hcju/isabelleps huggingface dataset:
frontier_of_eq_empty: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "(X frontier_of S = {}) = (closedin X S \<and> openin X S)"
coalesce_less: fixes a :: "real"   and c :: "real"   and b :: "real" assumes "a < c"   and "c < b" shows "g {a..c} \<^bold>* g {c..b} = g {a..b}"
rep_eq: fixes x :: "'a sq_mtx" shows "to_vec x\<^sup>\<dagger> = Finite_Cartesian_Product.transpose (to_vec x)"
bot_def: shows "DRA_Models.bot \<equiv> \<lambda>x. {}"
quat_inverse_eq_cnj: fixes q :: "quat" assumes "norm q = 1" shows "inverse q = Quaternions.cnj q"
Suc: shows "x \<le> (sqrtu x n)\<^sup>2"
cut_ge_not_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<notin> I" shows "x \<notin> I \<down>\<ge> t"
inj_matrix_vector_mult: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "invertible A" shows "inj ((*v) A)"
assms_4: shows "h \<turnstile> get_shadow_root host' \<rightarrow>\<^sub>r Some shadow_root_ptr"
tuple_prj: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "cospan f g"   and "seq \<p>\<^sub>1[f, g] h" shows "\<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle> = h"
fv\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('f, 'e) stateful_strand_step"   and A :: "('f, 'e) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t (a # A) = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> fv\<^sub>s\<^sub>s\<^sub>t A"
Abstract_Formulas_axioms: shows "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"
lm020: fixes A :: "('d \<times> 'c) set set"   and B :: "('d \<times> 'c) set set" shows "converse ` (A \<inter> B) = converse ` A \<inter> converse ` B"
inner_simps_6: fixes x :: "'b"   and r :: "real"   and y :: "'b" shows "x \<bullet> r *\<^sub>R y = r * (x \<bullet> y)"
set_zmset_def: fixes M :: "'b zmultiset" shows "set_zmset M = {x. zcount M x \<noteq> 0}"
f_Exec_State_Idle_append_replicate_NoMsg_gr0_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))"   and "0 < n" shows "output_fun (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c) = NoMsg"
SPLIT_def: fixes v :: "'a"   and w :: "'a" shows "Case_Labeling.SPLIT v w \<equiv> v = w"
rel_envir_def: fixes p :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" shows "rel_envir p f g = all_envir (pure p \<diamondop> f \<diamondop> g)"
add_node_set_nodes: fixes v :: "'b"   and G :: "'b list_graph" shows "set (nodesL (FiniteListGraph.add_node v G)) = set (nodesL G) \<union> {v}"
s_is: shows "S = characterize_root_list_p (pderiv (prod_list_var qs) * prod_list_var qs * [:real_of_int (- crb (prod_list_var qs)), 1:] * [:real_of_int (crb (prod_list_var qs)), 1:])"
interior_union_closed_intervals: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "b < c" shows "interior ({a..b} \<union> {c..d}) = interior {a..b} \<union> interior {c..d}"
subrel_runiq: fixes Q :: "('d \<times> 'e) set"   and P :: "('d \<times> 'e) set" assumes "runiq Q"   and "P \<subseteq> Q" shows "runiq P"
comp_inv_arr': fixes f :: "'a" assumes "C.iso f" shows "C.inv f \<cdot> f = C.dom f"
H_def: shows "H = P2 * D * Q2 @\<^sub>r E * Q2"
that: fixes ns' :: "'node SDG_node list" assumes "n ics-ns'\<rightarrow>\<^sub>d* n'" shows "thesis"
le_imp_minus_plus_hmset: fixes m :: "hmultiset"   and n :: "hmultiset"   and k :: "hmultiset" assumes "m \<le> n" shows "k \<le> k - m + n"
