hcju/isabelleps huggingface dataset:
eSubstAbs_nchotomy: fixes eY :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and eA :: "'d withERR"   and y :: "'var" shows "(\<exists>Y. eY = OK Y \<and> igWls MOD (asSort ys) Y) \<and> (\<exists>us s A. eA = OK A \<and> igWlsAbs MOD (us, s) A) \<or> eSubstAbs MOD ys eY y eA = ERR"
e_m1: shows "exp (- 1) \<le> 10 ^ 5 / 271828"
NList_def: fixes A :: "'b set"   and N :: "nat set" shows "NList A N \<equiv> \<Union> (nList A ` N)"
hn_refine_frame: fixes P' :: "assn"   and c :: "'d Heap"   and Q' :: "assn"   and R :: "'b \<Rightarrow> 'd \<Rightarrow> assn"   and P :: "assn"   and F :: "assn" assumes "hn_refine P' c Q' R (RCALL $ a $ b)"   and "P \<Longrightarrow>\<^sub>t F * P'" shows "hn_refine P c (F * Q') R (RCALL $ a $ b)"
select_convs_2: fixes obj_mark :: "bool"   and obj_fields :: "'a \<Rightarrow> 'b option"   and obj_payload :: "'a \<Rightarrow> 'c option"   and more :: "'d" shows "obj_fields \<lparr>obj_mark = obj_mark, obj_fields = obj_fields, obj_payload = obj_payload, \<dots> = more\<rparr> = obj_fields"
zero_min: fixes t :: "('x, nat) pp" shows "ord 0 t"
simple_match_valid_alt_hlp2: fixes x :: "16 word \<times> 16 word" shows "({p. simple_match_port x p} \<noteq> {}) = (case x of (s, e) \<Rightarrow> s \<le> e)"
choosefrom_basis: fixes B' :: "'a set" assumes "B' \<noteq> {}" shows "f B' \<in> B'"
one_eq_numeral_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(one = numeral.numeral one plus n) = (num.One = n)"
Cons_2: shows "separateAtoms (At # L) = (eq, les, leq, neq)"
case_2: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<Pi>\<^sub>1 set \<Rightarrow> 'a"   and x2 :: "\<Pi>\<^sub>1 set" shows "(case \<alpha>\<nu> x2 of \<omega>\<nu> x \<Rightarrow> f1 x | \<alpha>\<nu> x \<Rightarrow> f2 x) = f2 x2"
l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_axioms: shows "l_type_wf\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t type_wf"
take_inf_less: fixes t :: "'b llist"   and i :: "nat" assumes "t \<in> UNIV\<^sup>\<omega>" shows "t \<down> i < t"
rhseq: shows "(vec.span (columns A) = UNIV) = (\<forall>x. x \<in> vec.span (columns A))"
simps_2: fixes carrier :: "'b set"   and eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and more :: "'a" shows "gorder.more \<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr> = more"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (Ge (CN 0 c e)) = c"
poly_eval_cong: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and a :: "'b \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'c" assumes "p = q"   and "\<And>x. x \<in> indets q \<Longrightarrow> a x = b x" shows "poly_eval a p = poly_eval b q"
seqSubst2: fixes x :: "name"   and \<sigma> :: "(name list \<times> 'b list) list"   and T :: "'a" assumes "x \<sharp> \<sigma>"   and "x \<sharp> T" shows "x \<sharp> T[<\<sigma>>]"
wit_lsingleton: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "wit R (LCons x LNil) = LCons x LNil"
coboundedI1: fixes c :: "'c \<times> 'd"   and a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" assumes "less_eq_prod c a" shows "less_eq_prod c (sup_prod a b)"
rank_le_nc: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "rank A \<le> nc"
add_le_cancel_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus a c) (plus b c) = less_eq a b"
SigmaI: fixes a :: "'b"   and A :: "'b set"   and b :: "'c"   and B :: "'b \<Rightarrow> 'c set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
cblinfun_inv_def: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "cblinfun_inv A = (SOME B. B o\<^sub>C\<^sub>L A = id_cblinfun)"
sees_methods_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and Mm' :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "P \<turnstile> C sees_methods Mm"   and "P \<turnstile> C sees_methods Mm'" shows "Mm' = Mm"
hyps: assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = Gram_Schmidt_upt_k_iarrays (matrix_to_iarray A) k"
