hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a1 \<Rightarrow> a2" obtains \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>'__, App ea__ x__, Sa__)" and "a2 = (\<Gamma>'__, ea__, Arg x__ # Sa__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and y :: "Vars.var" and e :: "exp" and x :: "Vars.var" and S :: "stack_elem list" where "a1 = (\<Gamma>''__, Lam [y]. eb__, Arg xa__ # Sb__)" and "a2 = (\<Gamma>''__, eb__[y::=xa__], Sb__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and x :: "Vars.var" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>'''__, Var xb__, Sc__)" and "a2 = (delete xb__ \<Gamma>'''__, ec__, Upd xb__ # Sc__)" and "map_of \<Gamma>'''__ xb__ = Some ec__"   | x :: "Vars.var" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>''''__, ed__, Upd x # Sd__)" and "a2 = ((x, ed__) # \<Gamma>''''__, ed__, Sd__)" and "x \<notin> domA \<Gamma>''''__" and "isVal ed__"   | \<Delta> :: "(Vars.var \<times> exp) list" and \<Gamma> :: "(Vars.var \<times> exp) list" and S :: "stack_elem list" and e :: "exp" where "a1 = (\<Gamma>'''''__, Terms.Let \<Delta> e, Se__)" and "a2 = (\<Delta> @ \<Gamma>'''''__, e, Se__)" and "atom ` domA \<Delta> \<sharp>* \<Gamma>'''''__" and "atom ` domA \<Delta> \<sharp>* Se__"   | \<Gamma> :: "(Vars.var \<times> exp) list" and scrut :: "exp" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>''''''__, scrut ? e1__ : e2__, Sf__)" and "a2 = (\<Gamma>''''''__, scrut, Alts e1__ e2__ # Sf__)"   | \<Gamma> :: "(Vars.var \<times> exp) list" and b :: "bool" and e1 :: "exp" and e2 :: "exp" and S :: "stack_elem list" where "a1 = (\<Gamma>, Bool b, Alts e1 e2 # S)" and "a2 = (\<Gamma>, if b then e1 else e2, S)"
Inf_top_dual: fixes f :: "'b \<Rightarrow> 'c" assumes "Inf_dual f" shows "top_dual f"
L_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "L F g = {f. \<exists>c>0. \<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))}"
integrable_subalgebra: fixes f :: "'b \<Rightarrow> 'c"   and N :: "'b measure"   and M :: "'b measure" assumes "f \<in> borel_measurable N"   and "sets N \<subseteq> sets M"   and "space N = space M"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" shows "integrable N f = integrable M f"
suf_prolong: fixes w :: "'b list"   and r :: "'b list"   and z :: "'b list"   and s :: "'b list" assumes "w \<le>s r \<cdot> z"   and "r \<le>s s" shows "w \<le>s s \<cdot> z"
int_o_subst: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'h"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'e trm" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Struct.int intF \<xi> \<circ> subst \<rho> = Struct.int intF (Struct.int intF \<xi> \<circ> \<rho>)"
filtercomap_neq_bot_surj: fixes F :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "F \<noteq> bot"   and "surj f" shows "filtercomap f F \<noteq> bot"
product_simps_4: fixes A :: "('f, 'g, 'd) FSM"   and B :: "('f, 'g, 'e) FSM" shows "initial (product A B) = (initial A, initial B)"
add_strict_increasing2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_cancel_comm_monoid_add plus minus zero less_eq less"   and "less_eq zero a"   and "less b c" shows "less b (plus a c)"
polys_3: fixes qa :: "int poly"   and ria :: "root_info" assumes "(qa, ria) \<in> set polys" shows "0 < lead_coeff qa"
less_INF_D: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and i :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less y (Inf (f ` A))"   and "i \<in> A" shows "less y (f i)"
filterlim_at: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and b :: "'a"   and s :: "'a set"   and F :: "'b filter" assumes "class.topological_space open" shows "filterlim f (topological_space.at_within open b s) F = ((\<forall>\<^sub>F x in F. f x \<in> s \<and> f x \<noteq> b) \<and> filterlim f (topological_space.nhds open b) F)"
assms_2: shows "n\<^sub>0 = Max {i. f i \<le> n}"
eq: shows "restrict_indets_pp ` keys p = keys (restrict_indets p)"
