hcju/isabelleps huggingface dataset:
clI: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and x :: "'b"   and X :: "'b set" assumes "matroid carrier indep"   and "x \<in> carrier"   and "indep_system.lower_rank_of indep (insert x X) = indep_system.lower_rank_of indep X" shows "x \<in> matroid.cl carrier indep X"
Receive: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (??.local.update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t])"
wset_conv: fixes ls :: "'h"   and ts :: "'i"   and m :: "'j"   and ws :: "'g"   and "is" :: "'k" shows "wset (ls, (ts, m), ws, is) = ws"
left_cancel: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(a \<^bold>*\<^sub>o\<^sub>w b = a \<^bold>*\<^sub>o\<^sub>w c) = (b = c)"
local_order_carrier_closed: fixes e1 :: "('a \<times> ('a, 'b) operation) event"   and i :: "nat"   and e2 :: "('a \<times> ('a, 'b) operation) event" assumes "e1 \<sqsubset>\<^sup>i e2" shows "{e1, e2} \<subseteq> set (history i)"
f_dim: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs" shows "dim_vec (fs ! i) = n"
B: shows "decseq (\<lambda>n. emb I (J n) (B n))"
vrange_iff: fixes b :: "V"   and r :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r) = (\<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
supp_is_subset: fixes S :: "'a set"   and x :: "'b" assumes "S supports x"   and "finite S" shows "supp x \<subseteq> S"
homotopic_with: fixes X :: "'a topology"   and P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and Y :: "'b topology"   and p :: "'a \<Rightarrow> 'b"   and q :: "'a \<Rightarrow> 'b" assumes "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> P h = P k" shows "homotopic_with P X Y p q = (\<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x\<in>topspace X. h (0, x) = p x) \<and> (\<forall>x\<in>topspace X. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x))))"
has_derivative_mult_right: fixes g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and x :: "'c" assumes "(g has_derivative g') F" shows "((\<lambda>x. x * g x) has_derivative (\<lambda>x. x * g' x)) F"
access_matrix_pretty_ipv4_code_def: fixes c :: "parts_connection"   and rs :: "32 simple_rule list" shows "access_matrix_pretty_ipv4_code c rs \<equiv> if \<not> simple_firewall_without_interfaces rs then undefined else let W = build_ip_partition c rs; R = map getOneIp W; U = all_pairs R in (zip (map ipv4addr_toString R) (map ipv4addr_wordinterval_toString W), map (\<lambda>(x, y). (ipv4addr_toString x, ipv4addr_toString y)) (filter (\<lambda>(s, d). runFw s d c rs = Decision FinalAllow) (all_pairs R)))"
subcls1_irrefl: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C1D"   and "ws_prog G" shows "C \<noteq> D"
map_gpv'_comp: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'i"   and f' :: "'j \<Rightarrow> 'g"   and g' :: "'k \<Rightarrow> 'h"   and h' :: "'i \<Rightarrow> 'l"   and gpv :: "('j, 'k, 'l) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
OclAllInstances_generic_def: fixes fst_snd :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" shows "OclAllInstances_generic fst_snd H = (\<lambda>\<tau>. Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>Some ` (H ` ran (heap (fst_snd \<tau>)) - {None})\<rfloor>\<rfloor>)"
nonpos_Reals_mult_I2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<le>\<^sub>0"   and "b \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a * b \<in> \<real>\<^sub>\<le>\<^sub>0"
simps_2: fixes v :: "nat \<times> 'a"   and va :: "(nat \<times> 'a) list" shows "add_spvec [] (v # va) = v # va"