poly_cfs_subring: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP R)"   and "\<And>n. g n \<in> S" shows "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
fromlist_def: fixes L :: "(nat \<times> int \<times> nat) list" shows "fromlist L \<equiv> distinct (map \<alpha> L) \<and> (\<forall>u w v. (u, w, v) \<in> set L \<longrightarrow> u \<noteq> v)"
pick_same: fixes l :: "nat"   and xs :: "'b list" assumes "l < length xs" shows "pick (map (Pair 1) xs) (natural_of_nat l) = xs ! l"
summable_iff_shift: fixes f :: "nat \<Rightarrow> 'c"   and k :: "nat" shows "summable (\<lambda>n. f (n + k)) = summable f"
sum_squares_ge_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (plus (times x x) (times y y))"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "Def_Init_Exp.bval b s = Some True" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<rightarrow> (c\<^sub>1, s)"
simps_4: fixes f :: "'c"   and v :: "'c local.term"   and va :: "'c local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> v \<^bold>\<star> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> v \<^bold>\<star> va"
precompact_locally_finite_open_coverE: obtains W :: "nat \<Rightarrow> 'd set" where "manifold_eucl.dest.carrier = \<Union> (range W)" and "\<And>i. open (W i)" and "\<And>i. compact (closure (W i))" and "\<And>i. closure (W i) \<subseteq> manifold_eucl.dest.carrier" and "locally_finite_on manifold_eucl.dest.carrier UNIV W"
map_le_map_add: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "f \<subseteq>\<^sub>m g ++ f"
OclNot_not: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not (not X) = X"
der_g: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S)"
subcls_acyclic: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "ws_prog G" shows "(D, C) \<notin> (subcls1 G)\<^sup>+"
Generic_offending_list_def: fixes sinvar :: "'c list_graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c list_graph"   and nP :: "'c \<Rightarrow> 'd" shows "Generic_offending_list sinvar G nP = filter (\<lambda>f. (\<not> sinvar G nP \<and> sinvar (FiniteListGraph.delete_edges G f) nP) \<and> (\<forall>(e1, e2)\<in>set f. \<not> sinvar (FiniteListGraph.add_edge e1 e2 (FiniteListGraph.delete_edges G f)) nP)) (subseqs (edgesL G))"
oo: fixes U :: "'b set" assumes "openin (top_of_set T) U" shows "openin (top_of_set S) {x \<in> S. f x \<inter> U \<noteq> {}}"
epi_hom_induced_retraction_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "retraction_map X Y f" shows "hom_induced p X {} Y {} f \<in> epi (homology_group p X) (homology_group p Y)"
IH: fixes ys :: "('a \<times> 'a option) list" assumes "set xs = set ys"   and "insert_ops xs"   and "rga_ops ys" shows "interp_ins xs = interp_rga ys"
pass\<^sub>1_size: fixes hs :: "'b hp list" shows "size_hps (pass\<^sub>1 hs) = size_hps hs"
mmu_readable_def: fixes f :: "word3"   and asi :: "word8" shows "mmu_readable f asi \<equiv> if uint asi \<in> {8, 10} then if uint f \<in> {0, 1, 2, 3, 5} then True else False else if uint asi \<in> {9, 11} then if uint f \<in> {0, 1, 2, 3, 5, 6, 7} then True else False else False"
sinh_plus_cosh: fixes x :: "'a" shows "sinh x + cosh x = exp x"
assm: shows "\<exists>va u. v=va\<parallel>u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> (\<exists>v ua. u=v\<parallel>ua \<and> True \<and> 0 < \<parallel>ext ua\<parallel> \<and> len ua ts c = ext ua \<and> restrict ua (res ts) c = lan ua \<and> |lan ua| = 1)"
Units_l_inv_ex: fixes x :: "'a" assumes "x \<in> Units R" shows "\<exists>y\<in>carrier R. y \<otimes> x = \<one>"
pair_list_split: fixes l :: "('a \<times> 'b) list" obtains l1 :: "'a list" and l2 :: "'b list" where "l = zip l1 l2" and "length l1 = length l2" and "length l = length l2"
C2l: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.C2l_algebra plus times one zero less_eq less star"   and "x = times y x" shows "x = times (star y) x"
circ_right_top: fixes x :: "'a" shows "times_top_class.total (x\<^sup>\<circ>)"
ide_compE: fixes g :: "J.Arr"   and f :: "J.Arr" assumes "J.ide (g \<cdot>\<^sub>J f)" obtains "J.seq g f" and "J.seq f g" and "g \<cdot>\<^sub>J f = J.dom f" and "g \<cdot>\<^sub>J f = J.cod g"