subgraph_separation_min_size: fixes v :: "'a"   and n :: "nat"   and S :: "'a set" assumes "v \<noteq> v0"   and "v \<noteq> v1"   and "v \<in> V"   and "\<And>S. Separation G v0 v1 S \<Longrightarrow> Suc n \<le> card S"   and "Separation (remove_vertex v) v0 v1 S" shows "n \<le> card S"
decrease_key_def: fixes k0 :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" shows "decrease_key k0 f = Abs_poly_mapping (\<lambda>k. if k = k0 \<and> lookup f k \<noteq> (0::'d) then lookup f k - (1::'d) else lookup f k)"
poly_0: fixes x :: "'a" shows "poly 0 x = (0::'a)"
onorm_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_linear f"   and "bounded_linear g" shows "onorm (f \<circ> g) \<le> onorm f * onorm g"
ltrm_of_X_minus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "monom P (X_poly_minus R a (deg R (X_poly_minus R a))) (deg R (X_poly_minus R a)) = X_poly R"
reconstruct_reach_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "reconstruct_reach_code node_eq_impl node_hash_impl node_def_hash_size G_impl \<equiv> \<lambda>x xa. wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl (graph_restrict_left_impl (\<lambda>x f. f x) x (gi_E G_impl)) (gi_V0 G_impl) xa \<bind> (\<lambda>xb. RETURN (the xb))"
left_unique_mat_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "left_unique R" shows "left_unique (mat_rel R)"
Sup_lim: fixes b :: "nat \<Rightarrow> 'b"   and s :: "'b set"   and a :: "'b" assumes "\<And>n. b n \<in> s"   and "b \<longlonglongrightarrow> a" shows "a \<le> Sup s"
CFG_edge_Uses_pred_equal: fixes a :: "'a"   and s :: "'c"   and s' :: "'c" assumes "valid_edge a"   and "pred (kind a) s"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V" shows "pred (kind a) s'"
fe_inv_def: shows "fe_inv = (\<lambda>blues0 u0 onstack0 it (blues, reds, onstack, cyc). onstack = onstack0 \<and> gen_dfs_fe_inv E reachable onstack0 blues0 u0 it blues (cyc \<noteq> NO_CYC) \<and> cyc_post blues reds onstack u0 cyc)"
the_eq_trivial: fixes a :: "'a" shows "(THE x. x = a) = a"
spanning_subset_independent: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> U\<^sub>V\<^sub>S"   and "B \<subseteq> A"   and "\<not> local.dependent A"   and "A \<subseteq> local.span B" shows "A = B"
inj: fixes x :: "nat" shows "inj_on (\<lambda>l. replicate x R @ B # l) {l. length l = n - Suc x \<and> valid l}"
project_def: shows "project \<equiv> \<lambda>a ts. \<lparr>security_level = if a \<in> taints ts - untaints ts then 1 else 0, trusted = a \<in> untaints ts\<rparr>"
Inf_counterex_reducing: fixes N :: "'f set"   and D :: "'f" assumes "N \<inter> Bot = {}"   and "D \<in> N"   and "\<not> I_of N \<Turnstile> {D}"   and "\<And>C. \<lbrakk>C \<in> N; \<not> I_of N \<Turnstile> {C}\<rbrakk> \<Longrightarrow> D \<le> C" shows "\<exists>\<iota>\<in>Inf. prems_of \<iota> \<noteq> [] \<and> main_prem_of \<iota> = D \<and> set (side_prems_of \<iota>) \<subseteq> N \<and> I_of N \<Turnstile> set (side_prems_of \<iota>) \<and> \<not> I_of N \<Turnstile> {concl_of \<iota>} \<and> concl_of \<iota> < D"
lset_nth_member_inf: fixes xs :: "'a llist"   and A :: "'a set"   and n :: "nat" assumes "\<not> lfinite xs"   and "lset xs \<subseteq> A" shows "lnth xs n \<in> A"
LD'_def: fixes r :: "('a \<times> 'a) set"   and \<beta> :: "'a"   and \<alpha> :: "'a"   and \<sigma>1 :: "'a list"   and \<sigma>2 :: "'a list"   and \<sigma>3 :: "'a list"   and \<tau>1 :: "'a list"   and \<tau>2 :: "'a list"   and \<tau>3 :: "'a list" shows "LD' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<tau>1 \<tau>2 \<tau>3 = (LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LD_1' r \<alpha> \<beta> \<tau>1 \<tau>2 \<tau>3)"
funpow_simps_right_2: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f^(Suc n) = f^n \<circ> f"
degree_prod_list_eq: fixes ps :: "'a poly list" assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0" shows "degree (prod_list ps) = sum_list (map degree ps)"
