hcju/isabelleps huggingface dataset:
WT_converter_coinduct: fixes X :: "('a, 'b, 'c, 'd) converter \<Rightarrow> bool"   and conv :: "('a, 'b, 'c, 'd) converter"   and \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>" assumes "X conv"   and "\<And>conv q r conv'. \<lbrakk>X conv; q \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> \<I>' \<turnstile>g run_converter conv q \<surd> \<and> ((r, conv') \<in> results_gpv \<I>' (run_converter conv q) \<longrightarrow> r \<in> responses_\<I> \<I> q \<and> (X conv' \<or> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>))" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"
map_add_SomeD: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "(m ++ n) k = \<lfloor>x\<rfloor>" shows "n k = \<lfloor>x\<rfloor> \<or> n k = None \<and> m k = \<lfloor>x\<rfloor>"
fs1: shows "y' \<sharp> (M, M', P, P, y)"
da_elim_cases_18: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and C :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (Try c1 Catch(C vn) c2)\<guillemotright> A" obtains C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<lparr>lcl := lcl Env(VName vn \<mapsto> Class C)\<rparr>\<turnstile> insert (VName vn) B \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C1 \<inter> nrm C2" and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2"
finsum_empty: fixes f :: "'c \<Rightarrow> 'a list" shows "finsum (K [X]) f {} = \<zero>\<^bsub>K [X]\<^esub>"
LFP_fixed_point: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "LFP f \<in> fps L f"
ide_char: fixes a :: "'c" shows "Cop.ide a = C.ide a"
fin_g: fixes y :: "'b" assumes "y \<in> B" shows "g y \<noteq> \<top>"
if_P': fixes P :: "bool"   and x :: "'d"   and z :: "'d"   and y :: "'d" assumes "P"   and "x = z" shows "(if P then x else y) = z"
a_distinct_lists_code: fixes h :: "(_) heap" shows "local.a_all_ptrs_in_heap h = (\<forall>host\<in>fset (element_ptr_kinds h). h \<turnstile> ok get_shadow_root host \<longrightarrow> (case |h \<turnstile> get_shadow_root host|\<^sub>r of None \<Rightarrow> True | Some shadow_root_ptr \<Rightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h))"
from_list_correct: fixes xs :: "('key \<times> 'val) list"   and k :: "'key" shows "map_of xs k = get (from_list xs) k"
T_opt_def: fixes s :: "'c list"   and rs :: "'c list" shows "Move_to_Front.T_opt s rs = \<Sqinter> {Move_to_Front.T s rs as |as. length as = length rs}"
sel_2: fixes x1 :: "'a atom formula list"   and x2 :: "'a atom formula list" shows "dels (Effect x1 x2) = x2"
share_all_until_volatile_write_append_Ghost\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and L :: "nat set"   and A :: "nat set"   and R :: "nat set"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and W :: "nat set"   and \<D>' :: "bool"   and \<O>' :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> L = {}"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> A = {}"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}"   and "\<forall>j p is \<O> \<R> \<D> \<theta> sb. j < length ts \<longrightarrow> i \<noteq> j \<longrightarrow> ts ! j = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<inter> R = {}" shows "share_all_until_volatile_write (ts[i := (p', is', \<theta>', sb @ [Ghost\<^sub>s\<^sub>b A L R W], \<D>', \<O>')]) \<S> = share_all_until_volatile_write ts \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L"
comp_arr_dom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c" assumes "C.VV.arr f"   and "C.VV.dom f = a" shows "C.VV.comp f a = f"
image_chamber_system: shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> folding_g.\<C>"
