hcju/isabelleps huggingface dataset:
abc_comp_frist_steps_eq_pre: fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and B :: "abc_inst list" assumes "abc_notfinal (abc_steps_l (0, lm) A n) A"   and "A \<noteq> []" shows "abc_steps_l (0, lm) (A [+] B) n = abc_steps_l (0, lm) A n"
quo_lemma2: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_quo [x, y] = x div y"
asm: fixes z :: "ennreal" assumes "z \<in> ennreal ` A" shows "z \<le> y"
k_isNode_def: fixes a :: "'a"   and ec :: "'rep" shows "KBP.k_isNode a ec \<equiv> simAbs ec \<in> sim_equiv_class a ` jkbpC"
successor_in_\<alpha>n: fixes g :: "'g"   and n :: "'node" assumes "predecessors g n \<noteq> []"   and "invar g" shows "n \<in> set (\<alpha>n g)"
j: shows "j < dim_col (mat_of_cols (dim_row A) [col A 0] @\<^sub>c mat_of_cols (dim_row A) (map (col A) [1..<dim_col A]))"
has_field_derivative_Gamma: fixes z :: "'a"   and A :: "'a set" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(Gamma has_field_derivative Gamma z * Digamma z) (at z within A)"
sim: shows "weak_reduction_simulation (indRelTEQ TRel) (STCal Source Target)"
zero_divisor_def: fixes a :: "'a" shows "zero_divisor a \<equiv> \<exists>b. b \<noteq> (0::'a) \<and> a * b = (0::'a)"
JF2set_simps: fixes a :: "'d JF2" shows "JF2set a = F2set1 (dtor2 a) \<union> (\<Union> (JF1set ` F2set2 (dtor2 a)) \<union> \<Union> (JF2set ` F2set3 (dtor2 a)))"
le_Bits: fixes v :: "int"   and b :: "bool"   and w :: "int"   and c :: "bool" shows "(v BIT b \<le> w BIT c) = (v < w \<or> v \<le> w \<and> (\<not> b \<or> c))"
Abs_process_inverse: fixes y :: "('a list \<times> 'a set) set \<times> 'a list set" assumes "y \<in> process_set" shows "Rep_process (Abs_process y) = y"
vec_inv: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n"   and "v \<noteq> 0\<^sub>v n" shows "vec_inv v \<bullet> v = (1::'b)"
ipresIGOp_def: fixes h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('index, 'bindex, 'm, 'n, 'opSym, 'o, 'h, 'j, 'p) model_scheme" shows "ipresIGOp h hA MOD MOD' \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> h (igOp MOD delta inp binp) = igOp MOD' delta (lift h inp) (lift hA binp)"
assms: shows "\<forall>A\<in>carrier_mat 1 2. admits_triangular_reduction A"
poly_shift_times_monom_le: fixes n :: "nat"   and m :: "nat"   and c :: "'b"   and p :: "'b poly" assumes "n \<le> m" shows "poly_shift n (monom c m * p) = monom c (m - n) * p"
foldr_True_set: fixes f :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "foldr (\<lambda>x. (\<and>) (f x)) l True = (\<forall>x\<in>set l. f x)"
inv_partD_1: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m_i :: "nat"   and m_j :: "nat"   and i :: "nat"   and j :: "nat" assumes "inv_part n p A m_i m_j"   and "i < n"   and "j < n"   and "j < m_j" shows "p A i j"
Infty_neq_0_4: shows "0 \<noteq> - \<infinity>"
from_hma_to_hma\<^sub>v: fixes v :: "'b Matrix.vec" assumes "v \<in> carrier_vec CARD('c)" shows "HMA_Connect.from_hma\<^sub>v (HMA_Connect.to_hma\<^sub>v v) = v"
apx_norm_eq_spec: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "valid_dbm M"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "Approx\<^sub>\<beta> ([M]\<^bsub>v,n\<^esub>) = [DBM_Normalization.norm (FW M n) (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub>"
wordinterval_CIDR_split_prefixmatch: fixes r :: "'b wordinterval" shows "wordinterval_to_set r = \<Union> (prefix_to_wordset ` set (wordinterval_CIDR_split_prefixmatch r))"
borel_measurable_imp_continuous_limit: fixes h :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "h \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" obtains g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" where "\<And>n. continuous_on UNIV (g n)" and "AE x in lebesgue_on S. (\<lambda>n. g n x) \<longlonglongrightarrow> h x"