rapprox_rat_le1: fixes x :: "int"   and y :: "int"   and n :: "nat" assumes "0 \<le> x"   and "0 < y"   and "x \<le> y" shows "real_of_float (rapprox_rat n x y) \<le> 1"
curryI: fixes f :: "'a \<times> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "f (a, b)" shows "curry f a b"
monadic_nfoldli_refine_aux': fixes l' :: "'c list"   and S :: "'c set"   and \<Gamma> :: "assn"   and Rs :: "'e \<Rightarrow> 'd \<Rightarrow> assn"   and c :: "'d \<Rightarrow> bool Heap"   and c' :: "'e \<Rightarrow> bool nres"   and Rl :: "'c \<Rightarrow> 'f \<Rightarrow> assn"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> 'd Heap"   and Rl' :: "'c \<Rightarrow> 'f \<Rightarrow> assn"   and f' :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and Rl'' :: "'c \<Rightarrow> 'f \<Rightarrow> assn"   and l :: "'f list"   and s' :: "'e"   and s :: "'d" assumes "set l' \<subseteq> S"   and "\<And>s s'. hn_refine (\<Gamma> * hn_ctxt Rs s' s) (c s) (\<Gamma> * hn_ctxt Rs s' s) bool_assn (c' s')"   and "\<And>x x' s s'. x' \<in> S \<Longrightarrow> hn_refine (\<Gamma> * hn_ctxt Rl x' x * hn_ctxt Rs s' s) (f x s) (\<Gamma> * hn_ctxt Rl' x' x * hn_invalid Rs s' s) Rs (f' x' s')"   and "\<And>x x'. hn_ctxt Rl' x' x \<or>\<^sub>A hn_ctxt Rl x' x \<Longrightarrow>\<^sub>t hn_ctxt Rl'' x' x" shows "hn_refine (\<Gamma> * hn_ctxt (list_assn Rl) l' l * hn_ctxt Rs s' s) (imp_nfoldli l c f s) (\<Gamma> * hn_ctxt (list_assn Rl'') l' l * hn_invalid Rs s' s) Rs (monadic_nfoldli l' c' f' s')"
from_hma_to_hma\<^sub>m: fixes A :: "'d mat" assumes "A \<in> carrier_mat CARD('e) CARD('f)" shows "HMA_Connect.from_hma\<^sub>m (HMA_Connect.to_hma\<^sub>m A) = A"
SLC_sym: fixes r' :: "'r"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "\<exists>s\<^sub>3' s\<^sub>3. s\<^sub>3' \<approx> s\<^sub>3 \<and> (revision_step r' s\<^sub>2 s\<^sub>3' \<or> s\<^sub>2 = s\<^sub>3') \<and> (revision_step r s\<^sub>2' s\<^sub>3 \<or> s\<^sub>2' = s\<^sub>3)" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r s\<^sub>2' s\<^sub>3 \<or> s\<^sub>2' = s\<^sub>3) \<and> (revision_step r' s\<^sub>2 s\<^sub>3' \<or> s\<^sub>2 = s\<^sub>3')"
cross7_simps_36: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x - y) \<bullet> z = x \<bullet> z - y \<bullet> z"
map_FGl_def: fixes l1 :: "'q \<Rightarrow> 'ad"   and l2 :: "'r \<Rightarrow> 'ae"   and l3 :: "'x \<Rightarrow> 'aj"   and co1 :: "'s \<Rightarrow> 'af"   and co2 :: "'t \<Rightarrow> 'ag"   and co3 :: "'y \<Rightarrow> 'ak"   and co4 :: "'z \<Rightarrow> 'al"   and contra1 :: "'ah \<Rightarrow> 'u"   and contra2 :: "'ai \<Rightarrow> 'v"   and contra3 :: "'am \<Rightarrow> 'aa"   and contra4 :: "'an \<Rightarrow> 'ab" shows "map_FGl l1 l2 l3 co1 co2 co3 co4 contra1 contra2 contra3 contra4 = map_F (map_G l1 l2 co1 co2 contra1 contra2) l1 l3 co1 co3 co4 contra1 contra3 contra4"
foldSetD_determ: fixes A :: "'b set"   and x :: "'a"   and e :: "'a"   and y :: "'a" assumes "(A, x) \<in> foldSetD D (\<cdot>) e"   and "(A, y) \<in> foldSetD D (\<cdot>) e"   and "e \<in> D"   and "A \<subseteq> B" shows "y = x"
pdevs_val_One_pdevs: fixes e :: "nat \<Rightarrow> real" shows "pdevs_val e One_pdevs = (\<Sum>i<DIM('a). e i *\<^sub>R Basis_list ! i)"
map_of_rekey: fixes g :: "'a \<Rightarrow> 'f \<Rightarrow> 'e"   and xs :: "('a \<times> 'f) list"   and k :: "'b" shows "map_of (map (\<lambda>(k, v). (f k, g k v)) xs) k = map_option (g (inv f k)) (map_of xs (inv f k))"
eventually_atin: fixes P :: "'b \<Rightarrow> bool"   and X :: "'b topology"   and a :: "'b" shows "eventually P (atin X a) = (a \<notin> topspace X \<or> (\<exists>U. openin X U \<and> a \<in> U \<and> (\<forall>x\<in>U - {a}. P x)))"