boundOutputPar1Dest': fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' (Q \<parallel> R)"   and "xvec \<sharp>* yvec" obtains T :: "('d, 'e, 'f) psi" and p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set yvec" and "P = T \<parallel> (p \<bullet> R)" and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"
bound_len_concat: fixes w :: "'b list"   and k :: "nat" assumes "w \<in> {w. length w = k \<and> set w \<subseteq> L}" shows "cw_len_concat w \<le> k * max_len"
contours_in_ve_def: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "contours_in_ve ve = \<Union> {contours_in_d d |d. d \<in> ran ve}"
of_int_floor_cancel: fixes x :: "'b" shows "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n. x = of_int n)"
int_mult_le: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "0 \<le> k" shows "k * i \<le> k * j"
Quotient_nonempty_list: shows "Quotient (eq_onp (\<lambda>xs. xs \<noteq> [])) Abs_nonempty_list Rep_nonempty_list cr_nonempty_list"
cube_roots: fixes x :: "complex" shows "set (all_croots 3 x) = {y. y ^ 3 = x}"
inj_on_image_Pow: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "inj_on ((`) f) (Pow A)"
induct: fixes P :: "(fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option) \<Rightarrow> (key \<times> nat \<times> nat) signal \<Rightarrow> bool"   and a0 :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and a1 :: "(key \<times> nat \<times> nat) signal" assumes "\<And>runz A B Kab Ts Ta. P runz (Running [A, B] (Kab, Ts, Ta))"   and "\<And>runz A B Kab Ts Ta. P runz (Commit [A, B] (Kab, Ts, Ta))"   and "\<And>runz va. P runz (Running [] va)"   and "\<And>runz vb va. P runz (Running [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Running (vb # vd # vf # vg) va)"   and "\<And>runz va. P runz (Commit [] va)"   and "\<And>runz vb va. P runz (Commit [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Commit (vb # vd # vf # vg) va)" shows "P a0 a1"
ts_bc_def_3: shows "ts' = ts_c @ ts_b"
HK_2: shows "subgroup K (homology_group p (subtopology X S))"
simps_9: fixes x4 :: "ty" shows "ArrayT x4 \<noteq> NullT"
case_1: fixes f1 :: "'x"   and f2 :: "'y \<Rightarrow> 'x" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
euclidean_size_mod_ring_def: fixes x :: "'b mod_ring" shows "euclidean_size x = (if x = 0 then 0 else 1)"
add_minus_cancel: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "plus a (plus (uminus a) b) = b"
Quotient_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a"   and s :: "'a" assumes "Quotient R Abs Rep T" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
mem_vp_apow_mem_Vr: fixes v :: "'a \<Rightarrow> ant"   and N :: "ant"   and x :: "'a" assumes "valuation K v"   and "0 \<le> N"   and "x \<in> vp K v\<^bsup> Vr K v N\<^esup>" shows "x \<in> carrier (Vr K v)"
cSUP_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'c"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (g ` B)"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
cball_divide_subset: fixes d :: "real"   and x :: "'a"   and e :: "real" assumes "1 \<le> d" shows "cball x (e / d) \<subseteq> cball x e"
type_definition_errorable: shows "type_definition Rep_errorable Abs_errorable UNIV"
g1: shows "Seed\<^bsub>p\<^esub> [next_plane\<^bsub>p\<^esub>]\<rightarrow>* g"
rd_assign_def: fixes s :: "'d" shows "rd_assign s = action_of (R (true \<turnstile> \<lambda>(A, A'). ref A' = ref A \<and> tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = s))"
intros_1: fixes x :: "'b"   and xs :: "'b list" shows "ListMem x (x # xs)"
merge_not_top_s: fixes x :: "'a"   and ss :: "(nat \<times> 'a) list"   and c :: "'a list"   and pc :: "nat" assumes "x \<in> A"   and "snd ` set ss \<subseteq> A"   and "merge c pc ss x \<noteq> \<top>" shows "merge c pc ss x = map snd (filter (\<lambda>(p', t'). p' = pc + 1) ss) \<Squnion>\<^bsub>f\<^esub> x"