d_OUT_collect: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and x :: "'v" assumes "flow \<Delta>'' f" shows "d_OUT (\<lambda>(x, y). f (Edge x y, Vertex y)) x = d_OUT f (Vertex x)"
strict_pref_def: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c" shows "x \<^bsub>r\<^esub>\<prec> y \<equiv> x \<^bsub>r\<^esub>\<preceq> y \<and> (y, x) \<notin> r"
simps_1: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c" shows "dfs f [] = []"
lift_RealVal_def: fixes f :: "real \<Rightarrow> real" shows "lift_RealVal f \<equiv> \<lambda>x. case x of RealVal v \<Rightarrow> RealVal (f v) | _ \<Rightarrow> RealVal (f 0)"
dom_closed: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>" shows "\<guillemotleft>local.dom f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
ker_to_mimg: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "ker\<^bsub>M,mimg\<^bsub>R M,N\<^esub> f\<^esub> f = ker\<^bsub>M,N\<^esub> f"
flowD_finite_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow (residual_network f) f"   and "x \<noteq> sink (residual_network f)" shows "d_IN f x \<noteq> \<top>"
remove_child_removes_child: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "known_ptrs h"   and "type_wf h" shows "child \<notin> set children"
\<omega>\<^sub>O: shows "\<langle>\<omega>\<rangle>\<^sub>O = (\<lambda>z s l. l (\<lambda>n. \<langle>n\<rangle>\<^sub>N s z))"
aSh_def: fixes v :: "Value" shows "aSh v = (case v of shortV sh \<Rightarrow> sh)"
CH_card: fixes X :: "'a set" shows "card (CH X) = (\<Sum>h\<in>UNIV. card (Ch h X))"
fag_single_additionTr2: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and s :: "'e" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "s \<in> add_set f {a} 0" shows "s \<^sub>f+ \<^sub>i- a = z"
sup_state_opt_any_Some: fixes P :: "'a prog"   and s :: "ty list \<times> ty err list"   and X :: "(ty list \<times> ty err list) option" shows "P \<turnstile> \<lfloor>s\<rfloor> \<le>' X = (\<exists>s'. X = \<lfloor>s'\<rfloor> \<and> P \<turnstile> s \<le>\<^sub>i s')"
eq: shows "int (z - poly_deg f) = int z - int (poly_deg f)"
cut_greater_Min_eq_Diff: fixes I :: "'a set" shows "I \<down>> iMin I = I - {iMin I}"
exhaust: fixes y :: "'k \<times> 'l" obtains a :: "'k" and b :: "'l" where "y = (a, b)"
wellf_arity_nonzero: fixes f :: "recf" assumes "wellf f" shows "0 < arity f"
prime_Esigma_mult: fixes m :: "nat"   and n :: "nat" assumes "prime m"   and "prime n"   and "m \<noteq> n" shows "Esigma (m * n) = Esigma n * Esigma m"
pr_state: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and X :: "agent"   and n :: "nat"   and run :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "(NonceS (S (X, n, run)) = None \<longrightarrow> IntMapK (S (X, n, run)) = None) \<and> (IntMapK (S (X, n, run)) = None \<longrightarrow> ExtMapK (S (X, n, run)) = None) \<and> (ExtMapK (S (X, n, run)) = None \<longrightarrow> IntAgrK (S (X, n, run)) = None) \<and> (IntAgrK (S (X, n, run)) = None \<longrightarrow> ExtAgrK (S (X, n, run)) = None)"
nth_least_wellformed: fixes n :: "nat"   and A :: "'b set" assumes "enat n < esize A" shows "nth_least A n \<in> A"
proper_part_transitivity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "PP x y"   and "PP y z" shows "PP x z"
map_def: fixes f :: "'j" shows "\<chi>'.A.map f = (if J.arr f then a' else C.null)"
scene_union_idem: fixes X :: "'a scene" shows "X \<squnion>\<^sub>S X = X"
sums_ennreal: fixes f :: "nat \<Rightarrow> real"   and x :: "real" assumes "\<And>i. 0 \<le> f i"   and "0 \<le> x" shows "(\<lambda>i. ennreal (f i)) sums ennreal x = f sums x"
map_sel_1: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b fundef" shows "body (map_fundef f a) = map f (body a)"
lsemimodule5: fixes a :: "'a" shows "\<dd> \<circ> a = (0::'a)"
