hcju/isabelleps huggingface dataset:
Rep_ugraph: fixes x :: "'b ugraph" shows "Rep_ugraph x \<in> {(V, E). E \<subseteq> V \<times> V \<and> finite V \<and> sym E \<and> irrefl E}"
sunflower_def: fixes S :: "'b set set" shows "sunflower S = (\<forall>x. (\<exists>A B. A \<in> S \<and> B \<in> S \<and> A \<noteq> B \<and> x \<in> A \<and> x \<in> B) \<longrightarrow> (\<forall>A. A \<in> S \<longrightarrow> x \<in> A))"
t: shows "\<langle>\<Union> (range M);\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"
wf_RESTRICT_VARS: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and ks :: "'k list"   and vs :: "'k \<Rightarrow> nat list" assumes "local.wf idx \<phi>"   and "\<forall>k\<in>set ks. \<forall>v\<in>set (vs k). LESS k v idx" shows "local.wf idx (RESTRICT_VARS ks vs \<phi>)"
check_ext_l_dom_err_impl_def: fixes p :: "uint64" shows "check_ext_l_dom_err_impl p = ''There is already a polynomial with index '' @ show (nat_of_uint64 p)"
at_left_PInf: shows "at_left \<infinity> = filtermap ereal at_top"
init: shows "case (ws, s) of (wsa, sa) \<Rightarrow> \<exists>done. sa = foldl (\<lambda>x s. f s x) s done \<and> (Rel succs)\<^sup>* `` set ws = set done \<union> (Rel succs)\<^sup>* `` set wsa"
cis_add_pi: fixes x :: "real" shows "cis (pi + x) = - cis x"
simps_2: fixes T :: "nat \<Rightarrow> nat"   and x :: "nat" shows "min\<^sub>k T (Suc x) = (let k = min\<^sub>k T x in if T (Suc x) < T k then Suc x else k)"
emptyseq_def: shows "emptyseq \<equiv> \<lambda>s. \<forall>i. s i = s 0"
induct: fixes nr :: "nat"   and nc :: "nat"   and A :: "'b mat"   and zero :: "'b"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>i j. \<lbrakk>\<lbrakk>i < nr; j < nc; A $$ (i, j) = zero\<rbrakk> \<Longrightarrow> P i (Suc j); \<lbrakk>i < nr; j < nc; A $$ (i, j) \<noteq> zero\<rbrakk> \<Longrightarrow> P (Suc i) (Suc j)\<rbrakk> \<Longrightarrow> P i j" shows "P a0 a1"
rel_pmf_return_pmfI: fixes P :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h"   and y :: "'i" assumes "P x y" shows "rel_pmf P (return_pmf x) (return_pmf y)"
length_remdups_leq: fixes xs :: "'b list" shows "length (remdups xs) \<le> length xs"
alist_of_induct: fixes y :: "('c \<times> 'd) list"   and P :: "('c \<times> 'd) list \<Rightarrow> bool" assumes "y \<in> {xs. distinct (map fst xs) \<and> sorted (map snd xs)}"   and "\<And>x. P (alist_of x)" shows "P y"
widen_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and T :: "ty" shows "P \<turnstile> T \<le> T"
cprank_shallow_model: fixes m :: "real mat convnet"   and s :: "bool"   and Y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat"   and w :: "nat \<Rightarrow> real"   and y :: "nat" assumes "m = insert_weights s (shallow_model Y Z M N) w"   and "y < Y" shows "cprank (tensors_from_net m $ y) \<le> Z"
CH_def: fixes X :: "'a set" shows "Bossy'.CH X = (\<Union>h. BCh h X)"
flip: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "bounded_bilinear prod" shows "bounded_bilinear (\<lambda>x y. prod y x)"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'a list" assumes "i < |xs|" shows "take (Suc i) xs = take i xs @ [xs ! i]"
variation_cases_4: fixes x :: "real"   and y :: "real" assumes "x < 0"   and "y < 0" shows "variation x y = 0"
assms_1: shows "continuous_vnm (lotteries_on outcomes) \<R>"