iff_def: fixes p :: "fm"   and q :: "fm" shows "Parametric_Ferrante_Rackoff.iff p q \<equiv> if p = q then T else if p = NOT q \<or> NOT p = q then F else if p = F then not q else if q = F then not p else if p = T then q else if q = T then p else Iff p q"
r: fixes k :: "nat" assumes "k \<le> Suc (Suc n)"   and "\<And>i j. \<lbrakk>Suc k \<le> i; i \<le> Suc (Suc n); j < i\<rbrakk> \<Longrightarrow> f i \<noteq> f j" shows "\<exists>i j. i \<le> k \<and> j < i \<and> f i = f j"
Cons_3: shows "folding_g.gallery (C # (B # Bs) @ [D])"
that: fixes a :: "'a" assumes "a \<in> carrier G"   and "\<forall>x\<in>carrier G. ord x \<le> ord a" shows "thesis"
extended_distance_symmetric: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" shows "extended_distance x y = extended_distance y x"
HSem_restr: fixes ESem :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<rightarrow> 'b"   and h :: "('a \<times> 'c) list"   and \<rho> :: "'a \<Rightarrow> 'b" shows "has_ESem.HSem ESem h\<cdot>(\<rho> f|` (- domA h)) = has_ESem.HSem ESem h\<cdot>\<rho>"
filter_compliant_stateful_ACS_accu_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph"   and Es :: "('b \<times> 'b) list"   and accu :: "('b \<times> 'b) list" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu (list_graph_to_graph G) (get_spec M) accu Es"
new_document_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'w" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
lt_ex: fixes x :: "'a" shows "\<exists>y. y < x"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b list" shows "map_iarray f (IArray x) = IArray (map f x)"
hrelation_restr: fixes r :: "hf"   and x :: "hf" shows "hrelation (hrestrict r x)"
Abs_formula_eqvt: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "p \<bullet> Abs_formula t\<^sub>\<alpha> = Abs_formula (p \<bullet> t\<^sub>\<alpha>)"
\<C>pub_unfold: shows "{f. public f} \<inter> \<C> = {f. arity f = 0 \<and> public f}"
vrange_vempty: shows "\<R>\<^sub>\<circ> []\<^sub>\<circ> = []\<^sub>\<circ>"
dist_proj: fixes x :: "'d \<Rightarrow>\<^sub>F 'c"   and i :: "'d"   and y :: "'d \<Rightarrow>\<^sub>F 'c" shows "dist ((x)\<^sub>F i) ((y)\<^sub>F i) \<le> dist x y"
Norm_zero: fixes N :: "'c quasinorm" shows "Norm N (0::'c) = 0"
en_eq_8: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<diamond>\<^bold>\<not>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<not>\<lbrace>x,F\<rbrace> in v]"
abs_in_iff: fixes fr :: "'a \<Rightarrow> real"   and Fr :: "'a filter"   and gr :: "'a \<Rightarrow> real" shows "((\<lambda>x. \<bar>fr x\<bar>) \<in> \<Theta>[Fr](gr)) = (fr \<in> \<Theta>[Fr](gr))"
IH: fixes zs :: "bool list" assumes "xs \<preceq>\<^sub>b ys"   and "ys \<preceq>\<^sub>b zs" shows "xs \<preceq>\<^sub>b zs"
ideI: fixes a :: "'a" assumes "\<guillemotleft>a : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"   and "ide a" shows "hom'.ide a"
subgroup_kernel: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "subgroup (kernel G H h) G"
simps: fixes v :: "'a" shows "degree_vertex v = card (neighbors v)"
analz_Un_analz: fixes G :: "msg set"   and H :: "msg set" shows "analz (G \<union> analz H) = analz (G \<union> H)"
path_def: fixes S :: "'node set"   and mrk :: "'node set" shows "path S mrk \<equiv> {x. \<exists>s. s \<in> S \<and> (s, x) \<in> next O (Restr next (- mrk))\<^sup>*}"
simps_4: fixes v :: "('b \<times> nat) tree"   and va :: "'b \<times> nat"   and vc :: "('b \<times> nat) tree"   and vd :: "'b \<times> nat" shows "split \<langle>v, va, \<langle>vc, vd, \<langle>\<rangle>\<rangle>\<rangle> = \<langle>v, va, \<langle>vc, vd, \<langle>\<rangle>\<rangle>\<rangle>"
