hcju/isabelleps huggingface dataset:
bezout_iterate_preserves: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and i :: "'f"   and n :: "nat"   and b :: "'e"   and a :: "'f" assumes "echelon_form_upt_k A k"   and "is_bezout_ext bezout"   and "A $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "n < nrows A"   and "b < mod_type_class.from_nat k"   and "mod_type_class.to_nat i \<le> n"   and "k < ncols A"   and "is_zero_row_upt_k i k A" shows "bezout_iterate A n i (mod_type_class.from_nat k) bezout $ a $ b = A $ a $ b"
comp_eq_elim: fixes a :: "'i \<Rightarrow> 'h"   and b :: "'g \<Rightarrow> 'i"   and c :: "'j \<Rightarrow> 'h"   and d :: "'g \<Rightarrow> 'j" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_put_ptrs: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'v option) RShadowRoot_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put shadow_root_ptr shadow_root h = h'" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast (cast shadow_root_ptr)|}"
simps_3: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and T1 :: "('a, 'b) process_term"   and T2 :: "('a, 'b) process_term" shows "CH2 sys (CH T1 T2) = T2"
containsE: fixes A :: "'b set"   and x :: "'b" assumes "Predicate_Compile.contains A x" obtains A' :: "'b set" and x' :: "'b" where "A = A'" and "x = x'" and "x \<in> A"
simps_2: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalQuadratic var a b c d (Less p) = quadratic_sub_fm var a b c d (convertDerivative var p)"
assms_3: fixes i :: "'al" assumes "i \<in> A" shows "x \<le>\<^sub>o\<^sub>w i"
abelian_homology_group: fixes p :: "int"   and X :: "'a topology" shows "comm_group (homology_group p X)"
while_option_finite_increasing_Some: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "mono f"   and "finite UNIV"   and "s \<le> f s" shows "\<exists>P. while_option (\<lambda>A. f A \<noteq> A) f s = Some P"
awalk_verts_take_conv: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and n :: "nat" assumes "cas u p v" shows "awalk_verts u (take n p) = take (Suc n) (awalk_verts u p)"
fbrelation_fconverse: shows "fbrelation ((\<D>\<^sub>\<circ> f)\<inverse>\<^sub>\<bullet>)"
sum_UNIV: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'b)" shows "sum f S = sum f UNIV"
Not_eqvt: fixes p :: "perm"   and A :: "bool" shows "p \<bullet> (\<not> A) = (\<not> p \<bullet> A)"
ResF: fixes P :: "pi"   and \<alpha> :: "Early_Semantics.freeRes"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto>\<^sub>e \<alpha> \<prec> P'"   and "y \<sharp> \<alpha>" shows "<\<nu>y>P \<longmapsto>\<^sub>e \<alpha> \<prec> <\<nu>y>P'"
sigma_finite_subalgebra_is_sigma_finite: fixes M :: "'b measure"   and F :: "'b measure" assumes "sigma_finite_subalgebra M F" shows "sigma_finite_measure M"
closed_plane: fixes i :: "'b"   and c :: "real" shows "closed (plane i c)"
chain_def2: fixes S :: "nat \<Rightarrow> ('d, 'e) labeled_graph" shows "chain S = (\<forall>i j. i \<le> j \<longrightarrow> subgraph (S i) (S j))"
assms_2: shows "0\<^sub>h \<in> circline_set H"
option_rel_Some1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
finite_UNIV_set_def: shows "finite_UNIV = Phantom('a set) (of_phantom finite_UNIV)"