simps: fixes P :: "'a prog"   and a1 :: "String.literal"   and a2 :: "String.literal" shows "P \<turnstile> a1 \<prec>\<^sup>1 a2 = (\<exists>C D rest. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(D, rest)\<rfloor> \<and> C \<noteq> Object)"
ipurge_tr_aux_single_dom: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list" shows "ipurge_tr_aux I D {u} xs = ipurge_tr I D u xs"
x_1: shows "x * f $$ fls_subdegree f = (1::'a)"
affine_unop_def: fixes p :: "nat"   and a :: "real"   and b :: "real"   and d :: "real"   and X :: "(real \<times> real pdevs) \<times> real" shows "affine_unop p a b d X = (let ((x, xs), xe) = X; (ax, axe) = trunc_bound_eucl p (a * x); (y, ye) = trunc_bound_eucl p (ax + b); (ys, yse) = trunc_bound_pdevs p (scaleR_pdevs a xs) in ((y, ys), sum_list' p [truncate_up p (\<bar>a\<bar> * xe), axe, ye, yse, d]))"
prj_simps_8: shows "C.cod \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 = \<nu>.apex"
inverse_inverse_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (inverse a) = a"
h2: shows "lose t = [True] \<longrightarrow> ack t = [x] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = x"
column_Gram_Schmidt_matrix2: fixes k :: "'b"   and A :: "((real, 'b) vec, 'a) vec" shows "column k A = column k (Gram_Schmidt_matrix A) + (\<Sum>x\<in>{column i (Gram_Schmidt_matrix A) |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x)"
extend_preserves_model: fixes f :: "nat \<Rightarrow> bool list"   and C :: "fterm literal set"   and n :: "nat" assumes "wf_infpath f"   and "ground\<^sub>l\<^sub>s C"   and "\<not> falsifies\<^sub>c (f (Suc n)) C"   and "\<forall>l\<in>C. nat_of_fatom (get_atom l) \<le> n" shows "eval\<^sub>c HFun (extend f) C"
less_by_empty: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set" assumes "A = {}" shows "A \<subseteq> B"
vifunion_vempty_conv_1: fixes I :: "V"   and f :: "V \<Rightarrow> V" shows "(0 = \<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)) = (\<forall>i\<in>elts I. f i = 0)"
lzip_trans_fusion3: fixes g :: "'e \<Rightarrow> ('c, 'e) step"   and h :: "('d, 'f) lgenerator'"   and sg :: "'e"   and sh :: "'f" shows "lunstream (zip_raw g (lgenerator h)) (sg, sh, None) = lzip (lunstream g sg) (lunstream' h sh)"
qty_rem_comp_def: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c" shows "qty_rem_comp pf1 x = pf1(x := \<lambda>n w. 0)"
implied_frontier_def: fixes P :: "'loc \<Rightarrow> 't zmultiset"   and loc :: "'loc" shows "implied_frontier P loc = frontier (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_pos (P loc'))) (path_summary loc' loc))"
Suc_1: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R"   and "\<And>i j. \<lbrakk>i \<le> n; j \<le> n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "canonical_proj_ext I n ` carrier R = carrier (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
le_max_iff_disj: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and z :: "'h"   and x :: "'h"   and y :: "'h" assumes "class.linorder less_eq less" shows "less_eq z (ord.max less_eq x y) = (less_eq z x \<or> less_eq z y)"
mat_row_code: fixes k :: "'b"   and i :: "'a" shows "($) (mat_row k i) = (\<lambda>j. if i = j then k else (0::'b))"
non_empty_cycle_root_plus_converse: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>+ = x\<^sup>T\<^sup>+"
Var_def: fixes v :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and A :: "('c, 'd) action" shows "Var v A \<equiv> action_of (R (true \<turnstile> \<lambda>(A, A'). \<exists>a. tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = increase v a (alpha_rp.more A)) ;; action.relation_of A ;; R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = decrease v (alpha_rp.more A)))"
