hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "Gp sequent list"   and a2 :: "Gp sequent" assumes "(a1, a2) \<in> g3cp" obtains A :: "Gp form" and B :: "Gp form" where "a1 = [ (\<LM> A__  \<RM>) \<oplus> B__ \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> A__ \<and>* B__  \<RM> \<Rightarrow>* \<Empt>)"   | A :: "Gp form" and B :: "Gp form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Aa__  \<RM>,  \<Empt> \<Rightarrow>* \<LM> Ba__  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> Aa__ \<and>* Ba__  \<RM>)"   | A :: "Gp form" and B :: "Gp form" where "a1 = [ \<LM> Ab__  \<RM> \<Rightarrow>* \<Empt>,  \<LM> Bb__  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> Ab__ \<or>* Bb__  \<RM> \<Rightarrow>* \<Empt>)"   | A :: "Gp form" and B :: "Gp form" where "a1 = [ \<Empt> \<Rightarrow>* (\<LM> Ac__  \<RM>) \<oplus> Bc__]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> Ac__ \<or>* Bc__  \<RM>)"   | A :: "Gp form" and B :: "Gp form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Ad__  \<RM>,  \<LM> Bd__  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> Ad__ \<supset> Bd__  \<RM> \<Rightarrow>* \<Empt>)"   | A :: "Gp form" and B :: "Gp form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<LM> B  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> A \<supset> B  \<RM>)"
assms_1: shows "x1 \<in> carrier (\<G> n)"
badE: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "bad P f" obtains "\<And>i j. i < j \<Longrightarrow> \<not> P (f i) (f j)"
ide_char: fixes t :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "A2_B.ide t = (t \<noteq> A2_B.null \<and> functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (A2_B.Map t) \<and> A2_B.Dom t = A2_B.Map t \<and> A2_B.Cod t = A2_B.Map t)"
complex_div_cnj: fixes a :: "complex"   and b :: "complex" shows "a / b = a * cnj b / complex_of_real ((cmod b)\<^sup>2)"
empty_correct_1: shows "m1.\<alpha> (m1.empty ()) = Map.empty"
quotient_eq_iff: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and Y :: "'a set"   and x :: "'a"   and y :: "'a" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r"   and "x \<in> X"   and "y \<in> Y" shows "(X = Y) = ((x, y) \<in> r)"
COMPARE_intros_3: fixes x :: "real"   and y :: "real" assumes "x = y" shows "COMPARE x y \<equiv> EQ"
f2''_def: shows "f2'' = f2' * real l0 / real (l0 - 1)"
domain_very_strict: fixes x :: "'a" shows "(d x = (zero_class.zero::'a)) = (x = (zero_class.zero::'a))"
the_\<nu>_simps_5: shows "cod w.the_\<nu> = s\<^sub>1 \<star> w"
sinit_linear_asym_lemma1: fixes R :: "('b \<times> 'b) set"   and i :: "nat"   and j :: "nat"   and xs :: "'b seq"   and ys :: "'b seq" assumes "asym R"   and "i < j"   and "(sinit i xs, sinit i ys) \<in> lexord R"   and "(sinit j ys, sinit j xs) \<in> lexord R" shows "False"
hyps_4: assumes "i \<noteq> j0" shows "(f i, f j0) \<in> theta"
trace_core'_parametric: fixes S :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and E :: "'b \<Rightarrow> 'h \<Rightarrow> bool"   and IA :: "'c \<Rightarrow> 'i \<Rightarrow> bool"   and IU :: "'d \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (rel_core' S E IA IU (=) (=)) (rel_fun (rel_spmf S) (rel_fun (list_all2 (rel_sum E (rel_sum (rel_prod IA (=)) (rel_prod IU (=))))) (rel_prod (rel_fun E (=)) (rel_prod (rel_fun IA (=)) (rel_fun IU (=)))))) trace_core' trace_core'"
nonneg': shows "AE y in stock_measure t'. 0 \<le> eval_cexpr f (case_nat x \<rho>) y"
assms_2: shows "g \<turnstile> Entry g-ns\<rightarrow>m"
assms_2: shows "\<forall>x\<in>S - T. P x"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 n, $Br_if i] : ts _> ts'"
vertices_def: fixes labeled_graph :: "('e, 'd) labeled_graph" shows "vertices labeled_graph = (case labeled_graph of LG x1 x2 \<Rightarrow> x2)"
Trans: fixes T :: "Javatype"   and U :: "Javatype"   and V :: "Javatype" assumes "treach T U"   and "treach U V" shows "treach T V"