z_notin_0t: shows "z \<notin> path_image (subpath 0 t q)"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list"   and y :: "(nat \<times> nat) list" assumes "arcs x xa xb = y" obtains a :: "nat" and b :: "nat" where "x = aa__" and "xa = ba__" and "xb = []" and "y = [(aa__, ba__)]"   | a :: "nat" and b :: "nat" and xc :: "nat" and xs :: "nat list" where "x = a" and "xa = b" and "xb = xc # xs" and "y = (a, xc) # arcs xc b xs"
samequot_def: fixes a :: "'a"   and r :: "'b rexp" shows "samequot a r = Times (PLUS (sorted_list_of_set {\<guillemotleft>rderivs (replicate m a) r\<guillemotright> |m. True})) (Star (Atom (singleton a)))"
b_least2_le_arg: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" shows "b_least2 f x y \<le> y"
fps_unit_factor_def: fixes f :: "'b fps" shows "unit_factor f = fps_shift (subdegree f) f"
lappend_code_2: fixes xa :: "'b"   and x :: "'b llist"   and ys :: "'b llist" shows "(xa % x) $ ys = xa % x $ ys"
admS_def: shows "admS \<equiv> {R. \<bottom> \<in> R \<and> adm (\<lambda>x. x \<in> R)}"
assms: fixes x :: "'a" assumes "x \<in> sset xs" shows "f\<cdot>x \<noteq> \<bottom>"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "sup (sup a b) c = sup a (sup b c)"
prems: shows "Suc (bl2wc (a # c)) = 2 ^ m"
int_group_gen_by_one: shows "\<langle>{1}\<rangle>\<^bsub>int_group\<^esub> = carrier int_group"
mpt_factorE_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and proj :: "'b \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "mpt M T"   and "mpt.mpt_factor M T proj M2 T2" shows "AE x in M. proj (T x) = T2 (proj x)"
a: shows "steps0 (st, l, r) ap stp = (st'', l'', r'')"
fold_mset_union: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e"   and M :: "'d multiset"   and N :: "'d multiset" assumes "comp_fun_commute f" shows "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
assms_1: shows "K\<^bsub>n\<^esub> G"
ff_def: shows "ff = (\<lambda>a (c, d). (d * [:a:] + q1 * c, q2 * d))"
assms: shows "exec_step_input P C M pc ics = StepC2 C' Cs"
m_lcomm: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "z \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> (y \<oplus>\<^bsub>K [X]\<^esub> z) = y \<oplus>\<^bsub>K [X]\<^esub> (x \<oplus>\<^bsub>K [X]\<^esub> z)"
simps: fixes a :: "'h"   and r :: "('h \<times> 'i) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
openin_contains_ball: fixes T :: "'a set"   and S :: "'a set" shows "openin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x\<in>S. \<exists>e>0. ball x e \<inter> T \<subseteq> S))"
less_eq_downset_def: shows "(\<le>) \<equiv> id (\<lambda>X Y. Rep_downset X \<subseteq> Rep_downset Y)"
r: fixes \<phi> :: "real" shows "P (moebius_rotation \<phi>)"
ide_char: fixes a :: "'b" shows "B.ide a = (B.arr a \<and> B.dom a = a \<and> B.cod a = a)"
brc_cond_abs: fixes \<Sigma> :: "('a, unit) HashMap.hashmap \<times> 'a list \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap" shows "brc_cond \<Sigma> = (brc_\<alpha> \<Sigma> \<in> br'_cond)"
c_def: shows "c = (2::'a) * n / \<epsilon>"
induct: fixes x1 :: "'d set"   and x2 :: "'d set"   and CC :: "'d set set"   and F :: "'d program"   and P :: "'d set \<Rightarrow> 'd set \<Rightarrow> bool" assumes "(x1, x2) \<in> elt CC F"   and "\<And>A B. \<lbrakk>F \<in> A ensures B; A - B \<in> insert {} CC\<rbrakk> \<Longrightarrow> P A B"   and "\<And>A B C. \<lbrakk>(A, B) \<in> elt CC F; P A B; (B, C) \<in> elt CC F; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>S B. \<forall>A\<in>S. (A, B) \<in> elt CC F \<and> P A B \<Longrightarrow> P (\<Union> S) B" shows "P x1 x2"