invar_add_edge_aux: fixes g :: "'m"   and u :: "'v"   and v :: "'v"   and d :: "nat" assumes "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "invar (add_edge (u, v) d g)"
full_imp_complete: fixes n :: "nat"   and t :: "'b tree23" assumes "full n t" shows "complete t"
is_nth_power_upto_unit_conv_multiplicity: fixes n :: "nat"   and x :: "'a" assumes "0 < n" shows "is_nth_power_upto_unit n x = (\<forall>p. prime p \<longrightarrow> n dvd multiplicity p x)"
C: shows "(Owner n, \<lbrace>Num 3, PubKey C\<rbrace>) \<in> s"
continuous_on_def: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
swap_rows_mat_eq_permute: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n" shows "swaprows_mat n k l = mat n n (\<lambda>(i, j). 1\<^sub>m n $$ (Fun.swap k l id i, j))"
isNSCont_isCont_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "isNSCont f a = isCont f a"
the_enat_eSuc: fixes n :: "enat" assumes "n \<noteq> \<infinity>" shows "the_enat (eSuc n) = Suc (the_enat n)"
inverse_inverse_Pinfty_iff: fixes x :: "ereal" shows "(inverse x = \<infinity>) = (x = 0)"
numgcd_def: fixes t :: "MIR.num" shows "numgcd t = numgcdh t (maxcoeff t)"
noteq_sym: fixes a :: "'a array"   and b :: "'b array" assumes "a =!!= b" shows "b =!!= a"
ind_ring_morphisms_def: fixes V :: "'a set"   and W :: "'a set" shows "ind_ring_morphisms V W \<equiv> \<rho> (U \<inter> V) (U \<inter> W)"
sl3: shows "(abscissa B - abscissa C)\<^sup>2 + (ordinate B - ordinate C)\<^sup>2 \<in> radical_sqrt"
isomorphic_homology_imp_path_components: fixes X :: "'a topology"   and Y :: "'b topology" assumes "homology_group 0 X \<cong> homology_group 0 Y" shows "path_components_of X \<approx> path_components_of Y"
sum_measure_le_measure_inter3: fixes A :: "real set"   and B :: "real set"   and C :: "real set" assumes "A \<in> PZ.events"   and "B \<in> PZ.events"   and "C \<in> PZ.events" shows "PZ.prob A + PZ.prob B + PZ.prob C \<le> 2 + PZ.prob (A \<inter> B \<inter> C)"
Abs_viewres_inverse_Some: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "Rep_viewres (Abs_viewres (Some (a, t))) = Some (a, t)"
case1: fixes b :: "'b fps"   and a :: "'b fps" assumes "1 \<le> subdegree b" shows "fps_shift 1 (a * b) = a * fps_shift 1 b"
CatchMiss: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
var_lookup_in: fixes x :: "'b \<Longrightarrow> 'c"   and A :: "'c"   and A' :: "'d" shows "get\<^bsub>$x\<^esub> (A, A') = get\<^bsub>x\<^esub> A"
deadlock_mbisim_not_final_thread_pres: fixes t :: "'t"   and s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<in> r1.deadlocked s1 \<or> r1.deadlock s1"   and "r1.not_final_thread s1 t"   and "r1.final_thread s1 t \<Longrightarrow> r2.final_thread s2 t"   and "s1 \<approx>m s2" shows "r2.not_final_thread s2 t"
assms_2: fixes L :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" assumes "L \<in> extensions"   and "M \<lesssim> L" shows "law_restrict L = law_restrict M"
select_convs_3: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Cod \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Cod"
epiE: fixes f :: "'j"   and g :: "'j"   and g' :: "'j" assumes "J.epi f"   and "J.seq g f"   and "J.seq g' f"   and "g \<cdot>\<^sub>J f = g' \<cdot>\<^sub>J f" shows "g = g'"
nonmt: fixes K :: "'a set" assumes "K \<in> \<D>" shows "K \<inter> {x. x \<bullet> i = c} \<noteq> {}"
diamond_eq_characterisations_fun: shows "diamond_cube = diamond_cube_gen"
is_OK_returns_result_E: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog" assumes "h \<turnstile> ok f" obtains x :: "'as" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
prems_2: shows "\<Gamma>1 \<subseteq> \<Gamma>2"
cinsert_iff: fixes a :: "'f"   and b :: "'f"   and A :: "'f cset" shows "cin a (cinsert b A) = (a = b \<or> cin a A)"
fd_iso1: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" assumes "rdom x \<subseteq> rdom y" shows "relfdia z x \<subseteq> relfdia z y"