rderivs_snoc: fixes ws :: "'a list"   and w :: "'a"   and r :: "'b rexp" shows "rderivs (ws @ [w]) r = rderiv w (rderivs ws r)"
JFmap_unique: fixes u :: "'d JF1 \<Rightarrow> 'e JF1"   and f :: "'d \<Rightarrow> 'e"   and v :: "'d JF2 \<Rightarrow> 'e JF2" assumes "dtor1 \<circ> u = F1map f u v \<circ> dtor1"   and "dtor2 \<circ> v = F2map f u v \<circ> dtor2" shows "u = JF1map f \<and> v = JF2map f"
apath_imp_progressing: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "apath u p v" shows "progressing p"
Sigma_empty1: fixes B :: "'e \<Rightarrow> 'f set" shows "Sigma {} B = {}"
matrix_to_iarray_fst_Gauss_Jordan_in_ij_iarrays_PA: fixes i :: "'c"   and j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec"   and P :: "(('b, 'c) vec, 'c) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "matrix_to_iarray (fst (Gauss_Jordan_in_ij_PA (P, A) i j)) = fst (Gauss_Jordan_in_ij_iarrays_PA (matrix_to_iarray P, matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j))"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 k, $Load t tp a off] : ts _> ts'"
S_VALID_AS_VALID_IMP_MIN_IN_PLS: fixes s :: "('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "Inf (PLS s as) \<in> MPLS PROB"
img_is_subfield_1: fixes K :: "'a set" assumes "subfield K R"   and "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>" shows "inj_on h K"
permute_set_eq: fixes p :: "perm"   and X :: "'b set" shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
msg_distinct_96: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Enc X Y \<noteq> \<langle>X', Y'\<rangle>"
substEnv_psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "goodEnv rho'" shows "rho &[rho'] &[Y / y]_ys = rho &[rho' &[Y / y]_ys]"
scale_left_diff_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "(a - b) * x = a * x - b * x"
single_non_qAFresh_ordLess_var: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort" assumes "qGood X" shows "|{x. \<not> qAFresh xs x X}| <o |UNIV|"
assms: shows "(numeral m)\<^sup>2 \<le> 2 ^ (3 * n)"
simps_2: fixes x :: "'a" shows "invert (Neg x) = Pos x"
exec_call: shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> t"
factor_irreducible\<^sub>dI: fixes h :: "int poly" assumes "h dvd f"   and "p.dvdm u h"   and "0 < degree h"   and "degree h \<le> j'" shows "irreducible\<^sub>d h"
ind: fixes lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "aa + n \<le> ba"   and "aa < ba"   and "length lm1 = aa"   and "length lm2 = n"   and "length lm3 = ba - aa - n" shows "{\<lambda>nl. nl = lm1 @ lm2 @ lm3 @ 0 \<up> n @ lm4} mv_boxes aa ba n {\<lambda>nl. nl = lm1 @ 0 \<up> n @ lm3 @ lm2 @ lm4}"
merge_length_le_Suc: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs" shows "length us \<le> Suc (length vs)"
idx_find: fixes x :: "nat"   and i :: "'a extended" assumes "x < length (ext_ids s)"   and "ext_ids s ! x = i" shows "idx s i = Inr x"
epsclo_subset: fixes M :: "'b nfa"   and Q :: "hf set" assumes "nfa M" shows "nfa.epsclo M Q \<subseteq> nfa.states M"
eval_or_const_2: fixes \<phi> :: "tvl"   and \<psi> :: "tvl" shows "(eval_or \<phi> \<psi> = No) = (\<phi> = No \<and> \<psi> = No)"
intro: fixes set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')" shows "l_set_tag_name_get_disconnected_nodes_axioms set_tag_name_locs get_disconnected_nodes_locs"
orbits_def: fixes G :: "('e, 'f) partial_object_scheme"   and E :: "'d set"   and \<phi> :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "orbits G E \<phi> = {orbit G \<phi> x |x. x \<in> E}"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "'b" shows "map_tree f (Leaf x11 x12) = Leaf x11 (f x12)"
joinable_subst: fixes R :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. (x, y) \<in> R \<Longrightarrow> (P x, P y) \<in> R"   and "(x, y) \<in> joinable R" shows "(P x, P y) \<in> joinable R"