subst_cls_lists_comp_substs: fixes Cs :: "'a literal multiset list"   and \<tau>s :: "'s list"   and \<sigma>s :: "'s list" shows "Cs \<cdot>\<cdot>cl (\<tau>s \<odot>s \<sigma>s) = Cs \<cdot>\<cdot>cl \<tau>s \<cdot>\<cdot>cl \<sigma>s"
h: shows "[x].t\<^sub>1 = [y].t\<^sub>2"
cod_dom: fixes f :: "'a" assumes "B.arr f" shows "B.cod (B.dom f) = B.dom f"
degeneralize_ext_def: fixes G :: "('a, 'b) igb_graph_rec_scheme"   and ecnv :: "('a, 'b) igb_graph_rec_scheme \<Rightarrow> 'c" assumes "igb_graph G" shows "igb_graph.degeneralize_ext G ecnv \<equiv> if igbg_num_acc G = 0 then \<lparr>g_V = g_V G \<times> {0}, g_E = {((q, 0), q', 0) |q q'. (q, q') \<in> g_E G}, g_V0 = g_V0 G \<times> {0}, bg_F = g_V G \<times> {0}, \<dots> = ecnv G\<rparr> else \<lparr>g_V = g_V G \<times> {0..<igbg_num_acc G}, g_E = {((q, i), q', i') |i i' q q'. i < igbg_num_acc G \<and> (q, q') \<in> g_E G \<and> i' = (if i \<in> igbg_acc G q then (i + 1) mod igbg_num_acc G else i)}, g_V0 = g_V0 G \<times> {0}, bg_F = {(q, 0) |q. 0 \<in> igbg_acc G q}, \<dots> = ecnv G\<rparr>"
graph_homomorphism_def2: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'f) labeled_graph"   and f :: "('e \<times> 'f) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> univalent f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union (map_graph f G\<^sub>1) G\<^sub>2 = G\<^sub>2)"
N1_enabled_at_b: shows "\<turnstile> pc1 = #b \<longrightarrow> Enabled (<N1>_(x, y, sem, pc1, pc2))"
Inf_less_iff: fixes S :: "'a set"   and a :: "'a" shows "(Inf S < a) = (\<exists>x\<in>S. x < a)"
universe_absorbing: fixes x :: "i" shows "x \<oplus> u = u"
assms_8: shows "x \<le> - forest_components h \<sqinter> - - g"
execn_to_execn_strip_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> =n\<Rightarrow> t"
hyps_2: assumes "yvec \<sharp>* C"   and "xvec \<sharp>* yvec"   and "x mem xvec"   and "distinct yvec" shows "x \<sharp> [xvec yvec] \<bullet>\<^sub>v C"
prj_joint_monic: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and h' :: "'a" assumes "C.cospan f g"   and "C.seq \<p>\<^sub>1[f, g] h"   and "C.seq \<p>\<^sub>1[f, g] h'"   and "\<p>\<^sub>0[f, g] \<cdot> h = \<p>\<^sub>0[f, g] \<cdot> h'"   and "\<p>\<^sub>1[f, g] \<cdot> h = \<p>\<^sub>1[f, g] \<cdot> h'" shows "h = h'"
ev: shows "\<forall>\<^sub>F n in sequentially. a < a' n \<and> a' n < b' n \<and> b' n < b"
desargues_config_not_col_11: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col C A A'"
ldrop_lnull: fixes xs :: "'a llist"   and n :: "enat" assumes "lnull xs" shows "ldrop n xs = LNil"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>1)"
finite_lessThan_enat_iff: fixes m :: "enat" shows "finite {..<m} = (m \<noteq> \<infinity>)"
fic_crename: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname" assumes "fic (M[a\<turnstile>c>b]) c"   and "c \<sharp> (a, b)" shows "fic M c"
get_matrix_def: fixes data :: "rat mat \<times> nat list list \<times> rat list list" shows "get_matrix data = fst data"
Var: fixes rs :: "lam list"   and n :: "name" assumes "listsp WN rs" shows "WN (Var n \<degree>\<degree> rs)"
weak_convexI: fixes Pr :: "('b \<times> 'b) set" assumes "\<And>x y \<alpha> \<beta>. \<lbrakk>x \<succeq>[Pr] y; \<alpha> + \<beta> = 1; 0 < \<alpha>; 0 < \<beta>\<rbrakk> \<Longrightarrow> \<alpha> *\<^sub>R x + \<beta> *\<^sub>R y \<succeq>[Pr] y" shows "weak_convex_pref Pr"