elims_2: fixes x :: "('a \<times> 'b atom) set"   and xa :: "((nat \<Rightarrow> 'b option) \<times> (nat \<Rightarrow> 'b option)) \<times> (nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)" assumes "x \<Turnstile>\<^sub>i xa" obtains as :: "('a \<times> 'b atom) set" and bi :: "((nat \<Rightarrow> 'b option) \<times> (nat \<Rightarrow> 'b option)) \<times> (nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)" where "x = as" and "xa = bi" and "\<forall>I v. (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as \<longrightarrow> (I, v) \<Turnstile>\<^sub>i\<^sub>b bi"
wellformed2_sortedQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "all_in_list p l"   and "distinct p"   and "allNetsDistinct p"   and "singleCombinators p" shows "wellformed_policy2Pr (qsort p l)"
infiter_unfold_any: fixes c :: "'a::refinement_lattice"   and i :: "nat" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> = (c \<^sup>;^ (i::nat)) ; c\<^sup>\<infinity>"
QR_lemma_12: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.b p q + QR.d p q + QR.m p q = QR.r p q"
axioms_2: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Flow c s t f" shows "Flow_axioms c s t f"
iso_some_unit_1: fixes a :: "'a" assumes "weak_unit a" shows "local.iso (some_unit a)"
that: fixes ns' :: "'node SDG_node list" assumes "n is-ns'\<rightarrow>\<^sub>d* n'" shows "thesis"
B_A_step: fixes a' :: "'b"   and b' :: "'b"   and a :: "'a" assumes "B a' b'"   and "a \<sim> a'" shows "\<exists>b. A a b \<and> b \<sim> b'"
inv_upto_mono: fixes k :: "nat"   and p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and q :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "\<And>i j. \<lbrakk>i < n; j < k; p A i j\<rbrakk> \<Longrightarrow> q A i j"   and "local.inv_upto p A k" shows "local.inv_upto q A k"
Situation_def: fixes x :: "\<kappa>" shows "Situation x \<equiv> \<lparr>A!,x\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x,F\<rbrace> \<^bold>\<rightarrow> Propositional F)"
pwd_owner_v_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_owner_v n)"
char_one_nz: shows "principal_char (Characters (Characters G)) \<one>\<^bsub>Characters (Characters G)\<^esub> \<noteq> 0"
prems_4: shows "insert e S \<inter> (X \<union> Y) = {}"
prob_space_measure_pmf: fixes p :: "'b pmf" shows "prob_space (measure_pmf p)"
iso_order_closed: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "\<phi> \<in> Group.iso G H" shows "order G = order H"
insert_remove: fixes A :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
prems: shows "\<forall>s\<in>set (s # ss). s + s = (0::'a)"
ccFromList_filter: fixes P :: "Vars.var \<Rightarrow> bool"   and xs :: "Vars.var list" shows "ccFromList (filter P xs) = ccFromList xs G|` {x. P x}"
is_interval_cbox: fixes a :: "'a"   and b :: "'a" shows "is_interval (cbox a b)"
invc_baliR: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc l"   and "invc2 r" shows "invc (baliR l a r)"
arg: fixes s :: "('d, 'e) Term.term"   and ss :: "('d, 'e) Term.term list"   and f :: "'d" assumes "s \<in> set ss" shows "Fun f ss \<rhd> s"
desargues_config_2D_non_collinear_P_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {A', C', P} = 3"
assms_1: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>1"
fmempty_lookup: fixes x :: "'e" shows "fmlookup fmempty x = None"
set_eqI: fixes A :: "'e set"   and B :: "'e set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
wf_word: fixes n :: "nat"   and w :: "('a \<times> bool list) list" shows "wf_word n w = (w \<in> lists ((set \<circ> \<sigma> \<Sigma>) n))"
diff_0_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"
empty_mulex_on: fixes M :: "'b multiset"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "M \<noteq> {#}"   and "M \<in> multisets A" shows "mulex_on P A {#} M"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f"   and x :: "'i"   and m :: "'j" shows "thread_ok ts (NewThread t x m) = free_thread_id ts t"
prems: fixes i :: "'a" assumes "i \<in> insert j J" shows "f i \<noteq> 0"