set_set_locals: fixes l :: "lname \<Rightarrow> val option"   and l' :: "lname \<Rightarrow> val option"   and s :: "st" shows "set_locals l (set_locals l' s) = set_locals l s"
transpose_scalar: fixes k :: "'b"   and A :: "(('b, 'd) vec, 'c) vec" shows "Finite_Cartesian_Product.transpose (k *k A) = k *k Finite_Cartesian_Product.transpose A"
det_fst_Gauss_Jordan_upt_k_PA_eq_fst_Gauss_Jordan_upt_k_det_P: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "fst (Gauss_Jordan_upt_k_det_P A k) = det (fst (Gauss_Jordan_upt_k_PA A k))"
bounded_hashcode_def: fixes def_hashmap_size :: "'a itself \<Rightarrow> nat"   and hashcode :: "'a \<Rightarrow> uint32"   and n :: "uint32"   and x :: "'a" assumes "class.hashable def_hashmap_size" shows "hashable.bounded_hashcode hashcode n x = hashcode x mod n"
wellformed_item_def: fixes x :: "item" shows "wellformed_item x = (item_rule x \<in> \<RR> \<and> item_origin x \<le> item_end x \<and> item_end x \<le> length Doc \<and> item_dot x \<le> length (item_rhs x))"
subst_apply_strand_nil: fixes \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "[] \<cdot>\<^sub>s\<^sub>t \<delta> = []"
simps_3: fixes \<C> :: "t_context"   and t :: "t"   and uv :: "unop_f"   and ts :: "checker_type" shows "check_single \<C> (Unop_f t uv) ts = (if is_float_t t then type_update ts [TSome t] (Type [t]) else Bot)"
induct: fixes P :: "'a negation_type list \<Rightarrow> bool"   and a0 :: "'a negation_type list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (Pos x # xs)"   and "\<And>v xs. P xs \<Longrightarrow> P (Neg v # xs)" shows "P a0"
o2: shows "rel_generat A C (rel_fun R (rel_gpv'' A C R)) (IO out c) b2"
parent_edge: fixes v :: "'a" assumes "v \<in> V"   and "v \<noteq> root" shows "v \<rightarrow> parent v"
elims_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "xa__ = []"   | x :: "'b" and ys :: "'b list" where "xa__ = x # ys" and "Ball (ran\<^sub>l ys) (less_eq x) \<and> linorder.sorted less_eq ys"
Conj_rep_eq: fixes xset :: "('e, 'f, 'g) formula set['e]" assumes "finite (supp xset)" shows "Rep_formula (Conj xset) = Conj\<^sub>\<alpha> (map_bset Rep_formula xset)"
star_of_add: fixes x :: "'b"   and y :: "'b" shows "star_of (x + y) = star_of x + star_of y"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "('e, 'i, 'j, 'k) SLin_state_scheme" shows "P11 (s1, s2) = (initVals s2 = abortVals s1)"
smallomega_1_conv_filterlim: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "(f \<in> \<omega>[F](\<lambda>_. 1::'b)) = filterlim f at_infinity F"
orthogonal_def: fixes vs :: "'a vec list" shows "orthogonal vs \<equiv> \<forall>i j. i < length vs \<longrightarrow> j < length vs \<longrightarrow> (vs ! i \<bullet> vs ! j = (0::'a)) = (i \<noteq> j)"
L_satisfies_eqvt: fixes P\<^sub>L :: "('state, 'effect) L_state"   and \<phi> :: "'pred"   and p :: "perm" assumes "P\<^sub>L \<turnstile>\<^sub>L \<phi>" shows "p \<bullet> P\<^sub>L \<turnstile>\<^sub>L p \<bullet> \<phi>"
foldSet_not_depend: fixes D :: "'c set"   and E :: "'c set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "D \<subseteq> E" shows "foldSetD D f e \<subseteq> foldSetD E f e"
assms: shows "i < dim_vec (lpoly_to_vec (vec_to_lpoly v))"
child_c: shows "h \<turnstile> get_dom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r child_c"
merge_def: fixes qs :: "('a \<times> 'b) list"   and ps :: "('a \<times> 'b) list" shows "AList.merge qs ps = foldr (\<lambda>(k, v). AList.update k v) ps qs"
eq: shows "Jacobi (int a) (int p) = Legendre (int a) (int p)"
expr_of_expr_def: shows "expr_of = (\<lambda>e. e)"
