 1 / of_nat p powr s))"
cosh_add: fixes x :: "'a"   and y :: "'a" shows "cosh (x + y) = cosh x * cosh y + sinh x * sinh y"
in_eq_rel: fixes n :: "nat"   and t1 :: "'a MFOTL.trm"   and t2 :: "'a MFOTL.trm"   and v :: "'a option list" assumes "wf_tuple n (fv_trm t1 \<union> fv_trm t2) v"   and "trm.is_Const t1 \<or> trm.is_Const t2"   and "MFOTL.eval_trm (map the v) t1 = MFOTL.eval_trm (map the v) t2" shows "v \<in> eq_rel n t1 t2"
csl: shows "cs\<^bsup>\<rho>\<^esup> l = cs\<^bsup>\<rho>'\<^esup> l'"
antiparallel_edges_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "antiparallel_edges \<Delta> \<equiv> countable_network \<Delta>"
Mem_refl_E: fixes x :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (x IN x) H \<turnstile> A"
NS_DERIV_D: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a" assumes "(f has_field_derivative D) (at x)" shows "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow>\<^sub>N\<^sub>S D"
SD_preorderD_1: fixes q :: "'alt pmf"   and p :: "'alt pmf" assumes "SD (Pareto R) q p" shows "p \<in> lotteries_on alts"
shunt_bijective: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqinter> id z \<le> top \<and> top \<sqinter> z = top" shows "(x \<le> y \<sqinter> z) = (x \<sqinter> id z \<le> y)"
pullback_iso_self: fixes f :: "'a" assumes "C.iso f" shows "\<p>\<^sub>0[f, f] = \<p>\<^sub>1[f, f]"
delete_point_finite: fixes p :: "'a" shows "finite_incidence_system (del_point p) (del_point_blocks p)"
get_via_put: fixes x :: "'a \<Longrightarrow> 'b"   and s :: "'b" assumes "vwb_lens x" shows "get\<^bsub>x\<^esub> s = (THE v. put\<^bsub>x\<^esub> s v = s)"
l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf get_attribute get_attribute_locs"
split_indicator: fixes P :: "'a \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'a)) \<and> (x \<notin> S \<longrightarrow> P (0::'a)))"
nxt_either: fixes t :: "'b ttree"   and t' :: "'b ttree"   and x :: "'b" shows "nxt (t \<oplus>\<oplus> t') x = nxt t x \<oplus>\<oplus> nxt t' x"
lspasl_u_der: fixes h :: "'a" shows "h,0::'a\<triangleright>h"
elims: fixes x :: "nat \<times> nat"   and y :: "real" assumes "\<Phi> x = y" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "y = a * real n - b * real l"
invar_abs': fixes s :: "'a" assumes "s \<in> wa_invar WC" shows "\<alpha>1 s \<in> wa_invar WB"
is_rbt_rbt_unionwk: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "is_rbt t1"   and "is_rbt t2" shows "is_rbt (rbt_union_with_key f t1 t2)"
dbl_inc_simps_3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one one = numeral.numeral one plus (num.Bit1 num.One)"
element_ids_def: fixes list :: "('c \<times> 'd \<times> bool) list" shows "element_ids list \<equiv> set (map fst list)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<chi>.A.map D \<chi>"
Cantor_nf_unique_aux: fixes \<alpha> :: "V"   and \<alpha>s :: "V list"   and \<beta>s :: "V list"   and ms :: "nat list"   and ns :: "nat list" assumes "Ord \<alpha>"   and "list.set \<alpha>s \<subseteq> ON"   and "list.set \<beta>s \<subseteq> ON"   and "list.set ms \<subseteq> {0<..}"   and "list.set ns \<subseteq> {0<..}"   and "length \<alpha>s = length ms"   and "length \<beta>s = length ns"   and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"   and "sorted_wrt (\<lambda>x y. y < x) \<beta>s"   and "\<alpha> = Cantor_sum \<alpha>s ms"   and "\<alpha> = Cantor_sum \<beta>s ns" shows "\<alpha>s = \<beta>s \<and> ms = ns"
inverse_norm: fixes n :: "nat"   and w :: "'b word" shows "(word_of_nat n = w) = (unat w = n mod 2 ^ LENGTH('b))"
finite_intersection_of_inc: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "P S" shows "(finite intersection_of P) S"
bdd_above_image_mono: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "mono f"   and "bdd_above A" shows "bdd_above (f ` A)"
normalize_stable: fixes q :: "int"   and p :: "int" assumes "0 < q"   and "algebraic_semidom_class.coprime p q" shows "Rat.normalize (p, q) = (p, q)"
supp_3: fixes x3 :: "char list"   and x1 :: "NominalSequents.var"   and x2 :: "form" shows "supp (x3 \<nabla> [x1].x2) = supp x3 \<union> supp ([x1].x2)"