ChurchRosser_unique_final: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b1 :: "'a"   and b2 :: "'a" assumes "relation.is_ChurchRosser r"   and "r\<^sup>*\<^sup>* a b1"   and "r\<^sup>*\<^sup>* a b2"   and "relation.is_final r b1"   and "relation.is_final r b2" shows "b1 = b2"
assms_2: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto><X> Q"
C'FMT: shows "c' = {#[entry fg p]#} + c"
iso_defs_1: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "iso r r' f \<equiv> embed r r' f \<and> bij_betw f (Field r) (Field r')"
dsr4: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.domain_semiringl plus times one zero domain_op less_eq less" shows "domain_op zero = zero"
abs_eq: fixes x :: "'d list" shows "slist_of_list x = Abs_slist (rev (zip [0..<length x] x))"
mop_spec_rl: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd nres" assumes "P x" shows "mop P f x \<le> f x"
isomorphism_respects_weak_units: fixes a :: "'a"   and a' :: "'a" assumes "weak_unit a"   and "a \<cong> a'" shows "weak_unit a'"
arr_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxVxV.arr f = (arr (fst f) \<and> VxV.arr (snd f))"
h: shows "bij_betw h (UNIV // R) (hfset (ord_of (card (UNIV // R))))"
fds_nth_higher_deriv: fixes k :: "nat"   and F :: "'a fds" shows "fds_nth ((fds_deriv ^^ k) F) = (\<lambda>n. (- (1::'a)) ^ k * of_real (ln (real n)) ^ k * fds_nth F n)"
map_sum_if_distrib_else_2: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and e :: "bool"   and x :: "'e + 'f"   and y :: "'f" shows "map_sum f g (if e then x else Inr y) = (if e then map_sum f g x else Inr (g y))"
filtermap_fun_inverse: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and G :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "filterlim g F G"   and "filterlim f G F"   and "\<forall>\<^sub>F x in G. f (g x) = x" shows "filtermap f F = G"
interv_sum_list_conv_sum_set_nat: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "class.monoid_add plus zero" shows "sum_list (map f [m..<n]) = sum f (set [m..<n])"
pps_to_list_nth_lessI: fixes j :: "nat"   and i :: "nat"   and S :: "'t set" assumes "j < i"   and "i < card S" shows "pps_to_list S ! i \<prec>\<^sub>t pps_to_list S ! j"
exhaust: fixes y :: "'x \<times> 'y" obtains a :: "'x" and b :: "'y" where "y = (a, b)"
theI': fixes P :: "'f \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
Combination: fixes qs :: "nat list"   and pattern :: "nat list set"   and a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat"   and s :: "(nat list \<times> (bool list \<times> nat list + nat list)) pmf"   and ts :: "real"   and bit :: "real"   and opt :: "real" assumes "qs \<in> pattern"   and "a \<noteq> b"   and "{a, b} = {x, y}"   and "set qs \<subseteq> {a, b}"   and "inv_COMB s a [x, y]"   and "\<And>s h. \<lbrakk>a \<noteq> b; {a, b} = {x, y}; TS_inv s a [x, y]; set qs \<subseteq> {a, b}; qs \<in> pattern\<rbrakk> \<Longrightarrow> TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) s qs) (last qs) [x, y] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h)) s qs = ts"   and "\<And>s. \<lbrakk>a \<noteq> b; {a, b} = {x, y}; BIT_inv s a [x, y]; set qs \<subseteq> {a, b}; qs \<in> pattern\<rbrakk> \<Longrightarrow> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x, y] \<and> T\<^sub>p_on_rand' BIT s qs = bit"   and "\<lbrakk>a \<noteq> b; qs \<in> pattern\<rbrakk> \<Longrightarrow> real (T\<^sub>p [a, b] qs (OPT2 qs [a, b])) = opt"   and "qs \<in> pattern \<Longrightarrow> 2 / 10 * ts + 8 / 10 * bit \<le> 16 / 10 * opt" shows "T\<^sub>p_on_rand' (COMB []) s qs \<le> 16 / 10 * real (T\<^sub>p [a, b] qs (OPT2 qs [a, b])) \<and> inv_COMB (Partial_Cost_Model.config'_rand (COMB []) s qs) (last qs) [x, y]"
has_root_contra: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "point r"   and "x\<^sup>T ; r = 0" shows "x = 0"