cc_Suc: fixes j :: "nat" assumes "j < card X - 1" shows "int (aa (Suc j) + bb (Suc j)) = 2 + 2 * (- 1) ^ (card X - j) * (int (poly_deg f) - 1 gchoose (card X - Suc j)) + (\<Sum>i = j + 2..card X - 1. (- 1) ^ (i - j) * (int (aa i) gchoose (i - j) + (int (bb i) gchoose (i - j))))"
inj_list_decode: fixes A :: "nat set" shows "inj_on list_decode A"
types_preserved_if: fixes n :: "i32"   and tf :: "tf"   and e1s :: "b_e list"   and e2s :: "b_e list"   and es' :: "b_e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[$C ConstInt32 n, $b_e.If tf e1s e2s]\<rparr> \<leadsto> \<lparr>[$Block tf es']\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 n, $b_e.If tf e1s e2s] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [$Block tf es'] : ts _> ts'"
inj_map_graph: shows "inj (\<lambda>f. {(x, y). Some y = f x})"
setWatch1_def: fixes clause :: "nat"   and literal :: "Literal"   and state :: "State" shows "setWatch1 clause literal state = state\<lparr>getWatch1 := getWatch1 state(clause \<mapsto> literal), getWatchList := (getWatchList state)(literal := clause # getWatchList state literal)\<rparr>"
flush_all_until_volatile_write_update_other: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and a :: "nat"   and m :: "nat \<Rightarrow> nat"   and v :: "nat" assumes "\<forall>j<length ts. let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)" shows "flush_all_until_volatile_write ts (m(a := v)) = (flush_all_until_volatile_write ts m)(a := v)"
inject_1: fixes x2 :: "bool"   and y2 :: "bool" shows "(BoolVal x2 = BoolVal y2) = (x2 = y2)"
coboundedI2: fixes b :: "'b \<times> 'c"   and c :: "'b \<times> 'c"   and a :: "'b \<times> 'c" assumes "less_eq_prod b c" shows "less_eq_prod (inf_prod a b) c"
two_incomplete_designs_axioms: shows "two_incomplete_designs \<V> \<B> \<k> \<V> \<B>'"
rsr: fixes R :: "('a \<times> 'a) set" assumes "R \<subseteq> Id" shows "s2r (r2s R) = R"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_pseudo_ordered_set A less_eq" shows "connex A less_eq"
swapidseq_endswap: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "swapidseq n p"   and "a \<noteq> b" shows "swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
simps: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "('fun, 'var) Term.term" shows "(a1 \<turnstile>\<^sub>c a2) = ((\<exists>t M. a1 = M \<and> a2 = t \<and> t \<in> M) \<or> (\<exists>T f M. a1 = M \<and> a2 = Fun f T \<and> length T = arity f \<and> public f \<and> (\<forall>x. x \<in> set T \<longrightarrow> M \<turnstile>\<^sub>c x)))"
topspace_pullback_topology: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d topology" shows "topspace (pullback_topology A f T) = f -` topspace T \<inter> A"
card_of_UNION_ordLeq_infinite: fixes B :: "'a set"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "infinite B"   and "|I| \<le>o |B|"   and "\<forall>i\<in>I. |A i| \<le>o |B|" shows "|\<Union> (A ` I)| \<le>o |B|"
unitary_exhadamard_on_i: fixes k :: "nat" assumes "k < n" shows "unitary (tensor_P (hadamard_on_i k) (1\<^sub>m K))"
vec_to_iarray_least_non_zero_position_of_vector_from_index: fixes i :: "'b"   and A :: "('a, 'b) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray A)" shows "least_non_zero_position_of_vector_from_index (vec_to_iarray A) (mod_type_class.to_nat i) = mod_type_class.to_nat (LEAST n. A $ n \<noteq> (0::'a) \<and> i \<le> n)"
assms: shows "a * 2 = 1 + 2 * b"
parent_child_rel_parent_in_heap: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel"   and "(parent, child_ptr) \<in> parent_child_rel h" shows "parent |\<in>| object_ptr_kinds h"