word_sle_msb_le: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le>s y) = ((msb y \<longrightarrow> msb x) \<and> (msb x \<and> \<not> msb y \<or> x \<le> y))"
indexed_padd_index_free: fixes P :: "'d multiset \<Rightarrow> 'a"   and i :: "'d"   and Q :: "'d multiset \<Rightarrow> 'a" assumes "index_free P i"   and "index_free Q i" shows "index_free (P \<Oplus> Q) i"
a_kernel_def': fixes R :: "('d, 'e) ring_scheme"   and S :: "('f, 'g) ring_scheme"   and h :: "'d \<Rightarrow> 'f" shows "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
sc_mem: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and a :: "'i"   and m :: "'h" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<in> carrier M"
section_retraction_of_iso_2: fixes f :: "'s"   and g :: "'s" assumes "S.iso f"   and "S.ide (f \<cdot>\<^sub>S g)" shows "S.inverse_arrows f g"
lang_dual_rexp_dual_of: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "WS1S_Equivalence_Checking.wf \<Sigma> n r" shows "lang_dual \<Sigma> n (rexp_dual_of r) = lang \<Sigma> n r"
hequiv_names_rel: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set" assumes "Hintikka A H" shows "hequiv_rel A H \<subseteq> names H \<times> names H"
lens_comp_def: fixes Y :: "'e \<Longrightarrow> 'g"   and X :: "'g \<Longrightarrow> 'f" shows "Y ;\<^sub>L X = \<lparr>lens_get = get\<^bsub>Y\<^esub> \<circ> get\<^bsub>X\<^esub>, lens_put = \<lambda>\<sigma> v. put\<^bsub>X\<^esub> \<sigma> (put\<^bsub>Y\<^esub> (get\<^bsub>X\<^esub> \<sigma>) v)\<rparr>"
construct_lhs_vector_clean: fixes p :: "real poly"   and i :: "nat"   and signs :: "rat list list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "i < length signs" shows "of_rat (w_vec p qs signs $ i) = of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = signs ! i})"
no_further_steps: fixes s :: "nat"   and s' :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option" assumes "s \<noteq> s'" shows "\<not> visits s e s' r []"
sel_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (Plus r s) = (\<oo> r \<or> \<oo> s)"
borel_measurable_vimage: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and x :: "'d" assumes "f \<in> borel_measurable M" shows "f -` {x} \<inter> space M \<in> sets M"
mask_eqI: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "x && mask n = y && mask n"   and "x && ~~ mask n = y && ~~ mask n" shows "x = y"
vors_3: shows "vs1 = between (vertices f) ram1 ram2"
str_init1_def: fixes dummy :: "'d"   and y :: "('d, 'd IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool, 'd IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) F1"   and i :: "'d IIT" shows "str_init1 dummy y i = fst (snd (Rep_IIT i)) (F1map id (\<lambda>f. f i) (\<lambda>f. f i) y)"
preserves_reflects_arr: fixes f :: "'b" shows "D.arr (D.R f) = D.arr f"
inj_on_fst_vote_set: fixes v_hist :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and p :: "'e" shows "inj_on fst (vote_set v_hist {p})"
h8: shows "Sample req dt x1 stop lose ack i1 vc"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case NonStatic of Static \<Rightarrow> f1 | NonStatic \<Rightarrow> f2) = f2"
inject: fixes x1 :: "'a form multiset"   and x2 :: "'a form"   and y1 :: "'a form multiset"   and y2 :: "'a form" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
assms: shows "\<forall>X\<in>XX. well_related_set X (\<sqsubseteq>) \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. \<not> x \<sqsubseteq> y \<longrightarrow> y \<in> X)"
monoid_mat_def: fixes ty :: "'a itself"   and nr :: "nat"   and nc :: "nat" shows "monoid_mat ty nr nc \<equiv> \<lparr>carrier = carrier_mat nr nc, mult = (+), one = 0\<^sub>m nr nc\<rparr>"
prod_mset_empty: shows "\<Prod>\<^sub># {#} = (1::'a)"
int_not_and_bitANDN_num: fixes m :: "num"   and n :: "num" shows "NOT (numeral m) AND numeral n = (case bitANDN_num n m of None \<Rightarrow> 0 | Some n' \<Rightarrow> numeral n')"
whileAnnoFix_def: fixes b :: "'d set"   and I :: "'g \<Rightarrow> 'd set"   and V :: "'g \<Rightarrow> ('d \<times> 'd) set"   and c :: "'g \<Rightarrow> ('d, 'e, 'f) com" shows "whileAnnoFix b I V c = While b (c undefined)"