"case": fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(case return_id x of return_id x \<Rightarrow> f x) = f x"
streamValue6: fixes D :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'a" assumes "\<forall>j\<le>D. str (t + j + (i + k)) = x" shows "\<forall>j\<le>D. str (t + i + k + j) = x"
HH: shows "PairSet AgentSet (PairSet AgentSet G) \<union> PairSet AgentSet G \<subseteq> synth (analz (G \<union> H))"
add_diff_cancel_right': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
uadmit_prog_fml_ntadjoint'_2: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<phi> :: "('sf + 'b, 'sc, 'sz) formula" assumes "\<And>i. dsafe (\<sigma> i)"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>x\<in>{x. Inl (Inr x) \<in> SIGF \<phi>}. FVT (\<sigma> x))"   and "fsafe \<phi>" shows "fml_sem (adjointFO I \<sigma> \<nu>) \<phi> = fml_sem (adjointFO I \<sigma> \<omega>) \<phi>"
rev_finite_subset: fixes B :: "'q set"   and A :: "'q set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
ground_subst_ground_atm: fixes \<sigma> :: "'s"   and A :: "'a" assumes "is_ground_subst \<sigma>" shows "is_ground_atm (A \<cdot>a \<sigma>)"
eval_simps_3: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "cod (eval b c) = c"
bl_and_mask: fixes w :: "'b word"   and n :: "nat" shows "to_bl (w AND mask n) = replicate (LENGTH('b) - n) False @ drop (LENGTH('b) - n) (to_bl w)"
eq_commute: fixes a :: "'c"   and b :: "'c" shows "(a = b) = (b = a)"
WT_converter_parallel_converter2: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and \<I>1' :: "('m, 'n) \<I>"   and \<I>2' :: "('o, 'p) \<I>"   and conv2 :: "('m, 'n, 'o, 'p) converter" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>1', \<I>2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1', \<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<surd>"
distinct_4: fixes x31 :: "nat"   and x32 :: "int"   and x33 :: "Ferrack.num"   and x1 :: "int" shows "CN x31 x32 x33 \<noteq> C x1"
simps_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
E_Rcd: fixes fs :: "(char list \<times> trm) list"   and fs' :: "(char list \<times> trm) list" assumes "fs [\<longmapsto>] fs'" shows "Rcd fs \<longmapsto> Rcd fs'"
RedThrowNull: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" shows "extTA,P,t \<turnstile> \<langle>throw null,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
Mp_smult_2: fixes a :: "int"   and f :: "int poly" shows "smult a (Mp f) =m smult a f"
ag_inv_zero: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "-\<^sub>a\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>A\<^esub>"
Invokestatic_handlers: fixes P :: "'a prog"   and C :: "char list"   and pc :: "nat"   and xt :: "ex_table"   and pc' :: "nat"   and d' :: "nat"   and C\<^sub>0 :: "char list"   and n :: "char list"   and M :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d')\<rfloor>" shows "\<exists>(f, t, D, h, d)\<in>set (relevant_entries P (Invokestatic C\<^sub>0 n M) pc xt). P \<turnstile> C \<preceq>\<^sup>* D \<and> pc \<in> {f..<t} \<and> pc' = h \<and> d' = d"
is_array_list_lengthIA: fixes l :: "'b list"   and li :: "'b array \<times> nat" shows "is_array_list l li \<Longrightarrow>\<^sub>A \<up> (snd li = length l) * true"
span_explicit': fixes b :: "'a set" shows "More_Modules.ideal b = {\<Sum>v | f v \<noteq> (0::'a). f v * v |f. finite {v. f v \<noteq> (0::'a)} \<and> (\<forall>v. f v \<noteq> (0::'a) \<longrightarrow> v \<in> b)}"
inj_on_inv_into: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "B \<subseteq> f ` A" shows "inj_on (inv_into A f) B"
co12: shows "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