finite_dimension_imp_algebraic: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and F :: "'c set"   and x :: "'c" assumes "ring R"   and "subfield K R"   and "subring F R"   and "ring.finite_dimension R K F"   and "x \<in> F" shows "((\<lambda>K x. \<not> ring.transcendental R K x) over K) x"
XSe: shows "XSe = map (\<lambda>x. (x, 0)) XS"
bell00_index_2: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> $$ (1, 0) = 0"
ffact_add_diff_assoc: fixes a :: "'a"   and n :: "nat" shows "(a - of_nat n) * ffact n a + of_nat n * ffact n a = a * ffact n a"
rsp: shows "eq_onp (\<lambda>S. \<exists>m n. {m..n} = S) {} {}"
order_less_irrefl: fixes x :: "'d" shows "\<not> x < x"
RD_bounds_sublistD_i_b: fixes Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" shows "{length p - 1 |p. valid_path Pi p \<and> distinct p} \<noteq> {}"
fact_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 1 = one"
length_greater_0_conv: fixes xs :: "'g list" shows "(0 < length xs) = (xs \<noteq> [])"
incr_bv_incr_bv: fixes i :: "nat"   and k :: "nat"   and inc2 :: "nat"   and inc1 :: "nat"   and t :: "Core.term" assumes "i < k + 1" shows "incr_bv inc2 (k + inc1) (incr_bv inc1 i t) = incr_bv inc1 i (incr_bv inc2 k t)"
D: shows "D = b\<^sup>2 - 4 * a * c"
inorder_delete: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
atoms_res: fixes S :: "'b literal set set" shows "atoms_of_cnf (res S) \<subseteq> atoms_of_cnf S"
aff: fixes X :: "'a set" assumes "X \<in> \<F> - \<G>" shows "aff_dim X \<le> aff_dim T"
rev_finite_subset: fixes B :: "'k set"   and A :: "'k set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
ch_5: shows "r = (u to x' in v to z' in n) \<star> L"
newman_ingham_aux1: fixes R :: "real"   and z :: "complex" assumes "0 < R"   and "cmod z = R" shows "cmod (1 / z + z / complex_of_real (R\<^sup>2)) = 2 * \<bar>Re z\<bar> / R\<^sup>2"
SubspaceD1: fixes U :: "'v set" assumes "Subspace U" shows "VectorSpace (\<cdot>) U"
fps_power_equiv: fixes f :: "'a fps"   and g :: "'a fps"   and p :: "'a"   and x :: "nat" assumes "(f, g) \<in> fpsmodrel p" shows "(f ^ x, g ^ x) \<in> fpsmodrel p"
rhseq: shows "construct_lhs_vector_R p qs signs = vec_of_list (map (\<lambda>s. rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) signs)"
inj_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "inj f" shows "inj_on f S"
nth_Cons_Suc: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
unannotated: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "eval\<^sub>w_simple_abv (c, mem) (c', mem')" shows "\<langle>cxt_to_stmt E c, mds, mem\<rangle> \<leadsto> \<langle>cxt_to_stmt E c', mds, mem'\<rangle>"
S6_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S6 rmhist p = S rmhist True False False clkB rpcA histB histB p"
pdens: shows "has_subprob_density M (stock_measure (PRODUCT REAL REAL)) \<delta>"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<sim>\<^sup>s Q \<parallel> P"
dec_I2_Extend_SO: fixes \<AA> :: "interp"   and P :: "nat fset"   and X :: "WS1S_Nameful.so"   and bs2 :: "WS1S_Nameful.so list" shows "dec_I2 (Extend SO 0 \<AA> P) (X # bs2) = (dec_I2 \<AA> bs2)(X := P)"
run_af_G_letter_abs_eq_Abs_af_G_letter: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "run \<up>af\<^sub>G (Abs \<phi>) w i = Abs (run af_G_letter \<phi> w i)"
v'_as: shows "v' \<Turnstile>\<^sub>a\<^sub>s (snd ` (set (Atoms (preprocess' t start)) \<inter> I \<times> UNIV))"
root_unity_witness: fixes xs :: "complex list"   and n :: "nat"   and x :: "complex" assumes "(\<Prod>x\<leftarrow>xs. [:- x, 1:]) = monom 1 n - 1" shows "(x ^ n = 1) = (x \<in> set xs)"
lt_eq: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and x :: "'a"   and y :: "'a" assumes "linorder_on D cmp"   and "x \<in> D"   and "y \<in> D" shows "(cmp x y = LESS) = (cmp y x = GREATER)"
EQ_conv_set_ebounds: fixes x :: "'a"   and f :: "atom fm"   and xs :: "'a list" shows "(x \<in> EQ f xs) = (\<exists>e\<in>set (ebounds (DLO.atoms\<^sub>0 f)). x = xs ! e)"