Span_same_set: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> carrier R"   and "set Us = set Vs" shows "Span K Us = Span K Vs"
arr_char: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr f = (f \<noteq> concrete_category.Null \<and> |concrete_category.Dom f| <o natLeq \<and> |concrete_category.Cod f| <o natLeq \<and> concrete_category.Map f \<in> extensional (concrete_category.Dom f) \<inter> (concrete_category.Dom f \<rightarrow> concrete_category.Cod f))"
m_c2: fixes C1 :: "'av st option acom"   and C2 :: "'av st option acom" assumes "top_on\<^sub>c C1 (- vars C1)"   and "top_on\<^sub>c C2 (- vars C2)"   and "C1 < C2" shows "m\<^sub>c C2 < m\<^sub>c C1"
divmod_nat_code: fixes m :: "nat"   and n :: "nat" shows "divmod_nat m n = (let k = integer_of_nat m; l = integer_of_nat n in map_prod nat_of_integer nat_of_integer (if k = 0 then (0, 0) else if l = 0 then (0, k) else Code_Numeral.divmod_abs k l))"
ih_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (applyUnitPropagate state')) (getF (applyUnitPropagate state'))"   and "InvariantWatchListsUniq (getWatchList (applyUnitPropagate state'))"   and "InvariantWatchListsCharacterization (getWatchList (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesEl (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesDiffer (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantConflictClauseCharacterization (getConflictFlag (applyUnitPropagate state')) (getConflictClause (applyUnitPropagate state')) (getF (applyUnitPropagate state')) (getM (applyUnitPropagate state'))" shows "let state' = exhaustiveUnitPropagate (applyUnitPropagate state') in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
eq: shows "snd b \<oplus> (snd a \<oplus> fst c) = snd a \<oplus> (snd c \<oplus> fst b)"
lran_upd_outside_2: fixes h :: "int"   and i :: "int"   and a :: "int \<Rightarrow> 'a"   and x :: "'a"   and l :: "int" assumes "h \<le> i" shows "lran (a(i := x)) l h = lran a l h"
dg_in_ArrE: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
V_5: fixes j :: "nat" shows "closure (V j) \<subseteq> carrier"
card_of_unique2: fixes B :: "'a set"   and r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "card_order_on B r"   and "bij_betw f A B" shows "r =o |A|"
c: fixes n :: "'a" shows "continuous_on (cbox a b) (f n)"
betweenE_if_dist_leq: fixes A :: "'b"   and B :: "'b"   and X :: "'b" assumes "between (A, B) X"   and "dist A X \<le> dist B X" obtains u :: "real" where "1 / 2 \<le> u" and "u \<le> 1" and "X = u *\<^sub>R A + (1 - u) *\<^sub>R B"
grid_disjunct: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "d < length p" shows "grid (child p left d) ds \<inter> grid (child p right d) ds = {}"
assms: fixes v :: "'a" assumes "x = Some v" shows "RETURN v \<le> y"
last_index_less_size_conv: fixes xs :: "'b list"   and x :: "'b" shows "(last_index xs x < length xs) = (x \<in> set xs)"
vempty_Z_def: shows "[]\<^sub>\<circ> = set {x. x \<noteq> x}"
eq_matI: fixes B :: "'d mat"   and A :: "'d mat" assumes "\<And>i j. \<lbrakk>i < dim_row B; j < dim_col B\<rbrakk> \<Longrightarrow> A $$ (i, j) = B $$ (i, j)"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
fib_closed_form: fixes n :: "nat" shows "real (fib n) = (((1 + sqrt 5) / 2) ^ n - ((1 - sqrt 5) / 2) ^ n) / sqrt 5"
n2: shows "nprv (insert \<phi> F) (eql (Var x) t)"