bs2: fixes M1 :: "lam"   and M2 :: "lam"   and M3 :: "lam" assumes " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2"   and " M2 \<longrightarrow>\<^sub>\<beta> M3" shows " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M3"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.seq (trg\<^sub>B g) (trg\<^sub>B f)"
add_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
empty_empty: fixes Y :: "'b multiset" assumes "set_mset Y = {}" shows "Y = {#}"
oone_ordIso_oexp: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "r =o oone"   and "Well_order s" shows "r ^o s =o oone"
hom: fixes f :: "'c \<Rightarrow> 'a" assumes "continuous_on U f"   and "f ` U \<subseteq> s"   and "P f" shows "\<exists>c. homotopic_with_canon P U s f (\<lambda>x. c)"
assms_1: shows "f (0::'b) = (0::'a)"
eq_\<I>_generat_mono': fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and D :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "('a, 'c, 'd \<Rightarrow> 'e) generat"   and y :: "('b, 'c, 'd \<Rightarrow> 'f) generat"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and D' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and \<I>' :: "('c, 'd) \<I>" assumes "eq_\<I>_generat A \<I> D x y"   and "\<And>x y. A x y \<Longrightarrow> A' x y"   and "\<And>x y. D x y \<Longrightarrow> D' x y"   and "\<I> \<le> \<I>'" shows "eq_\<I>_generat A' \<I>' D' x y"
order_le_less: fixes x :: "'c"   and y :: "'c" shows "(x \<le> y) = (x < y \<or> x = y)"
form_def: fixes u :: "(nat \<times> nat) set"   and i :: "nat" shows "form u i \<equiv> \<exists>a b c d. u = {(a, b), (c, d)} \<and> form_split a b c d i"
simps_7: fixes i :: "int"   and a :: "Cooper.num" shows "zsplit0 (Mul i a) = (let (i', a') = zsplit0 a in (i * i', Mul i a'))"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
gauss_int_norm_0: shows "gauss_int_norm 0 = 0"
term_ok'_subst_bv1: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and u :: "Core.term"   and lev :: "nat" assumes "term_ok' (sig \<Theta>) f"   and "term_ok' (sig \<Theta>) u" shows "term_ok' (sig \<Theta>) (subst_bv1 f lev u)"
exec_step_ind_Pop: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI Pop) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
smallo_iff: fixes g1 :: "'a \<Rightarrow> real"   and g2 :: "'a \<Rightarrow> real"   and h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "g1 \<in> G"   and "g2 \<in> G"   and "h1 \<in> H"   and "h2 \<in> H" shows "((\<lambda>x. g1 x * h1 x) \<in> o[F](\<lambda>x. g2 x * h2 x)) = (g1 \<in> o[F](g2) \<or> g1 \<in> \<Theta>[F](g2) \<and> h1 \<in> o[F](h2))"
gauss_jordan_def: fixes A :: "'b mat"   and B :: "'b mat" shows "gauss_jordan A B \<equiv> gauss_jordan_main A B 0 0"
transition_type_safe: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and root' :: "(att \<times> char list, att, nat) env" assumes "root \<midarrow>x\<rightarrow> root'"   and "\<exists>att dir. root = Env att dir" shows "\<exists>att dir. root' = Env att dir"
dom_comp: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" shows "D.dom (g \<cdot>\<^sub>D f) = D.dom f"
Mem_Zero_iff: fixes i :: "name"   and t :: "tm"   and H :: "fm set" assumes "atom i \<sharp> t" shows "H \<turnstile> t EQ Zero IFF SyntaxN.All i (Neg (Var i IN t))"
continuous_on_matrix_vector_multl: fixes T :: "real set"   and A :: "real \<Rightarrow> (('b, 'c) vec, 'd) vec"   and s :: "('b, 'c) vec" assumes "matrix_continuous_on T A" shows "continuous_on T (\<lambda>t. A t *v s)"
prod_rel_comp: fixes A :: "('c \<times> 'g) set"   and B :: "('d \<times> 'h) set"   and C :: "('g \<times> 'e) set"   and D :: "('h \<times> 'f) set" shows "(A \<times>\<^sub>r B) O (C \<times>\<^sub>r D) = A O C \<times>\<^sub>r B O D"
HPhase1or2ReadThen_HInv4a2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4a2 s q" shows "HInv4a2 s' q"
prems: shows "paval' (Div a1 a2) ps = Some v"
omega_coinduct_eq_var2: fixes y :: "'a"   and x :: "'a" assumes "y = x \<cdot> y" shows "y \<le> x\<^sup>\<omega>"
hyps_2: fixes thesis :: "bool" assumes "\<And>q m. poly p (Fraction_Field.Fract a b) = Fraction_Field.Fract q (b ^ m) \<Longrightarrow> thesis"   and "set (Polynomial.coeffs p) \<subseteq> range (\<lambda>x. Fraction_Field.Fract x (1::'a))" shows "thesis"