eq_diff_eq: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "(a = c - b) = (a + b = c)"
SUP_ereal_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> ereal" assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"   and "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i" shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. Sup (range (f n)))"
R2_S2_False: fixes \<sigma> :: "bool"   and b0 :: "bool"   and b1 :: "bool"   and D2 :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf" assumes "(if \<sigma> then b0 else b1) = False" shows "spmf (R2 (b0, b1) \<sigma> \<bind> D2) True = spmf (funct_OT_12 (b0, b1) \<sigma> \<bind> (\<lambda>(out1, out2). S2 \<sigma> out2 \<bind> D2)) True"
3: fixes a :: "alt"   and b :: "alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "a \<noteq> b"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i. (a <\<^bsub>P i\<^esub> b) = (b <\<^bsub>P' i\<^esub> a)" shows "(a <\<^bsub>F P\<^esub> b) = (b <\<^bsub>F P'\<^esub> a)"
inv_mgp: fixes g :: "graph" assumes "Invariants.inv g" shows "minGraphProps g"
cases: fixes x :: "('a, 'b, 'c, 'e) SLin_state_scheme \<times> ('a, 'f, 'g, 'h) SLin_state_scheme" obtains s1 :: "('a, 'b, 'c, 'e) SLin_state_scheme" and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" where "x = (s1, s2)"
multi_self_add_other_not_self: fixes M :: "'c multiset"   and x :: "'c" shows "(M = add_mset x M) = False"
hyps_2: fixes z' :: "('a, 'b) form list" assumes "set z \<subseteq> set z'" shows "z' \<turnstile> p[App c []/0]"
rnatmod_def: fixes x :: "real"   and y :: "real" shows "rnatmod x y = real (nat \<lfloor>x\<rfloor> mod nat \<lfloor>y\<rfloor>)"
gso_carrier: fixes i :: "nat" assumes "i < length fs" shows "gso i \<in> carrier_vec n"
border_step: fixes xs :: "'b list"   and ys :: "'b list" shows "border xs ys = border (xs @ [ys ! length xs]) (ys @ [ys ! length xs])"
that: fixes a' :: "'edge"   and as' :: "'edge list" assumes "as = a' # as'"   and "n = sourcenode a'"   and "valid_edge a'"   and "targetnode a' -as'\<rightarrow>\<^sub>\<surd>* n'" shows "thesis"
cases: fixes a1 :: "nat \<Rightarrow> Syntax.type"   and a2 :: "nat \<Rightarrow> Syntax.type"   and a3 :: "ctxt"   and a4 :: "Syntax.type"   and a5 :: "Syntax.type" assumes "a1 , a2 \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t a3 : a4 \<Leftarrow> a5" obtains \<Gamma> :: "nat \<Rightarrow> Syntax.type" and \<Delta> :: "nat \<Rightarrow> Syntax.type" and T :: "Syntax.type" where "a1 = \<Gamma>''__" and "a2 = \<Delta>''__" and "a3 = \<diamond>" and "a4 = T" and "a5 = T"   | \<Gamma> :: "nat \<Rightarrow> Syntax.type" and \<Delta> :: "nat \<Rightarrow> Syntax.type" and E :: "ctxt" and T1 :: "Syntax.type" and T2 :: "Syntax.type" and U :: "Syntax.type" and t :: "trm" where "a1 = \<Gamma>" and "a2 = \<Delta>" and "a3 = E \<^sup>\<bullet> t" and "a4 = T2" and "a5 = U" and "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T1 \<rightarrow> T2 \<Leftarrow> U" and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1"
iso_iff_mon_epi: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "(f \<in> Group.iso G H) = (f \<in> mon G H \<and> f \<in> epi G H)"
