hcju/isabelleps huggingface dataset:
heap_is_wellformed_children_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "child |\<in>| node_ptr_kinds h"
wand_assn_def: fixes P :: "assn"   and Q :: "assn" shows "P -* Q \<equiv> Abs_assn (wand_raw (Rep_assn P) (Rep_assn Q))"
theNFT_def: fixes cl :: "('g, 'h, 'i) cmd list" shows "theNFT cl \<equiv> {n. n < length cl \<and> \<not> finished (cl ! n)}"
learn_consI2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i"   and "\<And>f n. f \<in> U \<Longrightarrow> \<forall>k\<le>n. \<psi> (the (s (f \<triangleright> n))) k = f k" shows "learn_cons \<psi> U s"
subset_empty: fixes A :: "'q set" shows "(A \<subseteq> {}) = (A = {})"
compact_complete_linorder: fixes X :: "nat \<Rightarrow> 'b" shows "\<exists>l r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> l"
le_refl: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "a \<in> carrier D" shows "a \<preceq>\<^bsub>D\<^esub> a"
sc_0_m: fixes m :: "'a" assumes "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<cdot>\<^sub>s m = \<zero>"
atLeastAtMost_insertL: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "insert m {Suc m..n} = {m..n}"
rep_eq: fixes x :: "'b mod_ring"   and xa :: "'b mod_ring" shows "Rep_mod_ring (x - xa) = (Rep_mod_ring x - Rep_mod_ring xa) mod int CARD('b)"
P: shows "pre @ [Broadcast (ir, Delete is e2)] prefix of k"
sorted_on_many: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and zs :: "'a list" assumes "Linear_order r"   and "(x, y) \<in> r"   and "sorted_on r (y # zs)" shows "sorted_on r (x # y # zs)"
1: shows "y\<^sup>\<star> = (1::'a) \<squnion> y"
s0_def: shows "s0 \<equiv> x0' - x (t 0)"
ord_le_group_order: fixes a :: "'a" assumes "finite (carrier G)"   and "a \<in> carrier G" shows "ord a \<le> order G"
except_Int: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and U :: "'b set"   and V :: "'b set" assumes "keys p \<subseteq> U \<union> V" shows "except p (U \<inter> V) = except p U + except p V"
invar_insert: fixes t :: "('b \<times> nat) tree"   and a :: "'b" assumes "invar t" shows "invar (insert a t)"
nprod_eq: fixes R :: "('c, 'd) Ring_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "Ring R"   and "\<forall>j\<le>n. f j \<in> carrier R"   and "\<forall>j\<le>n. g j \<in> carrier R"   and "\<forall>j\<le>n. f j = g j" shows "e\<Pi>\<^bsub>R,n\<^esub> f = e\<Pi>\<^bsub>R,n\<^esub> g"
nb_2: shows "\<forall>op. op \<in> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)} \<longrightarrow> \<not> lit_semantics \<A> op"
setops\<^sub>s\<^sub>s\<^sub>t_subst': fixes p :: "('a, 'b) Term.term \<times> ('a, 'b) Term.term"   and S :: "('a, 'b) stateful_strand_step list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "p \<in> setops\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>s\<in>setops\<^sub>s\<^sub>s\<^sub>t S. \<exists>X. set X \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t S \<and> p = s \<cdot>\<^sub>p rm_vars (set X) \<delta>"
Sup_union_distrib: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
RS_simps_3: shows "Maps.cod R\<^sub>0 = Maps.MkIde (src r)"
neval_loc_reach: fixes lc' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and n :: "nat"   and lc'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc' \<leadsto>\<^bsup>n\<^esup> lc''"   and "lc' \<in> loc_reach lc" shows "lc'' \<in> loc_reach lc"
in_ivp_sols: fixes s :: "'a"   and U :: "'a \<Rightarrow> real set" assumes "s \<in> S"   and "0 \<in> U s"   and "U s \<subseteq> T" shows "(\<lambda>t. \<phi> t s) \<in> Sols (\<lambda>t. f) U S 0 s"
c_def: fixes v :: "'a vec" shows "c v = (if v = f then 1 else 0)"
finite_jumpFs_subE: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and a' :: "real"   and b' :: "real" assumes "finite_jumpFs f a b"   and "a \<le> a'"   and "b' \<le> b" shows "finite_jumpFs f a' b'"