XY: fixes U :: "'a set" assumes "openin X U"   and "{x \<in> topspace X. f x \<in> f ` U} \<subseteq> U" shows "openin Y (f ` U)"
llength_ltakeWhile_eq_infinity': fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(llength (ltakeWhile P xs) = \<infinity>) = (\<not> lfinite xs \<and> (\<forall>x\<in>lset xs. P x))"
cprank_max1_prod_listE: fixes A :: "'b tensor" assumes "cprank_max1 A" obtains Bs :: "'b tensor list" and a :: "'b" where "\<And>B. B \<in> set Bs \<Longrightarrow> Tensor.order B = 1" and "a \<cdot> prod_list Bs = A"
ncoplanar_perm_23: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet D C B A"
commutative_squareI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "category C"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g"   and "partial_magma.arr C h \<and> partial_magma.arr C k \<and> partial_magma.dom C h = partial_magma.dom C k"   and "partial_magma.dom C f = partial_magma.cod C h"   and "C f h = C g k" shows "category.commutative_square C f g h k"
that: fixes xs' :: "'a list"   and xs'' :: "'a list"   and ys' :: "'b list"   and ys'' :: "'b list" assumes "xs = xs' @ x # xs''"   and "ys = ys' @ y # ys''"   and "length xs' = length ys'" shows "thesis"
independent_if_scalars_zero: fixes A :: "'c set" assumes "finite A"   and "\<And>f x. \<lbrakk>(\<Sum>x\<in>A. scaleC (f x) x) = (0::'c); x \<in> A\<rbrakk> \<Longrightarrow> f x = (0::'a)" shows "l.vs2.independent A"
traces_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_sub: fixes A :: "'b set" shows "{s. front_tickFree s \<and> set s \<subseteq> ev ` A \<union> {tick}} \<subseteq> T (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A)"
affine_hull_openin: fixes T :: "'a set"   and S :: "'a set" assumes "openin (top_of_set (affine hull T)) S"   and "S \<noteq> {}" shows "affine hull S = affine hull T"
pole_at_infinity: fixes f :: "complex \<Rightarrow> complex"   and l :: "complex" assumes "f holomorphic_on UNIV"   and "((inverse \<circ> f) \<longlongrightarrow> l) at_infinity" obtains a :: "nat \<Rightarrow> complex" and n :: "nat" where "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)"
cases: fixes x :: "'a \<times> 'b" obtains v :: "'a" and p :: "'b" where "x = (v, p)"
assms: shows "P \<Longrightarrow>\<^sub>\<tau> P'"
intros: fixes C :: "'a sequent"   and F :: "'a"   and Fs :: "'a form list"   and Ps :: "'a sequent list" assumes "C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* Em)" shows "leftPrincipal (Ps, C) (Compound F Fs)"
eq_\<I>_gpv_try_gpv_cong: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv1 :: "('a, 'c, 'd) gpv"   and gpv1' :: "('b, 'c, 'd) gpv"   and gpv2 :: "('a, 'c, 'd) gpv"   and gpv2' :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I> gpv1 gpv1'"   and "eq_\<I>_gpv A \<I> gpv2 gpv2'" shows "eq_\<I>_gpv A \<I> (TRY gpv1 ELSE gpv2) (TRY gpv1' ELSE gpv2')"
srtc_in_pmdl: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "relation.srtc (punit.red F) p q" shows "p - q \<in> local.punit.pmdl F"
min_set_min_product_helper: fixes x :: "'b fset"   and A :: "'b fset set"   and B :: "'b fset set" shows "(x \<in> min_set A \<otimes>\<^sub>m B) = (x \<in> A \<otimes>\<^sub>m B)"
g_def: fixes x :: "real" shows "g x = (if x \<le> 0 then f0 else f x)"
rem_noops_def: shows "rem_noops \<equiv> filter (\<lambda>op. op \<noteq> empty_sasp_action)"
assms_3: fixes x :: "'a" assumes "\<And>y. y < x \<Longrightarrow> Q y" shows "Q x"
cut_less_bound: fixes n :: "'a"   and I :: "'a set"   and t :: "'a" assumes "n \<in> I \<down>< t" shows "n < t"
substs_preserve_ground_terms: fixes t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_term t" shows "t \<lhd> \<sigma> = t"
hwlp: fixes a :: "'a"   and s :: "'s" assumes "a \<in> S s" shows "nearly_healthy (wlp (p a))"
osn_rreq: fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg rreq_rrep_sn \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:4, PAodv-:5} \<union> {PRreq-:n |n. True} \<longrightarrow> 1 \<le> osn \<xi>)"