that: fixes r :: "real"   and g :: "complex \<Rightarrow> complex" assumes "0 < r"   and "g holomorphic_on ball \<xi> r"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0" shows "thesis"
perm_set_eq: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs <~~> ys" shows "set xs = set ys"
closed_scaleC: fixes S :: "'d set"   and a :: "complex" assumes "closed S" shows "closed ((*\<^sub>C) a ` S)"
subst_cls_add_mset: fixes L :: "('a, 'b) Term.term literal"   and C :: "('a, 'b) Term.term literal multiset"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_cls (add_mset L C) \<sigma> = add_mset (subst_lit L \<sigma>) (subst_cls C \<sigma>)"
Gauss_Jordan_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k A k = snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])"
iso_class_def: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.iso_class f \<equiv> {f'. Maps.isomorphic f f'}"
sym_preproc_aux_induct: fixes P :: "'t list \<Rightarrow> 't list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'c) list \<Rightarrow> 't list \<times> ('t \<Rightarrow>\<^sub>0 'c) list \<Rightarrow> bool"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "\<And>ks fs. P ks [] fs (ks, fs)"   and "\<And>ks vs fs v vs'. \<lbrakk>vs \<noteq> []; v = ord_term_lin.Max (set vs); vs' = removeAll v vs; P (ks @ [v]) (fst (sym_preproc_addnew gs vs' fs v)) (snd (sym_preproc_addnew gs vs' fs v)) (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))\<rbrakk> \<Longrightarrow> P ks vs fs (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))" shows "P ks vs fs (sym_preproc_aux gs ks (vs, fs))"
eval_prop_entailment_1: fixes \<phi> :: "'a ltln" shows "(eval \<phi> = Yes) = {} \<Turnstile>\<^sub>P \<phi>"
cnj_transpose_is_dagger: fixes M :: "complex mat" shows "M\<^sup>\<star>\<^sup>t = M\<^sup>\<dagger>"
nat_to_ce_set_def: shows "nat_to_ce_set = (\<lambda>n. fn_to_set (pr_conv_1_to_2 (nat_to_pr n)))"
proj_state_succ: fixes a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and s :: "('c, 'd) fmap"   and vs :: "'c set" assumes "fst a \<subseteq>\<^sub>f s" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s a)"
exec_move_LAssI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and V :: "nat" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (V:=e) h s ta h' s'"
boundedI: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "b \<preceq>\<^sub>t a"   and "c \<preceq>\<^sub>t a" shows "ord_term_lin.max b c \<preceq>\<^sub>t a"
idmap_bij: fixes A :: "'c set" shows "bij_to (idmap A) A A"
wens_subset: fixes F :: "'b program"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens F act B - B \<subseteq> wp act B \<inter> awp F (B \<union> wens F act B)"
Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>2 {\<lambda>s. eSuc (P s)} SKIP {P}"
vec_of_list_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "vec_of_list (map f xs) = map_vec f (vec_of_list xs)"
carrier: fixes X :: "'b set" shows "carrier\<^bsub>discrete_top X\<^esub> = X"
sel_1: fixes x1 :: "'b" shows "atm_of (Pos x1) = x1"
cidr_split_def: fixes rs :: "'a wordinterval" shows "cidr_split rs \<equiv> map prefix_match_to_CIDR (wordinterval_CIDR_split_prefixmatch rs)"
ruler': fixes xs\<^sub>1 :: "'b list"   and ys :: "'b list"   and xs\<^sub>2 :: "'b list" assumes "xs\<^sub>1 \<le>p ys"   and "xs\<^sub>2 \<le>p ys" shows "xs\<^sub>1 \<bowtie> xs\<^sub>2"
underS_incl_iff: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(underS r a \<subseteq> underS r b) = ((a, b) \<in> r)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'c"   and el_step :: "'d el_step" shows "P (case el_step of LOC x \<Rightarrow> f1 x | ENV x \<Rightarrow> f2 x) = ((\<forall>x1. el_step = LOC x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. el_step = ENV x2 \<longrightarrow> P (f2 x2)))"
two: shows "1 \<noteq> 1\<^sup>\<star> \<cdot> 0"
Span_base_incl: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "set Us \<subseteq> Span K Us"
index_mult_mat_2: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_row (A * B) = dim_row A"