qe_infinitesimals_helper: fixes a :: "(real \<times> real \<times> real) list"   and k :: "real"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list" assumes "(\<forall>(d, e, f)\<in>set a. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"
induct: fixes P :: "Core.term \<Rightarrow> Core.term \<Rightarrow> (variable \<times> String.literal set \<Rightarrow> typ option) \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "Core.term"   and a2 :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "\<And>a T b U subs. P (Ct a T) (Ct b U) subs"   and "\<And>a T b U subs. P (Fv a T) (Fv b U) subs"   and "\<And>i j subs. P (Bv i) (Bv j) subs"   and "\<And>T t U u subs. (\<And>a. raw_match' T U subs = Some a \<Longrightarrow> P t u a) \<Longrightarrow> P (Abs T t) (Abs U u) subs"   and "\<And>f u f' u' subs. \<lbrakk>P f f' subs; \<And>a. raw_match_term f f' subs = Some a \<Longrightarrow> P u u' a\<rbrakk> \<Longrightarrow> P (f $ u) (f' $ u') subs"   and "\<And>v va vb vc uw_. P (Fv v va) (Ct vb vc) uw_"   and "\<And>v va vb uw_. P (Fv v va) (Bv vb) uw_"   and "\<And>v va vb vc uw_. P (Fv v va) (Abs vb vc) uw_"   and "\<And>v va vb vc uw_. P (Fv v va) (vb $ vc) uw_"   and "\<And>v va vb uw_. P (Bv v) (Ct va vb) uw_"   and "\<And>v va vb uw_. P (Bv v) (Fv va vb) uw_"   and "\<And>v va vb uw_. P (Bv v) (Abs va vb) uw_"   and "\<And>v va vb uw_. P (Bv v) (va $ vb) uw_"   and "\<And>v va vb vc uw_. P (Abs v va) (Ct vb vc) uw_"   and "\<And>v va vb vc uw_. P (Abs v va) (Fv vb vc) uw_"   and "\<And>v va vb uw_. P (Abs v va) (Bv vb) uw_"   and "\<And>v va vb vc uw_. P (Abs v va) (vb $ vc) uw_"   and "\<And>v va vb vc uw_. P (v $ va) (Ct vb vc) uw_"   and "\<And>v va vb vc uw_. P (v $ va) (Fv vb vc) uw_"   and "\<And>v va vb uw_. P (v $ va) (Bv vb) uw_"   and "\<And>v va vb vc uw_. P (v $ va) (Abs vb vc) uw_"   and "\<And>vb vc v va uw_. P (Ct vb vc) (Fv v va) uw_"   and "\<And>vb v va uw_. P (Bv vb) (Fv v va) uw_"   and "\<And>vb vc v va uw_. P (Abs vb vc) (Fv v va) uw_"   and "\<And>vb vc v va uw_. P (vb $ vc) (Fv v va) uw_"   and "\<And>va vb v uw_. P (Ct va vb) (Bv v) uw_"   and "\<And>va vb v uw_. P (Fv va vb) (Bv v) uw_"   and "\<And>va vb v uw_. P (Abs va vb) (Bv v) uw_"   and "\<And>va vb v uw_. P (va $ vb) (Bv v) uw_"   and "\<And>vb vc v va uw_. P (Ct vb vc) (Abs v va) uw_"   and "\<And>vb vc v va uw_. P (Fv vb vc) (Abs v va) uw_"   and "\<And>vb v va uw_. P (Bv vb) (Abs v va) uw_"   and "\<And>vb vc v va uw_. P (vb $ vc) (Abs v va) uw_"   and "\<And>vb vc v va uw_. P (Ct vb vc) (v $ va) uw_"   and "\<And>vb vc v va uw_. P (Fv vb vc) (v $ va) uw_"   and "\<And>vb v va uw_. P (Bv vb) (v $ va) uw_"   and "\<And>vb vc v va uw_. P (Abs vb vc) (v $ va) uw_" shows "P a0 a1 a2"
berlekamp_mat_closed_3: fixes u :: "'a mod_ring poly" shows "dim_col (local.berlekamp_mat u) = degree u"
open_map_def: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "open_map X1 X2 f \<equiv> \<forall>U. openin X1 U \<longrightarrow> openin X2 (f ` U)"
compact_imp_Lindelof_space: fixes X :: "'b topology" assumes "compact_space X" shows "Lindelof_space X"
assms: fixes x :: "'a" assumes "x \<in> I" shows "sets (M x) = sets N"
Hom_FopxC_map_simp: fixes gf :: "'d \<times> 'c" assumes "DopxC.arr gf" shows "Hom_FopxC.map gf = S.mkArr (HomC.set (F (D.cod (fst gf)), C.dom (snd gf))) (HomC.set (F (D.dom (fst gf)), C.cod (snd gf))) (\<phi>C (F (D.dom (fst gf)), C.cod (snd gf)) \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>C h \<cdot>\<^sub>C F (fst gf)) \<circ> \<psi>C (F (D.cod (fst gf)), C.dom (snd gf)))"