Lcm_int_def: fixes K :: "int set" shows "Lcm K = int (Lcm ((nat \<circ> abs) ` K))"
Aexp_restr_subst: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "x \<notin> S"   and "y \<notin> S" shows "Aexp_syn' a e[y::=x] f|` S = Aexp_syn' a e f|` S"
currentD_IN_eq_0: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f" shows "(d_IN f x = 0) = (\<forall>y. f (y, x) = 0)"
WTrt_elim_cases_9: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and v :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e\<bullet>F{D} := v : T" obtains C :: "char list" and T :: "ty" and T\<^sub>2 :: "ty" where "Ta__ = Void" and "P,E,h,sh \<turnstile> e : Class C" and "P \<turnstile> C has F,NonStatic:T in D" and "P,E,h,sh \<turnstile> v : T\<^sub>2__" and "P \<turnstile> T\<^sub>2__ \<le> T"   | T\<^sub>2 :: "ty" where "Ta__ = Void" and "P,E,h,sh \<turnstile> e : NT" and "P,E,h,sh \<turnstile> v : T\<^sub>2"
observable3_equiv_observable2: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set" shows "observable3 ob P = observable2 ob P"
close_app_comm: fixes A :: "'a list"   and B :: "'a list"   and x :: "'a" shows "close (A @ B) x = close A x @ close B x"
simps_6: fixes n :: "nat"   and f :: "recf" shows "arity (Mn n f) = n"
keys_monom_mult: fixes c :: "'c"   and t :: "'d"   and p :: "'d \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "keys (punit.monom_mult c t p) = punit.splus t ` keys p"
lm069: fixes x :: "'a"   and X :: "'a set"   and f :: "('a \<times> 'b) set" assumes "x \<in> X \<inter> Domain f"   and "runiq (f || X)" shows "(f || X) ,, x = f ,, x"
finite_measure_distr: fixes f :: "'a \<Rightarrow> 'c"   and M' :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M M'" shows "finite_measure (distr M M' f)"
topological_basisE: fixes "open" :: "'d set \<Rightarrow> bool"   and B :: "'d set set"   and O' :: "'d set"   and x :: "'d" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "open O'"   and "x \<in> O'" obtains B' :: "'d set" where "B' \<in> B" and "x \<in> B'" and "B' \<subseteq> O'"
36: fixes x :: "'a"   and y :: "'a" shows "(top \<squnion> x) \<sqinter> y = y \<squnion> x \<sqinter> y"
vimage_Compl: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set" shows "f -` (- A) = - f -` A"
size_rbl_add: fixes bl :: "bool list"   and cl :: "bool list" shows "length (rbl_add bl cl) = length bl"
trivial_phi: fixes g :: "'g"   and v :: "'val" assumes "trivial g v" shows "phi g v \<noteq> None"
inverse_arrows_sym: fixes f :: "'a"   and g :: "'a" shows "C.inverse_arrows f g = C.inverse_arrows g f"
bet2_le2__le_R2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and Po :: "'a"   and b :: "'a"   and A :: "'a"   and PO :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet a Po b"   and "Bet A PO B"   and "Tarski_neutral_dimensionless.Le Bet Cong Po a PO A"   and "Tarski_neutral_dimensionless.Le Bet Cong Po b PO B"   and "A \<noteq> PO"   and "B \<noteq> PO" shows "Tarski_neutral_dimensionless.Le Bet Cong a b A B"
c'_def: shows "f' c' = foldr f xs (f' b)"
preserves_left_adjoint: fixes f :: "'b" assumes "D.is_left_adjoint f" shows "C.is_left_adjoint (F.right_map f)"
lfp_eq_fixp: fixes f :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice" assumes "mono (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)" shows "lfp (f::'b::complete_lattice \<Rightarrow> 'b::complete_lattice) = ccpo_class.fixp f"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism Left_a.comp Left_a.comp (H\<^sub>L a') Left_a.map \<phi>" shows "equivalence_functor Left_a.comp Left_a.comp (H\<^sub>L a')"