implied_frontier_alt_def: fixes c :: "('loc, 't, 'b) configuration_scheme"   and loc :: "'loc" shows "implied_frontier_alt c loc = frontier (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_pts c loc'))))) (path_summary loc' loc))"
wbalanced_balance_list: fixes xs :: "'a list" shows "wbalanced (balance_list xs)"
sqrtl_def: fixes x :: "real"   and n :: "nat" shows "sqrtl x n = x / sqrtu x n"
odd_two_times_div_two_succ: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "plus (times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One)))) one = a"
ev_alw_alw_impl: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "ev (alw \<phi>) xs"   and "alw (\<lambda>xs. alw \<phi> xs \<longrightarrow> ev \<psi> xs) xs" shows "ev \<psi> xs"
refl_joinable: fixes R :: "('b \<times> 'b) set" assumes "refl R" shows "refl (joinable R)"
simps_2: fixes t :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle>\<^sub>s\<^sub>t) = {t}"
set_mset_add_mset_insert: fixes a :: "'a"   and A :: "'a multiset" shows "set_mset (a, A) = a \<triangleright> set_mset A"
Int_stable_def: fixes M :: "'b set set" shows "Int_stable M = (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M)"
induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes " x1 \<longrightarrow>\<^sub>\<beta>\<^sup>* x2"   and "\<And>t. P t t"   and "\<And>t s.  t \<longrightarrow>\<^sub>\<beta> s \<Longrightarrow> P t s"   and "\<And>t1 t2 t3. \<lbrakk> t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2; P t1 t2;  t2 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3; P t2 t3\<rbrakk> \<Longrightarrow> P t1 t3" shows "P x1 x2"
intros_3: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a llist" shows "llexord r LNil ys"
set_of_delete: fixes t :: "'b tree"   and x :: "'b" assumes "avl t"   and "is_ord t" shows "set_of (delete x t) = set_of t - {x}"
ext: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "COMP g f \<noteq> null" shows "seq g f"
pheap_del_min: fixes h :: "'b tree" assumes "is_root h"   and "pheap h" shows "pheap (del_min h)"
tranclp_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "less\<^sup>+\<^sup>+ = less"
eq: shows "(\<lambda>x. (f x, x)) = (\<lambda>(a, b). (b, a)) \<circ> (\<lambda>x. (x, f x))"
Seq_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e;; e') h (stk, loc, Suc (length (compE2 e) + pc), xcp) (stk', loc', Suc (length (compE2 e) + pc'), xcp')"
Sim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
t_def: shows "t \<equiv> harm n - ln (real (Suc n))"
continuous_on_cos_real: fixes a :: "real"   and b :: "real" shows "continuous_on {a..b} cos"
orbit_stabiliser_axioms: shows "orbit_stabiliser G (\<odot>)"
adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: shows "adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs Shadow_DOM.remove_child_locs set_disconnected_nodes_locs"
cancel_right_a: fixes b :: "'a"   and a :: "'a" shows "b r\<rightarrow> b * a = a"
induced_morphism_eval: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e"   and x :: "'d"   and C :: "('f set \<times> 'g) set"   and r :: "'f set \<times> 'g" assumes "ind_mor_btw_stalks X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f x"   and "C \<in> stalk.carrier_stalk is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y (f x)"   and "r \<in> C" shows "\<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub> C = direct_lim.class_of \<O>\<^sub>X \<rho>\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} (f \<^sup>\<inverse> X (fst r)) (\<phi>\<^sub>f (fst r) (snd r))"
4: shows "e\<^sup>T * top \<le> - - v"
map_of_Cons_code_2: fixes l :: "'b"   and v :: "'a"   and ps :: "('b \<times> 'a) list"   and k :: "'b" shows "map_of ((l, v) # ps) k = (if l = k then \<lfloor>v\<rfloor> else map_of ps k)"
fps_inverse_one_minus_fps_X: shows "inverse (1 - fps_X) = Abs_fps (\<lambda>n. 1::'b)"