ceuclidean_all_zero_iff: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and x :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "(\<forall>u\<in>CBasis. cinner x u = 0) = (x = zero)"
distinct_91: fixes x6 :: "arith_type"   and x7 :: "logic_type" shows "arith_type x6 \<noteq> logic_type x7"
simps: fixes x :: "natural \<times> 'b"   and xs :: "(natural \<times> 'b) list"   and i :: "natural" shows "pick (x # xs) i = (if i < fst x then snd x else pick xs (i - fst x))"
d_def: shows "st.d = prod_list (replicate (Suc (Suc k)) 2)"
fmupd_lookup: fixes a :: "'b"   and b :: "'a"   and m :: "('b, 'a) fmap"   and a' :: "'b" shows "m(a $$:= b) $$ a' = (if a = a' then Some b else m $$ a')"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) g"
strict_prefix_simps_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "strict_prefix (x # xs) (y # ys) = (x = y \<and> strict_prefix xs ys)"
psubseteq_dom_subset: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "f \<subseteq>\<^sub>p g" shows "pdom f \<subseteq> pdom g"
le_cong_r: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "x \<le> y"   and "y .=\<^bsub>\<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>\<^esub> z"   and "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "x \<le> z"
Red_F_imp_ex_non_Red_F: fixes Inf :: "'a inference set"   and Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and C :: "'a"   and N :: "'a set" assumes "calculus_with_standard_redundancy Inf Bot entails"   and "C \<in> calculus_with_standard_redundancy.Red_F entails N" shows "\<exists>CC\<subseteq>N - calculus_with_standard_redundancy.Red_F entails N. entails CC {C} \<and> (\<forall>C'\<in>CC. C' < C)"
leof_lift: fixes m :: "'a nres"   and F :: "'a nres" assumes "m \<le> F" shows "m \<le>\<^sub>n F"
is_down_eq: fixes f :: "'b word \<Rightarrow> 'c word" shows "is_down f = (target_size f \<le> source_size f)"
simps: fixes a1 :: "numeric_unop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type" shows "numeric_unop_type a1 a2 a3 = ((\<exists>\<tau>. a1 = UMinusOp \<and> a2 = \<tau> \<and> a3 = Real[1] \<and> \<tau> = Real[1]) \<or> (\<exists>\<tau>. a1 = UMinusOp \<and> a2 = \<tau> \<and> a3 = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1]) \<or> (\<exists>\<tau>. a1 = AbsOp \<and> a2 = \<tau> \<and> a3 = Real[1] \<and> \<tau> = Real[1]) \<or> (\<exists>\<tau>. a1 = AbsOp \<and> a2 = \<tau> \<and> a3 = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1]) \<or> (\<exists>\<tau>. a1 = FloorOp \<and> a2 = \<tau> \<and> a3 = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Real[1]) \<or> (\<exists>\<tau>. a1 = RoundOp \<and> a2 = \<tau> \<and> a3 = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Real[1]) \<or> (\<exists>\<tau>. a1 = numeric_unop.ToIntegerOp \<and> a2 = \<tau> \<and> a3 = Integer[1] \<and> \<tau> = UnlimitedNatural[1]))"
ctermsl_induct: fixes P :: "('b, 'c, 'd, 'a) seqp \<Rightarrow> bool"   and a0 :: "('b, 'c, 'd, 'a) seqp" assumes "\<And>l g p. P p \<Longrightarrow> P ({l}\<langle>g\<rangle> p)"   and "\<And>l u p. P p \<Longrightarrow> P ({l}\<lbrakk>u\<rbrakk> p)"   and "\<And>l s\<^sub>i\<^sub>p s\<^sub>m\<^sub>s\<^sub>g p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P ({l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q)"   and "\<And>l s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>i\<^sub>p\<^sub>s s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}send(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>l s\<^sub>d\<^sub>a\<^sub>t\<^sub>a p. P p \<Longrightarrow> P ({l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p)"   and "\<And>l u\<^sub>m\<^sub>s\<^sub>g p. P p \<Longrightarrow> P ({l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 \<oplus> p2)"   and "\<And>pn. P (call(pn))" shows "P a0"