content_division_lemma1: fixes \<D> :: "'a set set"   and S :: "'a set"   and a :: "'a"   and b :: "'a"   and i :: "'a" assumes "\<D> division_of S"   and "S \<subseteq> cbox a b"   and "i \<in> Basis"   and "\<And>K. K \<in> \<D> \<Longrightarrow> content K \<noteq> 0"   and "(\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})" shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> content (cbox a b)"
onode_arriveTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and s :: "'e"   and R :: "nat set"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'f"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and i' :: "nat"   and s' :: "'e"   and R' :: "nat set"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, NodeS i s R), ii\<not>ni:arrive(m), \<sigma>', NodeS i' s' R') \<in> onode_sos S" obtains "((\<sigma>, s), receive m, \<sigma>', s') \<in> S" and "ii = {i}" and "ni = {}" and "i' = i" and "R' = R"   | "\<sigma>' i = \<sigma> i" and "ii = {}" and "ni = {i}" and "i' = i" and "s' = s" and "R' = R"
Cs: fixes C :: "'a mat" assumes "C \<in> set Cs" shows "inv_all' uppert C \<and> ev_block (dim_col C) C \<and> dim_row C = dim_col C"
is_arith_prog_onD: fixes l :: "nat"   and start :: "int"   and step :: "nat"   and a :: "int"   and b :: "int"   and c :: "nat" assumes "is_arith_prog_on l start step a b"   and "c \<in> {0..<l}" shows "arith_prog start step c \<in> {a..b}"
single_valued_as_brE: fixes R :: "('d \<times> 'e) set" assumes "single_valued R" obtains \<alpha> :: "'d \<Rightarrow> 'e" and invar :: "'d \<Rightarrow> bool" where "R = br \<alpha> invar"
eq_iff: fixes a :: "'f"   and b :: "'f" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
HAInitStates_HAStates: fixes A :: "('d, 'e, 'f) hierauto" shows "HAInitStates A \<subseteq> HAStates A"
cspan_to_conjugate_space: fixes X :: "'d set" shows "cspan (to_conjugate_space ` X) = to_conjugate_space ` cspan X"
WTrt_elim_cases_19: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'c"   and e :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (insync(a) e) T" obtains z :: "htype" where "J_heap_base.WTrt typeof_addr P h E e T" and "typeof_addr h a = \<lfloor>z\<rfloor>"
Smith_normal_form_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Smith_normal_form_upt_k A k = ((\<forall>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < k \<and> mod_type_class.to_nat b + 1 < k \<longrightarrow> A $ a $ b dvd A $ (a + (1::'d)) $ (b + (1::'c))) \<and> isDiagonal_upt_k A k)"
poly_0_coeff_0: fixes p :: "'c poly" shows "poly p (0::'c) = poly.coeff p 0"
that: fixes rhs :: "pterm" assumes "rhs' = pterm_to_sterm rhs"   and "(name, rhs) |\<in>| rs"   and "rest = Rewriting_Sterm.compile (rs |-| {|(name, rhs)|})" shows "thesis"
deriv_is_axiom: fixes s :: "(nat \<times> form) list" assumes "is_axiom (s_of_ns s)" shows "deriv s = {(0, s)}"
isolate_var_0: fixes x :: "nat" shows "isolate_variable_sparse (Var x) x 0 = 0"
infinite_imp_nonempty: fixes S :: "'g set" assumes "infinite S" shows "S \<noteq> {}"
s\<^sub>2's\<^sub>3: shows "(s\<^sub>2', s\<^sub>3') \<in> [\<leadsto>]\<^sup>="
comm_monoid_axioms: shows "Groups.comm_monoid (*) (1::'a)"
min_PInf: fixes x :: "ereal" shows "min \<infinity> x = x"