collision_restrict: shows "delayed_safe_distance.collision {0..} = delayed_safe_distance.collision {0..max ego2.t_stop delayed_safe_distance.other.t_stop}"
ipt_tcp_flags_Any: fixes pkt :: "tcp_flag set" shows "match_tcp_flags ipt_tcp_flags_Any pkt"
simps_2: fixes v :: "'d"   and va :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set" shows "v # va \<otimes>\<^bsub>\<alpha>\<^esub> [] = {v # va}"
stirling_integral_1_unfold: fixes m :: "nat" assumes "0 < m" shows "stirling_integral 1 s = stirling_integral m s / of_nat m - (\<Sum>k = 1..<m. complex_of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k))"
lincomb_clean: fixes A :: "'c set"   and Z :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "A \<subseteq> carrier V"   and "Z \<subseteq> carrier V"   and "finite A"   and "finite Z"   and "a \<in> A \<rightarrow> carrier K"   and "a \<in> Z \<rightarrow> {\<zero>\<^bsub>K\<^esub>}" shows "lincomb a (A \<union> Z) = lincomb a A"
induct: fixes P :: "variable \<times> typ \<Rightarrow> nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "variable \<times> typ"   and a1 :: "nat"   and a2 :: "Core.term" assumes "\<And>vT n v T. P vT n (Fv v T)"   and "\<And>vT n T t. P vT (n + 1) t \<Longrightarrow> P vT n (Abs T t)"   and "\<And>vT n f u. \<lbrakk>P vT n f; P vT n u\<rbrakk> \<Longrightarrow> P vT n (f $ u)"   and "\<And>uu_ uv_ v va. P uu_ uv_ (Ct v va)"   and "\<And>uu_ uv_ v. P uu_ uv_ (Bv v)" shows "P a0 a1 a2"
psi_inner_psi'_l: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "inner_prod (grover_state.\<psi> n) (grover_state.psi'_l n f l) = grover_state.alpha_l n f l * complex_of_real (cos (grover_state.\<theta> n f / 2)) - grover_state.beta_l n f l * complex_of_real (sin (grover_state.\<theta> n f / 2))"
inorder2_join_adj: fixes ts :: "'a tree23s" assumes "not_T ts" shows "inorder2 (join_adj ts) = inorder2 ts"
retract_of_space_topspace: fixes X :: "'b topology" shows "topspace X retract_of_space X"
wf_until_aux_def: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and n :: "nat"   and R :: "'b list set"   and pos :: "bool"   and \<phi> :: "'b MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'b MFOTL.formula"   and aux :: "(nat \<times> 'b option list set \<times> 'b option list set) list"   and ne :: "nat" shows "wf_until_aux \<sigma> n R pos \<phi> I \<psi> aux ne = list_all2 (\<lambda>x i. case x of (t, r1, r2) \<Rightarrow> t = \<tau> \<sigma> i \<and> qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. if pos then \<forall>k\<in>{i..<ne + length aux}. MFOTL.sat \<sigma> (map the v) k \<phi> else \<exists>k\<in>{i..<ne + length aux}. MFOTL.sat \<sigma> (map the v) k \<phi>) r1 \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < ne + length aux \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> MFOTL.sat \<sigma> (map the v) j \<psi> \<and> (\<forall>k\<in>{i..<j}. if pos then MFOTL.sat \<sigma> (map the v) k \<phi> else \<not> MFOTL.sat \<sigma> (map the v) k \<phi>)) r2) aux [ne..<ne + length aux]"
finite_emeasure_space: fixes M :: "'c measure" assumes "finite_measure M" shows "emeasure M (space M) \<noteq> top"
exp_total: fixes y :: "real" assumes "0 < y" shows "\<exists>x. exp x = y"
countable: fixes A :: "'a set"   and M :: "'a measure" assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"   and "countable A" shows "A \<in> sets M"
pmf_Pi_empty: fixes dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" shows "Pi_pmf {} dflt p = return_pmf (\<lambda>_. dflt)"
z_1: shows "((\<lambda>x. if x \<in> i then f x else (0::'a)) has_integral z) (cbox a b)"