l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<equiv> l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs \<and> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
representation_def: fixes basis :: "'d set"   and v :: "'d" shows "representation basis v = (if independent basis \<and> v \<in> span basis then SOME f. (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> 0} \<and> (\<Sum>v | f v \<noteq> 0. f v *\<^sub>R v) = v else (\<lambda>b. 0))"
closure_insert: fixes x :: "'b"   and s :: "'b set" shows "closure (insert x s) = insert x (closure s)"
sel_11: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "expr.type (StaticOperationCall x61 x62 x63) = x61"
C_eq_RS2: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy1_strong p" shows "Cp (list2FWpolicy (removeShadowRules2 p)) = Cp (list2FWpolicy p)"
str_eq_alt: fixes A :: "'b list set" shows "\<approx>A = {(x, y). fold Deriv x A = fold Deriv y A}"
diffeomorphism_axioms: shows "Differentiable_Manifold.diffeomorphism k charts1 charts2 f f'"
PHI'_3: shows "\<forall>x\<in>PHI''. x \<in> Csafe"
add_mset_remove_trivial: fixes x :: "'a"   and M :: "'a multiset" shows "add_mset x M - {#x#} = M"
gauss_cnj_diff: fixes a :: "gauss_int"   and b :: "gauss_int" shows "gauss_cnj (a - b) = gauss_cnj a - gauss_cnj b"
WTrt_elim_cases_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and v :: "char list"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> v:=e : T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Void" and "E v = \<lfloor>T\<rfloor>" and "P,E,h \<turnstile> e : T'" and "P \<turnstile> T' \<le> T"
1: shows "\<guillemotleft>F y : F y \<rightarrow>\<^sub>C F y\<guillemotright>"
add_data_cache_def: fixes state :: "'b sparc_state"   and va :: "word32"   and word :: "word32"   and byte_mask :: "word4" shows "add_data_cache state va word byte_mask \<equiv> let tag = ucast (va >> 12); offset0 = ucast va AND 4092; offset1 = offset0 OR 1; offset2 = offset0 OR 2; offset3 = offset0 OR 3; byte0 = ucast (word >> 24); byte1 = ucast (word >> 16); byte2 = ucast (word >> 8); byte3 = ucast word; s0 = if byte_mask AND 8 >> 3 = 1 then dcache_mod (tag, offset0) byte0 state else state; s1 = if byte_mask AND 4 >> 2 = 1 then dcache_mod (tag, offset1) byte1 s0 else s0; s2 = if byte_mask AND 2 >> 1 = 1 then dcache_mod (tag, offset2) byte2 s1 else s1; s3 = if byte_mask AND 1 = 1 then dcache_mod (tag, offset3) byte3 s2 else s2 in s3"
lb1_def: shows "lb1 = (\<lambda>n. \<Prod>p\<in>{p. prime p \<and> real p \<le> ln (real n)}. 1 - 1 / real p)"
right_inverse: fixes a :: "'e" shows "a + - a = (0::'e)"
pcompose_0': fixes p :: "'a poly" shows "p \<circ>\<^sub>p 0 = [:coeff p 0:]"
Myhill_Nerode: fixes A :: "'a list set" shows "(\<exists>r. A = lang r) = finite (UNIV // \<approx>A)"
wf_dom_inA: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w" shows "ss \<in> list n A"
simps_1: fixes \<mu> :: "'a" shows "Inv \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
out_arcs_def: fixes G :: "('d, 'c) pre_digraph"   and v :: "'d" shows "out_arcs G v \<equiv> {e \<in> arcs G. tail G e = v}"
cycle_free_up_to_loop_dest: fixes i :: "nat"   and n :: "nat"   and xs :: "nat list"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "set xs \<subseteq> {0..k}"   and "cycle_free_up_to m k n" shows "\<one> \<le> len m i i xs"
cones_map_vcomp: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and D' :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and f :: "'b"   and a' :: "'b" assumes "diagram J C D"   and "diagram J C D'"   and "natural_transformation J C D D' \<tau>"   and "Limit.cone J C D a \<chi>"   and "partial_magma.in_hom C f a' a" shows "(\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D' (partial_magma.cod C f) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) f else partial_magma.null C) (vertical_composite.map J C \<chi> \<tau>) = vertical_composite.map J C ((\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D (partial_magma.cod C f) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) f else partial_magma.null C) \<chi>) \<tau>"