trivial_reduced_homology_group_contractible_space: fixes X :: "'a topology"   and p :: "int" assumes "contractible_space X" shows "trivial_group (reduced_homology_group p X)"
tLength_g0_conv: fixes t :: "'a Trace" shows "(0 < tLength t) = (\<exists>s t'. t = t' \<leadsto> s \<and> tLength t = Suc (tLength t'))"
order_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
j_le_f: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma> x y" shows "j (\<langle>x\<rangle>, \<langle>y\<rangle>) \<le> f (x, y)"
le_Suc_base: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s" shows "a j \<le> Suc (base j)"
okRead1_def: fixes f :: "nat \<Rightarrow> real"   and x :: "real"   and ppred :: "nat \<Rightarrow> bool" shows "okRead1 f x ppred = (\<forall>l m. ppred l \<and> ppred m \<longrightarrow> \<bar>f l - f m\<bar> \<le> x)"
new_thread_memory: fixes t :: "'t"   and s :: "'x \<times> 'm"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "'x \<times> 'm"   and t' :: "'t"   and x :: "'x"   and m :: "'m" assumes "t \<turnstile> s -ta\<rightarrow> s'"   and "NewThread t' x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "m = snd s'"
finite_Union_vars_term: fixes ts :: "('b, 'a) Term.term list" shows "finite (\<Union> (vars_term ` set ts))"
BVG_assign: fixes x :: "variable"   and \<theta> :: "trm" shows "BVG (x := \<theta>) = (if \<forall>I \<omega>. term_sem I \<theta> \<omega> = \<omega> x then {} else {x})"
simps_28: fixes x5 :: "String.literal" shows "Class x5 \<noteq> NT"
intros_4: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta> \<down> Suc n"
EndPhase0_HInv3_t: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' t"   and "HInv3_inner s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_inner s' p q d"
Abs_eqD: fixes x :: "nat"   and y :: "nat" assumes "word_of_nat x = word_of_nat y"   and "x \<in> unats LENGTH('b)"   and "y \<in> unats LENGTH('b)" shows "x = y"
weakBisimCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto><(X \<union> weakBisim)> S"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> (S, R) \<in> X" shows "P \<approx> Q"
maxcoeff_pos: fixes t :: "MIR.num" shows "0 \<le> maxcoeff t"
add_mset_lt_le_lt: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a \<le> b"   and "A < B" shows "add_mset a A < add_mset b B"
intros_4: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
exp_conj_def: fixes a :: "'c \<Rightarrow> real"   and b :: "'c \<Rightarrow> real" shows "a && b \<equiv> \<lambda>s. a s .& b s"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and xa :: "'b ccsubspace" shows "space_as_set (x *\<^sub>S xa) = closure ((*\<^sub>V) x ` space_as_set xa)"
\<omega>_dec_Cons_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<beta>s :: "V list" shows "sorted_wrt (\<lambda>x y. y \<le> x) (\<alpha> # \<beta> # \<beta>s) = (\<beta> \<le> \<alpha> \<and> sorted_wrt (\<lambda>x y. y \<le> x) (\<beta> # \<beta>s))"
spr_repVal_def: shows "spr_repVal \<equiv> envVal \<circ> snd"
fst_swap: fixes x :: "'d \<times> 'c" shows "fst (prod.swap x) = snd x"
split_spec_coll_def: shows "split_spec_coll = split_spec"
pos_ord_prefix: shows "\<forall>x y p q r. (q, p) \<in> pos_ord x y \<longrightarrow> (q @ r, p) \<in> pos_ord x y"
ubx_axioms: shows "ubx F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>"
not_outside: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "preference carrier relation"   and "x \<succeq>[relation] y" shows "x \<in> carrier"