Rf_eq_Rf_diff_Rf: fixes N :: "'a literal multiset set" shows "Rf N = Rf (N - Rf N)"
lebesgue_mpoly_zero_set: fixes p :: "real mpoly"   and n :: "nat" assumes "p \<noteq> 0"   and "vars p \<subseteq> {..<n}" shows "{f \<in> space (lborel_f n). insertion f p = 0} \<in> null_sets (lborel_f n)"
legal_zpassign_def: fixes A :: "'v pvar \<Rightarrow> zhmultiset" shows "legal_zpassign A = (\<forall>x. zhmset_of (min_passign x) \<le> A x)"
assms_2: shows "range (poly.coeff p) \<subseteq> P[- {x}]"
A_def: shows "A = sum_list (ss @ [s]) `\<rightarrow> C0"
entry_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.entry prob = ifc_problem.entry prob"
proper_ideal: fixes I :: "'a set" assumes "ideal R I"   and "1\<^sub>r \<notin> I" shows "I \<noteq> carrier R"
converse_rtranclpE: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and z :: "'e" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'e" where "r x y" and "r\<^sup>*\<^sup>* y z"
borel_measurable_continuous_onI: fixes f :: "'d \<Rightarrow> 'e" assumes "continuous_on UNIV f" shows "f \<in> borel_measurable borel"
f1'c_le_f1: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real" assumes "Table0 f1 f2 f1' f2' e c" shows "f1' / c \<le> f1"
Set_filter_insert: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a set" shows "Set.filter P (insert x A) = (if P x then insert x (Set.filter P A) else Set.filter P A)"
extender_Sucset: fixes k :: "nat"   and S :: "'b formula set" shows "extender k S \<subseteq> extender (Suc k) S"
notTerminalSucNotTerminal: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and n :: "nat"   and gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "\<not> terminal subs (f (Suc n))"   and "branch subs gamma f" shows "\<not> terminal subs (f n)"
q: shows "q \<in> More_Modules.ideal (insert p (set_mset A \<union> polynomial_bool))"
norm_scaleR: shows "norm \<circ> (*\<^sub>R) r = (*\<^sub>R) \<bar>r\<bar> \<circ> norm"
not_less_iff_gr_or_eq: fixes x :: "'c word"   and y :: "'c word" shows "(\<not> x <s y) = (y <s x \<or> x = y)"
Gromov_product_le_dist_1: fixes e :: "'b"   and x :: "'b"   and y :: "'b" shows "Gromov_product_at e x y \<le> dist e x"
length_replicate: fixes n :: "nat"   and x :: "'a" shows "length (replicate n x) = n"
length_interpret_floatariths: fixes slp :: "floatarith list"   and xs :: "real list" shows "length (interpret_floatariths slp xs) = length slp"
assms: shows "xvec \<sharp>* (\<alpha> \<prec> P)"
AssertAllState'''_axioms_def: fixes ass_bnd :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and chk :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" shows "AssertAllState'''_axioms ass_bnd chk update eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update \<equiv> ass_bnd = Update.assert_bound update \<and> chk = PivotUpdateMinVars.check eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"
ltln_strong_to_weak_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi>"
interpret_slp_map_Var: fixes "is" :: "nat list"   and xs :: "real list" assumes "\<And>i. i < length is \<Longrightarrow> i \<le> is ! i"   and "\<And>i. i < length is \<Longrightarrow> is ! i - i < length xs" shows "interpret_slp (map Var is) xs = rev (map2 (\<lambda>i e. xs ! (e - i)) [0..<length is] is) @ xs"
simps_4: fixes delta :: "'h"   and inp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option"   and xs' :: "'f"   and x' :: "'g" shows "(qOp delta inp binp #= qVar xs' x') = False"
n_kat_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<cdot> t z = (0::'a)) = (!x \<cdot> y \<cdot> !z = !x \<cdot> y)"