mf_fm: fixes x :: "'a \<Rightarrow> 'd"   and M :: "'d measure" assumes "x \<in> space (Pi\<^sub>M J (\<lambda>_. M))" shows "mf (fm x) = x"
distinct_9: fixes x11 :: "('a, 'c) trm"   and x12 :: "('a, 'c) trm"   and x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula" shows "Geq x11 x12 \<noteq> (\<langle> x61 \<rangle> x62)"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" shows "lex_ext_unbounded f [] [] = (False, True)"
empty_setinterleaving: fixes t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list" assumes "[] setinterleaves ((t, u), A)" shows "t = []"
Red_F_Bot: fixes Bot :: "'b set"   and Inf :: "'b inference set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I :: "'b set \<Rightarrow> 'b inference set"   and Red_F :: "'b set \<Rightarrow> 'b set"   and B :: "'b"   and N :: "'b set" assumes "calculus Bot Inf entails Red_I Red_F"   and "B \<in> Bot"   and "entails N {B}" shows "entails (N - Red_F N) {B}"
isDERIV_inner_iff: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and i :: "nat"   and vs :: "real list" assumes "length xs = length ys" shows "isDERIV i (inner_floatariths xs ys) vs = ((\<forall>k<length xs. isDERIV i (xs ! k) vs) \<and> (\<forall>k<length ys. isDERIV i (ys ! k) vs))"
match_ex_table_pcsD: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and D :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', D)\<rfloor>" shows "pc \<in> pcs xt"
lunit'_simps_4: fixes f :: "'a" assumes "B.ide f" shows "B.dom (B.lunit' f) = f"
hom_pivot_positions: fixes A :: "'a mat" shows "pivot_positions (mat\<^sub>h A) = pivot_positions A"
OK_lift2_OK: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c err"   and y :: "'e" shows "OK x \<squnion>\<^bsub>lift2 f\<^esub> OK y = x \<squnion>\<^bsub>f\<^esub> y"
abs_power2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a 2) = power.power one times a 2"
certificate_valid_pubSK: fixes i :: "nat"   and SKi :: "nat"   and evs :: "event list" assumes "cert (CA i) SKi onlySig (priSK RCA) \<in> parts (knows Spy evs)"   and "evs \<in> set_mr" shows "SKi = pubSK (CA i)"
Cd_range': fixes x :: "'x"   and d :: "'d"   and X :: "'x set" assumes "x \<in> Cd d X" shows "x \<in> X \<inter> Field (Pd d)"
star_inductr_var_eq: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x = y" shows "y \<cdot> x\<^sup>\<star> \<le> y"
is_red_alt: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and a :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.is_red F a = (\<exists>b. punit.red F a b)"
OrR_inv: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F, G, \<Delta>"
mod_add_self1: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and b :: "'e"   and a :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus b a) b = modulo a b"
simps_4: fixes M :: "('fun, 'var) Term.term set"   and uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d = (\<lambda>\<I>. Unifier \<I> t t' \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d \<I>)"
ideal_npow_closed: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e"   and n :: "nat" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "0 < n" shows "x^\<^bsup>R n\<^esup> \<in> I"
exH_k_dim: fixes k :: "nat" assumes "k < n" shows "exH_k k \<in> carrier_mat N N"
iterate_add_to_map_correct: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and upd :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'a"   and it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and M :: "'b \<Rightarrow> 'c option" assumes "map_update \<alpha> invar upd"   and "invar m"   and "map_iterator it M" shows "\<alpha> (iterate_add_to_map m upd it) = \<alpha> m ++ M \<and> invar (iterate_add_to_map m upd it)"