that: fixes K :: "'a set" assumes "openin (top_of_set S) K"   and "compact K"   and "C \<subseteq> K"   and "K \<subseteq> U" shows "thesis"
get_ancestors_is_l_get_ancestors: shows "l_get_ancestors Shadow_DOM.get_ancestors"
prems: shows "\<forall>a\<in>set (f # fs). is_literal_formula a"
t: shows "\<langle>f S\<rangle>\<^sub>t \<in> timpl_closure \<langle>f T\<rangle>\<^sub>t TI"
assms_4: shows "final_iteration M2 M1 \<Omega> V m i"
lm118: fixes l :: "'b list"   and X :: "'b set" shows "(!) l ` set (concat (map (\<lambda>n. if l ! n \<in> X then [n] else []) [0..<length l])) \<subseteq> X \<inter> set l"
Rwhile_def: shows "Rwhile = while_option (\<lambda>S. \<box> \<notin> S \<and> \<not> res S \<subseteq> S) (\<lambda>S. res S \<union> S)"
dict_def: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt"   and i :: "indi" shows "dict f i \<equiv> \<forall>P\<in>Prof. \<forall>a. a \<noteq> f P \<longrightarrow> a <\<^bsub>P i\<^esub> f P"
last_i_take: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "last (f \<Down> Suc n) = f n"
trees_cyl_map_in_sets: fixes t :: "'d tree"   and f :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "\<And>x. x \<in> set_tree t \<Longrightarrow> f x \<in> sets M" shows "trees_cyl (map_tree f t) \<in> sets (tree_sigma M)"
subspace: fixes A :: "'a set" assumes "closed_csubspace A" shows "csubspace A"
rel_distinct_18: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y6 :: "'b rexp" shows "\<not> rel_rexp R (Star y6) One"
better_AndL2_substn: fixes y :: "name"   and x :: "name"   and M :: "trm"   and c :: "coname"   and P :: "trm" assumes "y \<sharp> [x].M" shows "AndL2 x.M y{y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P z'.AndL2 x.M z')"
assms: shows "x \<notin> \<int>\<^sub>\<le>\<^sub>0"
language_Reverse_nfa: shows "nfa.language (Reverse_nfa MS) = rev ` language"
monom_mult_list_inv: fixes m1 :: "('b \<times> nat) list"   and m2 :: "('b \<times> nat) list" assumes "monom_inv m1"   and "monom_inv m2" shows "monom_inv (monom_mult_list m1 m2)"
dependent_live_vars_cdep_dependent_live_vars: fixes n'' :: "'node"   and as'' :: "'edge list"   and n' :: "'node"   and V' :: "'var"   and as' :: "'edge list"   and as :: "'edge list" assumes "n'' -as''\<rightarrow>\<^sub>c\<^sub>d n'"   and "(V', as', as) \<in> dependent_live_vars n''" shows "(V', as', as @ as'') \<in> dependent_live_vars n'"
shift_snth_less: fixes i :: "nat"   and xs :: "'b list"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "i < length xs" shows "(xs @- \<sigma>) i = xs ! i"
ordering1: shows "\<forall>n. (finite {a, b, c, d} \<longrightarrow> n < card {a, b, c, d}) \<longrightarrow> (if n = 0 then a else if n = 1 then b else if n = 2 then c else d) \<in> {a, b, c, d}"
pred_inject: fixes P :: "'b \<Rightarrow> bool"   and a :: "'c \<Rightarrow> 'b \<times> 'c" shows "pred_state P (State a) = pred_fun (\<lambda>_. True) (pred_prod P top) a"
Disj_EH_3: fixes A :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and H :: "fm set"   and B :: "fm"   and Ba :: "fm" assumes "insert A (insert A1 (insert A2 H)) \<turnstile> B"   and "insert Ba (insert A1 (insert A2 H)) \<turnstile> B" shows "insert A1 (insert A2 (insert (A OR Ba) H)) \<turnstile> B"
resultant_const_2: fixes p :: "'c poly"   and a :: "'c" shows "resultant p [:a:] = a ^ degree p"
simps_2: fixes s :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list"   and x :: "'a" shows "shows_sep s sep [x] = s x"