inf_bool_def: fixes P :: "bool"   and Q :: "bool" shows "P \<squnion> Q = (P \<and> Q)"
KL_div_pos_gen: fixes S :: "'b set"   and a :: "'b \<Rightarrow> real"   and d :: "'b \<Rightarrow> real" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < a i"   and "\<And>i. i \<in> S \<Longrightarrow> 0 < d i"   and "sum a S = 1"   and "sum d S = 1" shows "0 \<le> KL_div S a d"
convex_rel_interior_iff2: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "S \<noteq> {}" shows "(z \<in> rel_interior S) = (\<forall>x\<in>affine hull S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
useful_tautologies_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
constrains_def: fixes A :: "'c set"   and B :: "'c set" shows "A co B \<equiv> {F. \<forall>act\<in>Acts F. act `` A \<subseteq> B}"
vimage_UNIV: fixes f :: "'c \<Rightarrow> 'd" shows "f -` UNIV = UNIV"
Units_inv_Units: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<in> Units G"
JNF_4: shows "P * iP = 1\<^sub>m n"
eq_fin_concat_start: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* (w @ w\<^sub>1) (w @ w\<^sub>2) = (traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2"
IH: fixes n :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "\<not> sink (token_run x (n + m))"   and "\<not> sink (token_run y (n + m))"   and "older_seniors x n = older_seniors x (n + m)"   and "older_seniors y n \<subseteq> older_seniors x n" shows "older_seniors y n = older_seniors y (n + m)"
simps_1: fixes x2 :: "'f"   and y2 :: "'f" shows "(Some x2 = Some y2) = (x2 = y2)"
lower_rank_of_def: fixes carrier' :: "'a set" shows "\<E>'\<^sub>1.lower_rank_of carrier' \<equiv> Min {card B |B. \<E>'\<^sub>1.basis_in carrier' B}"
counit_simps_3: shows "cod \<epsilon> = trg (h \<star> f)"
assms_3: shows "sources \<nu> \<inter> targets \<mu> \<noteq> {}"
cert_okD4: fixes c :: "'c list"   and n :: "nat"   and T :: "'c"   and B :: "'c"   and A :: "'c set"   and pc :: "nat" assumes "cert_ok c n T B A"   and "pc < n" shows "c ! pc \<noteq> T"
extend_map_cell_complex_to_sphere_cofinite: fixes \<F> :: "'a set set"   and S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "S \<subseteq> \<Union> \<F>"   and "closed S"   and "convex T"   and "bounded T"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X"   and "continuous_on S f"   and "f ` S \<subseteq> rel_frontier T" obtains C :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite C" and "disjnt C S" and "continuous_on (\<Union> \<F> - C) g" and "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
is_sublist_same_len: fixes xs :: "'a list"   and ys :: "'a list" assumes "|xs| = |ys|" shows "is_sublist xs ys = (xs = ys)"
subalgebra_axioms_def: fixes K :: "'a set"   and V :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "subalgebra_axioms K V R \<equiv> \<forall>k v. k \<in> K \<longrightarrow> v \<in> V \<longrightarrow> k \<otimes>\<^bsub>R\<^esub> v \<in> V"
plus_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f + g = (\<lambda>x. f x + g x)"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "L.inverse_arrows f g" obtains "L.ide (L.comp g f)" and "L.ide (L.comp f g)"
interleave_ConsE: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "x # xs \<in> ys \<otimes> zs" obtains xs :: "'b list" where "ysb__ = x # xs" and "xsb__ \<in> xs \<otimes> zs"   | ys :: "'b list" where "zs = x # ys" and "xsb__ \<in> ysb__ \<otimes> ys"
gState_progress_rel_rtrancl_absorb: fixes prog :: "program"   and g :: "'a gState_scheme" assumes "gState_inv prog g" shows "(gState_progress_rel prog)\<^sup>* `` {g} = gState_progress_rel prog `` {g}"