sym_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sim> y"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq> z"
false_to_true_imp_ex_pos: fixes I :: "'b set"   and J :: "'b set"   and C :: "'b literal multiset" assumes "I \<subseteq> J"   and "\<not> I \<Turnstile> C"   and "J \<Turnstile> C" shows "\<exists>A\<in>J. Pos A \<in># C"
connected_card_eq_iff_nontrivial: fixes S :: "'b set" assumes "connected S" shows "uncountable S = (\<nexists>a. S \<subseteq> {a})"
least_0: shows "(LEAST n. n \<in> UNIV) = (0::'a)"
comb'_def: fixes l :: "'basis euclidean_space" shows "comb' l = comb (Rep_euclidean_space l \<circ> abs)"
map_cong: fixes x :: "('a, 'b, 'c) gpv"   and ya :: "('a, 'b, 'c) gpv"   and f1 :: "'a \<Rightarrow> 'd"   and g1 :: "'a \<Rightarrow> 'd"   and f2 :: "'b \<Rightarrow> 'e"   and g2 :: "'b \<Rightarrow> 'e" assumes "x = ya"   and "\<And>z1. z1 \<in> results'_gpv ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> outs'_gpv ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_gpv f1 f2 x = map_gpv g1 g2 ya"
simps_7: fixes r :: "real"   and x :: "octo" shows "Im6 (r *\<^sub>R x) = r * Im6 x"
double_sum_aux: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and n :: "nat" shows "(\<Sum>k = 1..n. \<Sum>j = 1..n. f k * g j) = (\<Sum>k = 1..n. \<Sum>j = 1..n. (f k * g j + f j * g k) / 2)"
assms_3: shows "(r, Store e2 i r) \<in> set (node_deliver_messages xs)"
a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and uu :: "unit" shows "local.a_get_owner_document\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr uu = Heap_Error_Monad.bind (get_host shadow_root_ptr) (\<lambda>host. local.CD.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^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 host) ())"
ta_upd_proj_simps_5: fixes ta :: "('b, 'a, 'c, 'd, 'e, 'f) thread_action"   and obs :: "'f" shows "\<lbrace>ta_update_obs ta obs\<rbrace>\<^bsub>i\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) (G \<circ> F) (G \<circ> F) (G \<circ> F)"
transitive: fixes A :: "'a trm"   and A' :: "'a trm"   and A'' :: "'a trm" assumes "A >>\<^sup>* A'"   and "A' >> A''" shows "A >>\<^sup>* A''"
locally_compact: fixes s :: "'c set" shows "locally compact s = (\<forall>x\<in>s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (top_of_set s) u \<and> compact v)"
lcm_proj2_if_dvd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a b" shows "lcm a b = normalize b"
assms_2: fixes i :: "nat" shows "Y (Suc i) \<sqsubseteq> Y i"
weakFreeTransition_def: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<equiv> P \<Longrightarrow>\<alpha> \<prec> P' \<or> \<alpha> = \<tau> \<and> P = P'"
WT_gpv_outs_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>"
monad_reader_state_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and ask :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('e \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'e \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_reader_state return bind ask get put" shows "monad_reader_state (return_writer return) (bind_writer return bind) (ask_writer ask) (get_writer get) (put_writer put)"
wf4: shows "mat (row_length B2) (length B2) B2"
remove_arg: fixes i :: "nat" assumes "head t = head s"   and "i < num_args t"   and "args s = take i (args t) @ drop (Suc i) (args t)" shows "P"
netgmap_netgmap_not_rhs: fixes i :: "nat"   and s2 :: "'c net_state"   and sr :: "'c \<Rightarrow> 'd \<times> 'e"   and s1 :: "'c net_state" assumes "i \<notin> net_ips s2" shows "(fst (netgmap sr s1) ++ fst (netgmap sr s2)) i = fst (netgmap sr s1) i"