assms: fixes S :: "'a set set" shows "f (\<Union> S) = \<Union> (f ` S)"
v11_vlrestriction_vsubset: fixes f :: "V"   and A :: "V"   and B :: "V" assumes "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> A)"   and "B \<subseteq>\<^sub>\<circ> A" shows "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> B)"
continuous_on_componentwise: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on S f = (\<forall>i\<in>Basis. continuous_on S (\<lambda>x. f x \<bullet> i))"
lmap_eq_lmap_conv_llist_all2: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist"   and g :: "'d \<Rightarrow> 'b"   and ys :: "'d llist" shows "(lmap f xs = lmap g ys) = llist_all2 (\<lambda>x y. f x = g y) xs ys"
l_null: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<zero>\<^bsub>K [X]\<^esub> \<otimes>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"
WTrtRI: 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 T :: "ty"   and e' :: "char list exp"   and T' :: "ty"   and C :: "char list"   and Cs :: "char list list" assumes "P,E,h,sh \<turnstile> e : T"   and "P,E,h,sh \<turnstile> e' : T'"   and "\<forall>C'\<in>set (C # Cs). is_class P C'"   and "\<not> sub_RI e'"   and "\<forall>C'\<in>set (C # Cs). not_init C' e"   and "\<forall>C'\<in>set Cs. \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>"   and "\<exists>sfs. sh C = \<lfloor>(sfs, Processing)\<rfloor> \<or> sh C = \<lfloor>(sfs, Error)\<rfloor> \<and> e = THROW NoClassDefFoundError"   and "distinct (C # Cs)"   and "supercls_lst P (C # Cs)" shows "P,E,h,sh \<turnstile> RI (C,e) ; Cs \<leftarrow> e' : T'"
HUnion_hunion: fixes A :: "hf"   and B :: "hf" shows "\<Squnion>(A \<squnion> B) = \<Squnion>A \<squnion> \<Squnion>B"
runit_naturality: fixes f :: "'d" assumes "arr f" shows "\<r>[cod f] \<cdot> (f \<otimes> \<I>) = f \<cdot> \<r>[local.dom f]"
subcls_confluent: fixes C :: "char list"   and C' :: "char list"   and P :: "'b prog"   and C'' :: "char list" assumes "P \<turnstile> C \<preceq>\<^sup>* C'"   and "P \<turnstile> C \<preceq>\<^sup>* C''" shows "P \<turnstile> C' \<preceq>\<^sup>* C'' \<or> P \<turnstile> C'' \<preceq>\<^sup>* C'"
g_def: shows "g = (\<lambda>s. h (s - t))"
plus_eoracle_def: fixes eoracle1 :: "'v \<Rightarrow> 'p \<Rightarrow> (('r \<times> 't list) \<times> 'v) spmf"   and eoracle2 :: "'v \<Rightarrow> 'q \<Rightarrow> (('s \<times> 'u list) \<times> 'v) spmf"   and state :: "'v" shows "plus_eoracle eoracle1 eoracle2 state \<equiv> map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) \<circ> (eoracle1 \<oplus>\<^sub>O eoracle2) state"
weight_ins_edge: fixes u :: "'a"   and v :: "'a"   and g :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "u \<noteq> v"   and "(u, v) \<notin> edges g" shows "weight w (ins_edge (u, v) g) = w {u, v} + weight w g"
satisfies_bounded_nFNot: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b nFNot \<phi>) = (\<AA> \<Turnstile>\<^sub>b FNot \<phi>)"
Rel_match_Rel: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f" assumes "Rel_match R x y" shows "Transfer.Rel R x y"
mk_mat_def: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "mk_mat nr nc f \<equiv> \<lambda>(i, j). if i < nr \<and> j < nc then f (i, j) else undef_mat nr nc f (i, j)"
remove_comp_cls_val_process0: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and y :: "'c" assumes "portfolio pf1" shows "\<forall>w. cls_val_process Mkt (qty_rem_comp pf1 y) 0 w = cls_val_process Mkt pf1 0 w - prices Mkt y 0 w * pf1 y (Suc 0) w"