syncHdAdd1: fixes r :: "'a event list"   and t :: "'a event list"   and A :: "'a event set"   and u :: "'a event list"   and a :: "'a event" assumes "r setinterleaves ((t, u), A) \<and> a \<notin> A \<and> hd t \<in> A" shows "a # r setinterleaves ((t, a # u), A)"
Quotient_real: shows "Quotient realrel Real.Real rep_real cr_real"
vector_derivative_unique_within_closed_interval: fixes a :: "real"   and b :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and f'' :: "'a" assumes "a < b"   and "x \<in> cbox a b"   and "(f has_vector_derivative f') (at x within cbox a b)"   and "(f has_vector_derivative f'') (at x within cbox a b)" shows "f' = f''"
AddSA_CompFun_ran: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "ran (CompFun (A [++] (S, SA))) = insert {} (insert (insert SA (the (CompFun A S))) (ran (CompFun A) - {the (CompFun A S)}))"
is_faithful: fixes f :: "'b"   and f' :: "'b" assumes "D.par f f'"   and "F.right_map f = F.right_map f'" shows "f = f'"
meet_comm: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "x \<sqinter>\<^bsub>L\<^esub> y = y \<sqinter>\<^bsub>L\<^esub> x"
ind: fixes c :: "nat" assumes "distinct ls"   and "last ls = v"   and "v \<in> set ls" shows "nextElem ls c v = c"
bin_nth_numeral_unfold_2: fixes x :: "num"   and n :: "nat" shows "bit (numeral (num.Bit1 x)) n = (0 < n \<longrightarrow> bit (numeral x) (n - 1))"
non_constructive_selector_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and G :: "nat \<times> ('d \<times> nat \<times> nat) set" shows "non_constructive_selector Rs G \<equiv> let wl = worklist G Rs in if wl = {} then None else Some (SOME (R, f). valid_selection Rs G R f)"
core_subsetD: fixes c :: "'a"   and B :: "'a set" assumes "c \<in> core B" shows "c \<in> B"
self_codomain_iff_ide: fixes a :: "'a" shows "(a \<in> Right.codomains a) = Right.ide a"
open_Diff: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "closed T" shows "open (S - T)"
inf_llist_lnth: fixes xs :: "'c llist" assumes "\<not> lfinite xs" shows "inf_llist (lnth xs) = xs"
has_integral_integral_eqI: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set"   and k :: "'c" assumes "f integrable_on s"   and "integral s f = k" shows "(f has_integral k) s"
from_list_def: fixes l :: "(('a \<times> 'a) \<times> nat) list" shows "G.from_list l = foldr (\<lambda>(e, d). G_add_edge e d) l G_empty"
trancl_impl: fixes set_of :: "'b \<Rightarrow> 'c set"   and memb :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and empty :: "'b"   and un :: "'c list \<Rightarrow> 'b \<Rightarrow> 'b"   and gen_succ :: "('c \<times> 'c) list \<Rightarrow> 'c list \<Rightarrow> 'c list"   and rel :: "('c \<times> 'c) list"   and as :: "'c list" assumes "set_access_gen set_of memb empty un gen_succ" shows "set_of (trancl_impl gen_succ un memb empty rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set rel)\<^sup>+}"
simps_1: fixes P :: "'a prog"   and C :: "char list"   and Cs :: "char list list" shows "supercls_lst P (C # Cs) = ((\<forall>C'\<in>set Cs. P \<turnstile> C' \<preceq>\<^sup>* C) \<and> supercls_lst P Cs)"
closure_complete: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "lang n r = lang n s"   and "local.wf n r"   and "local.wf n s" shows "\<exists>N. closure ([(init r, init s)], {(post (init r), post (init s))}) = Some ([], N)"
sdrop_smap: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b stream" shows "sdrop n (smap f s) = smap f (sdrop n s)"
subdegree_eq_0_iff: fixes f :: "'b fps" shows "(subdegree f = 0) = (f = 0 \<or> f $ 0 \<noteq> (0::'b))"
add_0_left: fixes a :: "'c" shows "bot \<squnion> a = a"
S5: fixes y :: "i" shows "cod y \<cdot> y \<cong> y"