add_mult_distrib2: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> E" shows "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
g: fixes z :: "complex" assumes "z \<in> S" shows "f z = exp (g z)"
image_Collect_subsetI: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
simps_2: fixes p :: "'b \<Rightarrow> bool"   and a :: "'b"   and S :: "'b list" shows "\<guillemotleft>p\<guillemotright> (a # S) = p a"
ma_of_rat_def: shows "ma_of_rat \<equiv> map_fun id Abs_mini_alg (\<lambda>x. (x, 0, 0))"
simps_11: fixes x11 :: "'a"   and x12 :: "'b"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x11 \<leftarrow> x12 \<noteq> x41 ;; x42"
length_def: fixes as :: "'b iarray" shows "IArray.length as = length (IArray.list_of as)"
bprv_prv: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>" shows "prv \<phi>"
solves_ode_def: fixes y :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set" shows "(y solves_ode f) T X = ((y has_vderiv_on (\<lambda>t. f t (y t))) T \<and> y \<in> T \<rightarrow> X)"
kmp_SPEC_def: fixes s :: "'b list"   and t :: "'b list" shows "kmp_SPEC s t = SPEC (\<lambda>x. case x of None \<Rightarrow> \<nexists>i. sublist_at s t i | Some i \<Rightarrow> sublist_at s t i \<and> (\<forall>ii<i. \<not> sublist_at s t ii))"
fls_integral_X_power: fixes n :: "nat" shows "fls_integral (fls_X ^ n) = fls_const (inverse (of_nat (Suc n))) * fls_X ^ Suc n"
map_of_delete: fixes k' :: "'d"   and k :: "'d"   and al :: "('d \<times> 'e) list" assumes "k' \<noteq> k" shows "map_of (delete k al) k' = map_of al k'"
union_fold_mset_add_mset: fixes A :: "'c multiset"   and B :: "'c multiset" shows "A + B = fold_mset add_mset A B"
SCallThrowParams: fixes es :: "char list exp list"   and vs :: "val list"   and e :: "char list exp"   and es' :: "char list exp list"   and C :: "char list"   and M :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "es = map Val vs @ throw e # es'" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
ipresVar_def: fixes h :: "('e, 'f, 'varSort, 'var, 'g) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"
induced_automorphism_selfcomp_halftrivial_f: shows "fixespointwise (\
 \<circ> \
) (\<Union> (f \<turnstile> X))"
card_union: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "card (A \<union> B) = card A + card B - card (A \<inter> B)"
Collect_mem_eq: fixes A :: "'a set" shows "_Collect x (x \<in> A) = A"
simps: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat" shows "insert2 x (t, n) = (case ins2 n 0 x t of Same2 \<Rightarrow> (t, n) | Bal2 t' \<Rightarrow> (t', n + 1))"
conv_backward_finite: fixes x :: "'a" shows "backward_finite x = forward_finite (x\<^sup>T)"
deleteAround_nextVertex_nextVertex: fixes f :: "face"   and g :: "graph"   and a :: "nat"   and b :: "nat"   and ps :: "(nat \<times> nat) list" assumes "f \<in> set (facesAt g a)"   and "|vertices f| = 4" shows "(f \<bullet> (f \<bullet> a), b) \<notin> set (deleteAround g a ps)"
ord_eq_less_trans: fixes a :: "'d"   and b :: "'d"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d" assumes "a = b"   and "less b c" shows "less a c"
unitary_isometry: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "unitary U" shows "isometry U"
freshEnv_swapEnv_exchange1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "goodEnv rho" shows "freshEnv xs x2 (rho &[x1 \<and> x2]_xs) = freshEnv xs x1 rho"
fls_const_nth: fixes c :: "'a"   and n :: "int" shows "fls_nth (fls_const c) n = (if n = 0 then c else (0::'a))"
f0: shows "dim_vec w = dim_vec (- \<langle>v|w\<rangle> / \<langle>v|v\<rangle> \<cdot>\<^sub>v v)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_hom hom"   and "inj_semigroup_add_hom hom" shows "inj_monoid_add_hom hom"
vector_eq: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x)"
HComplex_add: fixes x1 :: "real star"   and y1 :: "real star"   and x2 :: "real star"   and y2 :: "real star" shows "HComplex x1 y1 + HComplex x2 y2 = HComplex (x1 + x2) (y1 + y2)"
ord_p_wf_on: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfp_on (\<prec>\<^sub>p) (dgrad_p_set d m)"