measure_bind: fixes f :: "'a stream \<Rightarrow> 'c measure"   and s :: "'a"   and N :: "'c measure"   and X :: "'c set" assumes "K1.T.random_variable s (subprob_algebra N) f"   and "X \<in> sets N" shows "Sigma_Algebra.measure (K1.T s \<bind> f) X = K1.T.expectation s (\<lambda>x. Sigma_Algebra.measure (f x) X)"
dist_along_quasiconvex: fixes C :: "real"   and G :: "'a set"   and p :: "'a"   and x :: "'a"   and y :: "'a" assumes "quasiconvex C G"   and "p \<in> proj_set x G"   and "y \<in> G" shows "dist x p + dist p y \<le> dist x y + 4 * deltaG TYPE('a) + 2 * C"
Quotient_rel_rep: 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 a :: "'b"   and b :: "'b" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep b) = (a = b)"
assms_4: shows "(\<lambda>n. Norm (\<LL> p M) (f n)) \<longlonglongrightarrow> l"
\<sigma>1_2: shows "strategy_attracts p \<sigma>1 (attractor p K) K"
closed_UNIV: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open UNIV"
record_simps_1: fixes f :: "'h \<Rightarrow> 'f"   and x :: "('h, 'g) foo" shows "foo.field_1 (foo.update_field_1 f x) = f (foo.field_1 x)"
mono_greaterThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b" assumes "class.order less_eq less" shows "order.mono_set less_eq (ord.greaterThan less B)"
r_dettimeI: fixes x :: "nat"   and i :: "nat"   and t :: "nat" assumes "x < e_length (block i t) \<and> x \<noteq> gap i t"   and "\<And>T. x < e_length (block i T) \<and> x \<noteq> gap i T \<Longrightarrow> t \<le> T" shows "dettime i x \<down>= t"
weak_formula_eqvt: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and x :: "('a, 'd, 'b) formula"   and p :: "perm" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition"   and "indexed_weak_nominal_ts.weak_formula \<tau> x" shows "indexed_weak_nominal_ts.weak_formula \<tau> (p \<bullet> x)"
observable2_def: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set" shows "observable2 ob P \<equiv> \<exists>Q. P = ob -` Q"
P2: shows "Coplanar A B C Y \<and> Bet R Y S"
all_forward: fixes P' :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x::'b. (P'::'b \<Rightarrow> bool) x"   and "\<And>x::'b. (P'::'b \<Rightarrow> bool) x \<Longrightarrow> (P::'b \<Rightarrow> bool) x" shows "\<forall>x::'b. (P::'b \<Rightarrow> bool) x"
HInfinite_HFinite_disj: fixes x :: "'a star" shows "x \<in> HInfinite \<or> x \<in> HFinite"
max_flow_min_cut'''_bounded: assumes "\<And>y. \<not> edge \<Delta>''' (sink \<Delta>''') y"   and "\<And>x. \<not> edge \<Delta>''' x (source \<Delta>''')"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>'''\<^esub> \<Longrightarrow> 0 < capacity \<Delta>''' e" shows "\<exists>f S. flow \<Delta>''' f \<and> MFMC_Network.cut \<Delta>''' S \<and> orthogonal \<Delta>''' f S"
almost_full_onD: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and f :: "nat \<Rightarrow> 'd" assumes "almost_full_on P A"   and "\<And>i. f i \<in> A" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
uminus_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "uminus_ms_aux (MSLCons x xs) = MSLCons (- fst x, snd x) (uminus_ms_aux xs)"
less_than_inc: fixes x :: "nat" shows "x \<le> inc x"
E_valid_1: fixes G :: "('a, 'b) graph" assumes "valid_graph G" shows "fst ` edges G \<subseteq> nodes G"
map_ugraph_finite: fixes G :: "nat set \<times> nat set set"   and f :: "nat \<Rightarrow> nat" assumes "finite_graph G" shows "finite_graph (map_ugraph f G)"
steps_smaller_set: fixes A :: "'a set"   and m :: "nat"   and l :: "nat"   and f :: "'a \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "finite A"   and "m \<le> l" shows "steps A f l u v = steps {x \<in> A. m \<le> f x} f l u v"
trans_refl: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "preference carrier relation" shows "preorder_on carrier relation"