invariant3p_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'a set" shows "invariant3p r I = (\<forall>s tl s'. s \<in> I \<longrightarrow> r s tl s' \<longrightarrow> s' \<in> I)"
pi_approx2_def: fixes n :: "nat" shows "pi_approx2 n = 48 * arctan_approx (6 * n) (1 / 18) + 32 * arctan_approx (4 * n) (1 / 57) - 20 * arctan_approx (3 * n) (1 / 239)"
pell_valuation_def: fixes D :: "nat" assumes "pell D" shows "pell.pell_valuation D = (\<lambda>x. case case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y) of (a, b) \<Rightarrow> a + b * sqrt (real D))"
abs_eq_iff: fixes x :: "'b ptrm"   and y :: "'b ptrm" shows "(abs_trm x = abs_trm y) = x \<approx> y"
tendsto_principal_singleton: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "(f \<longlongrightarrow> f x) (principal {x})"
on_meridian_in_hyp2_rep: fixes a :: "proj2" assumes "a \<in> hyp2"   and "proj2_incident a meridian" shows "\<exists>y. \<bar>y\<bar> < 1 \<and> a = proj2_abs (vector [0, y, 1])"
eq_sym: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim> \<psi>" shows "\<psi> \<sim> \<phi>"
P14_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c r\<rightarrow> a \<le> c r\<rightarrow> b"
per_union_commute: fixes R :: "('a \<times> 'a) set"   and i :: "'a"   and j :: "'a" shows "per_union R i j = per_union R j i"
rel_generat_outputD: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Out :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A Out R x y"   and "\<not> is_Pure x \<or> \<not> is_Pure y" shows "Out (generat.output x) (generat.output y)"
minus_mult_minus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) (uminus b) = times a b"
update_correct_2: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'a"   and k :: "'b"   and v :: "'c" assumes "map_update \<alpha> invar update"   and "invar m" shows "invar (update k v m)"
p_prod_tau_alpha: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y = x \<parallel> \<nu> y + \<nu> x \<parallel> y + \<tau> x \<parallel> \<tau> y"
endpoint_in_S_S_hyp2_distinct: fixes p :: "proj2"   and a :: "proj2" assumes "p \<in> S"   and "a \<in> hyp2 \<union> S"   and "p \<noteq> a" shows "endpoint_in_S p a \<noteq> p"
induct: fixes x1 :: "'Var \<Rightarrow> Sec option"   and x2 :: "('Var, 'AExp, 'BExp) Stmt"   and x3 :: "'Var \<Rightarrow> Sec option"   and P :: "('Var \<Rightarrow> Sec option) \<Rightarrow> ('Var, 'AExp, 'BExp) Stmt \<Rightarrow> ('Var \<Rightarrow> Sec option) \<Rightarrow> bool" assumes "\<turnstile> x1 {x2} x3"   and "\<And>\<Gamma>. P \<Gamma> Stop \<Gamma>"   and "\<And>\<Gamma>. P \<Gamma> Skip \<Gamma>"   and "\<And>x \<Gamma> e t. \<lbrakk>x \<notin> dom \<Gamma>; \<Gamma> \<turnstile>\<^sub>a e \<in> t; t \<sqsubseteq> dma x\<rbrakk> \<Longrightarrow> P \<Gamma> (x \<leftarrow> e) \<Gamma>"   and "\<And>x \<Gamma> e t. \<lbrakk>x \<in> dom \<Gamma>; \<Gamma> \<turnstile>\<^sub>a e \<in> t\<rbrakk> \<Longrightarrow> P \<Gamma> (x \<leftarrow> e) (\<Gamma>(x \<mapsto> t))"   and "\<And>\<Gamma> e c\<^sub>1 c\<^sub>2 \<Gamma>'. \<lbrakk>\<Gamma> \<turnstile>\<^sub>b e \<in> High  \<longrightarrow> (\<forall>mds. mds_consistent mds \<Gamma> \<longrightarrow> c\<^sub>1 \<sim>\<^bsub>mds\<^esub> c\<^sub>2) \<and> (\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High); \<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'; P \<Gamma> c\<^sub>1 \<Gamma>'; \<turnstile> \<Gamma> {c\<^sub>2} \<Gamma>'; P \<Gamma> c\<^sub>2 \<Gamma>'\<rbrakk> \<Longrightarrow> P \<Gamma> (Stmt.If e c\<^sub>1 c\<^sub>2) \<Gamma>'"   and "\<And>\<Gamma> e c. \<lbrakk>\<Gamma> \<turnstile>\<^sub>b e \<in> Low ; \<turnstile> \<Gamma> {c} \<Gamma>; P \<Gamma> c \<Gamma>\<rbrakk> \<Longrightarrow> P \<Gamma> (While e c) \<Gamma>"   and "\<And>\<Gamma>' \<Gamma> upd c \<Gamma>''. \<lbrakk>\<Gamma>' = \<Gamma> \<oplus> upd; \<turnstile> \<Gamma>' {c} \<Gamma>''; P \<Gamma>' c \<Gamma>''; c \<noteq> Stop; \<forall>x. to_total \<Gamma> x \<sqsubseteq> to_total \<Gamma>' x\<rbrakk> \<Longrightarrow> P \<Gamma> (c@[upd]) \<Gamma>''"   and "\<And>\<Gamma> c\<^sub>1 \<Gamma>' c\<^sub>2 \<Gamma>''. \<lbrakk>\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'; P \<Gamma> c\<^sub>1 \<Gamma>'; \<turnstile> \<Gamma>' {c\<^sub>2} \<Gamma>''; P \<Gamma>' c\<^sub>2 \<Gamma>''\<rbrakk> \<Longrightarrow> P \<Gamma> (c\<^sub>1 ;; c\<^sub>2) \<Gamma>''"   and "\<And>\<Gamma>\<^sub>1 c \<Gamma>\<^sub>1' \<Gamma>\<^sub>2 \<Gamma>\<^sub>2'. \<lbrakk>\<turnstile> \<Gamma>\<^sub>1 {c} \<Gamma>\<^sub>1'; P \<Gamma>\<^sub>1 c \<Gamma>\<^sub>1'; \<Gamma>\<^sub>2 \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>1; \<Gamma>\<^sub>1' \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>2'\<rbrakk> \<Longrightarrow> P \<Gamma>\<^sub>2 c \<Gamma>\<^sub>2'" shows "P x1 x2 x3"
ownsD: fixes s :: "event list"   and r :: "room"   and g :: "guest" assumes "Trace.owns s r = \<lfloor>g\<rfloor>" shows "\<exists>s\<^sub>1 s\<^sub>2 g c. s = s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1 \<and> no_Check_in s\<^sub>2 r"
rel_frontier_convex_hull_cases: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "rel_frontier (convex hull S) = \<Union> {convex hull (S - {x}) |x. x \<in> S}"
parallel_HSem_ind_different_ESem: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> bool"   and \<rho> :: "'b \<Rightarrow> 'c"   and h :: "('b \<times> 'f) list"   and ESem1 :: "'f \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and \<rho>2 :: "'d \<Rightarrow> 'e"   and h2 :: "('d \<times> 'g) list"   and ESem2 :: "'g \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho> ++\<^bsub>domA h\<^esub> evalHeap h (\<lambda>e. ESem1 e\<cdot>y)) (\<rho>2 ++\<^bsub>domA h2\<^esub> evalHeap h2 (\<lambda>e. ESem2 e\<cdot>z))" shows "P (has_ESem.HSem ESem1 h\<cdot>\<rho>) (has_ESem.HSem ESem2 h2\<cdot>\<rho>2)"
JF2col_Suc: fixes n :: "nat"   and x :: "'d JF2" shows "JF2col (Suc n) x = F2set1 (dtor2 x) \<union> (\<Union> (JF1col n ` F2set2 (dtor2 x)) \<union> \<Union> (JF2col n ` F2set3 (dtor2 x)))"
fin: fixes A :: "'a vec set" assumes "fin_dim"   and "lin_indpt A"   and "A \<subseteq> carrier_vec n" shows "finite A"
lmap_lprefix: fixes xs :: "'c llist"   and ys :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "lprefix xs ys" shows "lprefix (lmap f xs) (lmap f ys)"
map_rexp_ACI_inv: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp"   and y :: "'d rexp" assumes "map_rexp f x ~ y" shows "\<exists>z. x ~~ z \<and> y = map_rexp f z"
intro: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP" shows "ND_Rules_Inst freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"
orthogonal_proj_set: fixes y :: "'b"   and C :: "'b set"   and a :: "'b" assumes "y \<in> C"   and "finite C"   and "pairwise orthogonal C" shows "orthogonal (a - proj_onto a C) y"