chine_pentagon: shows "Chn ((f \<star> \<lparr>Chn = fghk.\<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((g \<star> h) \<star> k), Cod = Cod (g \<star> h \<star> k)\<rparr>) \<bullet> \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f (g \<star> h) k, Dom = Dom ((f \<star> g \<star> h) \<star> k), Cod = Cod (f \<star> (g \<star> h) \<star> k)\<rparr> \<bullet> (\<lparr>Chn = fghk.\<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> \<star> k)) = Chn (\<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g (h \<star> k), Dom = Dom ((f \<star> g) \<star> h \<star> k), Cod = Cod (f \<star> g \<star> h \<star> k)\<rparr> \<bullet> \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 (f \<star> g) h k, Dom = Dom (((f \<star> g) \<star> h) \<star> k), Cod = Cod ((f \<star> g) \<star> h \<star> k)\<rparr>)"
subts_of_irred_trms_are_irred: fixes y :: "'a trm"   and S :: "'a eclause set"   and x :: "'a trm"   and p :: "indices list" assumes "trm_rep y S \<noteq> y" shows "subterm x p y \<longrightarrow> trm_rep x S \<noteq> x"
inj_map_prod': fixes f :: "'k \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'n"   and X :: "('k \<times> 'm) set" assumes "inj f"   and "inj g" shows "inj_on (map_prod f g) X"
filter_inf_closed: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'c set"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "ord.filter less_eq F"   and "x \<in> F"   and "y \<in> F" shows "inf x y \<in> F"
null_sets_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "\<Omega> \<in> sets M" shows "(A \<in> null_sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> null_sets M)"
asymp_equiv_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" shows "(f \<sim>[F] g) = ((\<lambda>x. if f x = (0::'d) \<and> g x = (0::'d) then 1::'d else f x / g x) \<longlongrightarrow> (1::'d)) F"
positive_p: fixes x :: "'a" shows "0 \<le> p x"
j_def: shows "j \<equiv> if v \<in> Basis then v else - v"
Aexp_lam_simp: fixes x :: "Vars.var"   and e :: "exp"   and n :: "Arity" shows "Aexp_syn' n (Lam [x]. e) = env_delete x (Aexp_syn' (pred\<cdot>n) e)"
ring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "Ring_Hom.ring_hom hom \<equiv> semiring_hom hom"
set_shadow_root_get_shadow_root: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option"   and h' :: "(_) heap" assumes "ShadowRootClass.type_wf h"   and "h \<turnstile> set_shadow_root ptr shadow_root_ptr_opt \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r shadow_root_ptr_opt"
elims_2: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and xb :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = va__" and "xa = []" and "xb = v'a__" and "va__ = v'a__ \<and> v'a__ \<in> nodes G"   | v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and ps :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, w, v2) # ps" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> edges G \<and> (v1, w, v2) \<notin> set ps \<and> (v2, w, v1) \<notin> set ps \<and> valid_unMultigraph.is_trail G v2 ps v'"
A: shows "(\<lambda>v. cexpr_sem (case_nat v \<sigma>) e) = (\<lambda>v. cexpr_sem (case_nat v \<sigma>') e)"
ring_homomorphism_axioms: shows "ring_homomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
not_blt: fixes n :: "nat"   and p :: "fm" assumes "boundslt n p" shows "boundslt n (not p)"
causal_program_history: fixes sb\<^sub>1 :: "'a memref list"   and sb\<^sub>2 :: "'a memref list" assumes "sb = sb\<^sub>1 @ sb\<^sub>2" shows "\<exists>is. instrs sb\<^sub>2 @ Write volatile a sop A L R W # is\<^sub>s\<^sub>b = is @ prog_instrs sb\<^sub>2"
span_is_submodule: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n" shows "LinearCombinations.submodule class_ring (local.span S) V"
set_card: fixes a :: "'t" assumes "S'.ide a" shows "|S'.set a| <o \<AA>"
Gp_carrier: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "carrier (\<natural>H) = H"