asm: shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
lessThan_subset_iff: fixes x :: "'b"   and y :: "'b" shows "({..<x} \<subseteq> {..<y}) = (x \<le> y)"
sumRes: fixes x :: "name"   and P :: "pi"   and Q :: "pi" shows "<\<nu>x>(P \<oplus> Q) \<sim> <\<nu>x>P \<oplus> <\<nu>x>Q"
inj_ring_hom_axioms: shows "inj_ring_hom (inv hom)"
all_ideals: shows "{I. ideal I R} = {{\<zero>}, carrier R}"
topspace_prod_topology: fixes X :: "'c topology"   and Y :: "'d topology" shows "topspace (prod_topology X Y) = topspace X \<times> topspace Y"
igSwapInp_def: fixes MOD :: "('f, 'g, 'h, 'i, 'j, 'k, 'e, 'l, 'm) model_scheme"   and zs :: "'h"   and z1 :: "'k"   and z2 :: "'k"   and inp :: "'d \<Rightarrow> 'e option" shows "igSwapInp MOD zs z1 z2 inp \<equiv> lift (igSwap MOD zs z1 z2) inp"
summable_imp_abs_convergent_prod: fixes f :: "nat \<Rightarrow> 'e" assumes "summable (\<lambda>i. norm (f i - (1::'e)))" shows "abs_convergent_prod f"
wfs_s\<^sub>L\<^sub>0: shows "wfs \<Gamma>\<^sub>0 s\<^sub>L\<^sub>0"
steps_to_trace: fixes x :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and y :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" assumes "x \<Rightarrow>\<^sub>d\<^sup>* y" shows "\<exists>c k. trace c 0 k \<and> c 0 = x \<and> c k = y"
monoD: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "continuous F"   and "x \<le> y" shows "F x \<le> F y"
reflexive_inf_star: fixes y :: "'a"   and x :: "'a" assumes "top \<le> y" shows "y \<sqinter> top = top \<squnion> y \<sqinter> (x \<sqinter> top)"
poly_mult_monom': fixes p :: "'a list"   and a :: "'a"   and n :: "nat" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R" shows "poly_mult (local.monom a n) p = local.normalize (map ((\<otimes>) a) p @ replicate n \<zero>)"
echelon_form_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "echelon_form A = echelon_form_upt_k A (ncols A)"
simps_12: fixes x31 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and x32 :: "'i"   and x33 :: "'j"   and x34 :: "(('g \<Rightarrow> 'h option) \<Rightarrow> 'h option) list"   and x2 :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool" shows "x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 \<noteq> (x2)\<^sub>\<surd>"
Arr_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Arr C (free_monoidal_category.rep C f)"
assms: shows "xs \<in> filter P ys \<otimes> filter P zs"
set_inline1_lift_spmf1: fixes p :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e) spmf"   and gpv :: "('d, 'i, 'h) gpv"   and s :: "'e" shows "set_spmf (inline1 (\<lambda>s x. lift_spmf (p s x)) gpv s) \<subseteq> range Inl"
simps: fixes \<Sigma> :: "'c set"   and \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and F :: "'b set"   and n :: "nat" shows "mojmir_def.\<S> \<Sigma> \<delta> q\<^sub>0 w F n = F \<union> {q. \<exists>j\<ge>the (mojmir_def.smallest_accepting_rank \<Sigma> \<delta> q\<^sub>0 w F). semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some j}"
intro: fixes A :: "V"   and f :: "V" assumes "nop_v11 A (1\<^sub>\<nat>) f" shows "unop_v11 A f"
assms: shows "(f, h) \<in> [comp_PRE S1 Q P X]\<^sub>f R \<rightarrow> S"
WhileFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> Normal s"
arith_simps_53: fixes m :: "num"   and n :: "num" shows "numeral m - - numeral n = numeral (m + n)"
lfp_fusion3: fixes g :: "'d \<rightarrow> 'e"   and f :: "'d \<rightarrow> 'd"   and h :: "'e \<rightarrow> 'e" assumes "g oo f = h oo g"   and "g\<cdot>\<bottom> = \<bottom>" shows "g\<cdot>(fix\<cdot>f) = fix\<cdot>h"
simps_2: fixes l :: "('b \<times> 'c) tree"   and a :: "'b"   and uu :: "'c"   and r :: "('b \<times> 'c) tree" shows "Tree2.set_tree \<langle>l, (a, uu), r\<rangle> = {a} \<union> Tree2.set_tree l \<union> Tree2.set_tree r"