fmap_ext: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "\<And>x. m $$ x = n $$ x" shows "m = n"
cfinitely_spanned: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.cfinite_dim scaleR scaleC plus zero minus uminus" shows "\<exists>S. finite S \<and> cspan S = UNIV"
assms_2: fixes n :: "nat" assumes "n < length a" shows "a ! n \<sqsubseteq>\<^bsub>r\<^esub> b ! n"
that: fixes x :: "'a" assumes "[\<phi> x in v]" shows "thesis"
refine: fixes ga :: "('V, 'W) graph"   and v0 :: "'V"   and g :: "'G" assumes "dijkstraC_fixg g_ops mr_ops qw_ops ga v0 g" shows "(RETURN (cdijkstra g v0), Dijkstra.mdijkstra ga v0) \<in> \<langle>\<langle>v_rel, \<langle>v_rel \<times>\<^sub>r w_rel \<times>\<^sub>r v_rel\<rangle>list_rel \<times>\<^sub>r w_rel\<rangle>mr.rel\<rangle>nres_rel"
assms: fixes x :: "'b" assumes "x \<in> set xs"   and "\<not> P x" shows "f x = (0::'a)"
compress_pos_interfaces_None: fixes ifces :: "iface list"   and p_i :: "char list" assumes "compress_pos_interfaces ifces = None" shows "\<not> (\<forall>i\<in>set ifces. match_iface i p_i)"
Rp_1: shows "integrable M (\<lambda>x. fp x * real_cond_exp M F g x)"
bottom_closed: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order_bottom L" shows "\<bottom>\<^bsub>L\<^esub> \<in> carrier L"
merge_sat1: fixes A :: "assn"   and A' :: "assn"   and Am :: "assn" assumes "A \<or>\<^sub>A A' \<Longrightarrow>\<^sub>t Am" shows "A \<or>\<^sub>A Am \<Longrightarrow>\<^sub>t Am"
dg_prod_3_def: fixes \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" shows "\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC> = (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>3\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | i = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>)"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = a \<sqinter> b" shows "a \<le> b"
init_seg_of_def: shows "init_seg_of = {(r, s). r \<subseteq> s \<and> (\<forall>a b c. (a, b) \<in> s \<and> (b, c) \<in> r \<longrightarrow> (a, b) \<in> r)}"
comm_source_nondet_stimcomm: fixes B :: "'a"   and C :: "'a"   and A :: "'a" assumes "B \<rightarrow>\<^sub>\<S> C" shows "B + A \<rightarrow>\<^sub>\<S> C"
coeff_int_eq_0: fixes i :: "int"   and p :: "'b poly" assumes "i < 0 \<or> int (degree p) < i" shows "coeff_int p i = (0::'b)"
ccompare_finite_4_def: shows "ccompare \<equiv> None"
new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'w" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
unique_factorization_m_factor: fixes f :: "int poly"   and g :: "int poly"   and c :: "int"   and hs :: "int poly multiset"   and n :: "nat" assumes "unique_factorization_m (f * g) (c, hs)"   and "comm_monoid_mult_class.coprime (lead_coeff (f * g)) p"   and "p.square_free_m (f * g)"   and "n \<noteq> 0"   and "m = p ^ n" shows "\<exists>fs gs. unique_factorization_m f (lead_coeff f, fs) \<and> unique_factorization_m g (lead_coeff g, gs) \<and> Mf (c, hs) = Mf (lead_coeff f * lead_coeff g, fs + gs) \<and> image_mset Mp fs = fs \<and> image_mset Mp gs = gs"
lm05: fixes P :: "('c \<times> 'd) set" assumes "inj_on fst P" shows "runiq P"
GroupHom_restrict0_subgroup: fixes H :: "'m set" assumes "VecGroup.Subgroup H" shows "GroupHom H (T \<down> H)"
ttree_join_is_either: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "t \<squnion> t' = t \<oplus>\<oplus> t'"
nunit_factors: fixes a :: "'a"   and as :: "'a list" assumes "a \<notin> Units G"   and "factors G as a" shows "0 < length as"
completely_multiplicative_function_inverse: fixes f :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f" shows "completely_multiplicative_function (\<lambda>n. inverse (f n))"
that_3: assumes "ord a b = GREATER"   and "b < a" shows "thesis"