Rats_divide: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a / b \<in> \<rat>"
vv: shows "v \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])"
assms: shows "x \<notin> \<Union> (set ` set xss)"
Abs_IIT_inverse: fixes y :: "((((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) set \<times> (((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) set) \<times> (('d, ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool, ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) F1 \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<times> (('d, ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool, ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) F2 \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool)" assumes "y \<in> UNIV" shows "Rep_IIT (Abs_IIT y) = y"
new_keys_not_analzd: fixes evs :: "event list"   and K :: "nat" assumes "evs \<in> kerbV"   and "K \<in> symKeys"   and "Key K \<notin> used evs" shows "K \<notin> keysFor (analz (knows Spy evs))"
le_infI: fixes x :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "x \<subseteq># a"   and "x \<subseteq># b" shows "x \<subseteq># a \<inter># b"
norm_nth_le: fixes x :: "('c, 'd) vec"   and i :: "'d" shows "norm (x $ i) \<le> norm x"
vec_plus_comm: fixes x :: "'a list"   and y :: "'a list" shows "vec_plus x y = vec_plus y x"
remap_rta: fixes f :: "'c \<Rightarrow> 'd" shows "ranked_tree_automaton (ta_remap f TA') A"
empty_replace_proj_dual7: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as' :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "sat_precond_as s (as @ as')" shows "sat_precond_as (exec_plan s as) as'"
lunit_in_vhom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.trg f = b" shows "Span.in_hom (Span.lunit f) (Span.hcomp b f) f"
IH: assumes "i < \<^bold>|r \<^sup>@ k\<^bold>|" shows "(r \<^sup>@ k) ! i = r ! (i mod \<^bold>|r\<^bold>|)"
sset_sconst: fixes x :: "'b" shows "sset (sconst x) = {x}"
DynCastNull: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
hf_equalityI: fixes a :: "hf"   and b :: "hf" assumes "\<And>x. (x \<^bold>\<in> a) = (x \<^bold>\<in> b)" shows "a = b"
even_numeral: fixes n :: "num" shows "even (numeral (num.Bit0 n))"
p_mono: fixes a :: "real"   and v :: "real"   and x :: "real"   and y :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "x \<le> y"   and "y \<le> movement.t_stop a v" shows "movement.p a v s0 x \<le> movement.p a v s0 y"
succ_fail: shows "succ AB w (p1, p2) = {FAIL}"
ide_dom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.dom C f)"
assms: shows "agree_on G f\<^sub>1 f\<^sub>2"
filter_IFS_no_violations_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations G (get_impl M) = TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations (list_graph_to_graph G) (get_spec M) (edgesL G)"
assms_3: shows "cfg \<in> R_G.cfg_on s'"
newton_step_def: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int" assumes "hensel p f a" shows "hensel.newton_step p f x = x \<ominus>\<^bsub>padic_int p\<^esub> padic_integers.divide p (UP_cring.to_fun (padic_int p) f x) (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) x)"
distinct_48: fixes x7 :: "CTypeId" shows "CClassT x7 \<noteq> ByteT"