hcju/isabelleps huggingface dataset:
swapsthrough: fixes xs :: "'a list"   and q :: "'a"   and entf :: "nat" assumes "distinct xs"   and "q \<in> set xs" shows "index (swaps [index xs q - entf..<index xs q] xs) q = index xs q - entf"
Red_I_Sup_subset_Red_I_Liminf: fixes Ns :: "'f set llist" assumes "chain (\<rhd>) Ns" shows "Red_I (Sup_llist Ns) \<subseteq> Red_I (Liminf_llist Ns)"
listToFuns_def: fixes f :: "'c \<Rightarrow> 'd list" shows "listToFuns f \<equiv> listToFun \<circ> map (\<lambda>x. (x, f x))"
taylor_deg_1_expansion: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and fa :: "'a"   and f'a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c = to_fun (shift 2 (T\<^bsub>a\<^esub> f)) (b \<ominus> a)"   and "fa = to_fun f a"   and "f'a = deriv f a" shows "to_fun f b = fa \<oplus> f'a \<otimes> (b \<ominus> a) \<oplus> c \<otimes> (b \<ominus> a) [^] 2"
Neg_Forall_in_extend: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat"   and P :: "(nat, 'a) form" assumes "extend S C f n \<union> {f n} \<in> C"   and "Neg (Forall P) = f n" shows "Neg (P[App (SOME k. k \<notin> \<Union> (params ` (extend S C f n \<union> {f n}))) []/0]) \<in> extend S C f (Suc n)"
vseq_implies_hpar_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.seq \<nu> \<mu>" shows "trg\<^sub>C \<nu> = trg\<^sub>C \<mu>"
col_mat_plus: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and i :: "nat"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m1"   and "mat nr nc m2"   and "i < nc" shows "col (mat_plusI pl m1 m2) i = vec_plusI pl (col m1 i) (col m2 i)"
equal: fixes equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
has_field_derivative_Beta1: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" assumes "x \<notin> \<int>\<^sub>\<le>\<^sub>0"   and "x + y \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "((\<lambda>x. Beta x y) has_field_derivative Beta x y * (Digamma x - Digamma (x + y))) (at x within A)"
crename_interesting3: fixes M :: "trm"   and a :: "coname"   and c :: "coname"   and x :: "name"   and y :: "name" shows "M[a\<turnstile>c>c][x\<turnstile>n>y] = M[x\<turnstile>n>y][a\<turnstile>c>c]"
sw_simps2: fixes xs :: "'b"   and y :: "'a"   and x :: "'a" shows "x @xs[y \<and> x]_xs = y"
zmset_subset_eq_zcount: fixes A :: "'a zmultiset"   and B :: "'a zmultiset"   and a :: "'a" assumes "A \<subseteq>#\<^sub>z B" shows "zcount A a \<le> zcount B a"
min_term_punit_def: shows "min_term_punit \<equiv> ordered_term.min_term fst"
ideal_generated_unit: fixes u :: "'a" assumes "u dvd (1::'a)" shows "ideal_generated {u} = UNIV"
prim_vc_1: fixes g :: "'a"   and r :: "'a" assumes "prim_precondition g r" shows "prim_spanning_invariant bot r g r"
to_hma_from_hma\<^sub>v: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.to_hma\<^sub>v (Mod_Type_Connect.from_hma\<^sub>v v) = v"
pe: shows "length (snd a) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set s \<and> y \<in> set s \<and> x < y}. ALG_P (snd a) x y s)"
lookup_dflt_map_pair: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c"   and k :: "'a"   and v :: "'b" assumes "distinct (map fst xs)"   and "snd (f (k, 0::'b)) = (0::'c)"   and "\<And>a b. (fst (f a) = fst (f b)) = (fst a = fst b)" shows "lookup_dflt (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_dflt xs k))"
prems_14: shows "InvariantVarsF (getF state') F0 Vbl"
has_lower_bound: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set"   and V :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I"   and "V \<in> I" shows "\<exists>W\<in>I. W \<subseteq> U \<inter> V"
swap_atom_simps_1: fixes a :: "atom"   and b :: "atom" assumes "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> a = b"
null_eqI: fixes n :: "'a \<times> 'a" assumes "\<And>f. S.VV.comp n f = n \<and> S.VV.comp f n = n" shows "n = S.VV.null"
sSum_eq: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and H1 :: "'a set"   and h :: "'a" assumes "R module N"   and "free_generator R M H"   and "H1 \<subseteq> H"   and "h \<in> H - H1" shows "fgs R M H1 \<minusplus> fgs R M {h} = fgs R M (H1 \<union> {h})"
Quadratic_Reciprocity: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "Legendre (int p) (int q) * Legendre (int q) (int p) = (- 1) ^ ((p - 1) div 2 * ((q - 1) div 2))"
relcomppI: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h"   and s :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and c :: "'i" assumes "r a b"   and "s b c" shows "(r OO s) a c"
hyps_1: fixes i :: "nat" assumes "n = j - i"   and "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> p \<and> occurs_on (t ! k) \<noteq> q"   and "channel cid = Some (p, q)" shows "msgs (S t i) cid = msgs (S t j) cid \<and> cs (S t i) cid = cs (S t j) cid"
points_index_pair: fixes y :: "'a"   and x :: "'a" assumes "y \<in> \<V>"   and "x \<in> \<V>"   and "x \<noteq> y" shows "int (size (filter_mset ((\<subseteq>) {x, y}) \<B>)) = \<Lambda>"
plossless_gpv_ContD: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv"   and out :: "'b"   and c :: "('e, 'b, 'd) rpv"   and input :: "'d" assumes "plossless_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv \<I> (c input)"
IdTr2: fixes F :: "('e, 'f) Group_scheme" assumes "Group F" shows "gbij\<^bsub>F,F\<^esub> I\<^bsub>F\<^esub>"
primepow_prime_power: fixes p :: "'b"   and n :: "nat" assumes "prime p" shows "primepow (p ^ n) = (0 < n)"
real_tendsto_divide_at_top: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g at_top F" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
2: shows "(supp x - set as) \<inter> supp p = {}"
group_incl_imp_subgroup: fixes H :: "'c set" assumes "H \<subseteq> carrier M"   and "Group.group (add_monoid M\<lparr>carrier := H\<rparr>)" shows "subgroup H (add_monoid M)"
add: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M"   and "a \<inter> b = {}" shows "\<mu> (a \<union> b) = \<mu> a + \<mu> b"
simps: fixes a1 :: "'s"   and a2 :: "'s stream" shows "enabled a1 a2 = (\<exists>\<omega> s. a1 = s \<and> a2 = \<omega> \<and> enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
Interleaves_swap: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "xs \<cong> {ys, zs, P} = xs \<cong> {zs, ys, \<lambda>w ws. \<not> P w ws}"
sort_quicksort_by_rel: shows "sort = quicksort_by_rel (\<le>) []"
part2: fixes i :: "'n"   and j :: "'n" shows "((Bij_Nat.to_nat i, Bij_Nat.to_nat j) \<in> (graph_of_mat a)\<^sup>+) = ((i, j) \<in> G\<^sup>+)"
subseteq_guards_Guard: fixes c :: "('d, 'e, 'f) com"   and f :: "'f"   and g :: "'d set"   and c' :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Guard f g c'" shows "(c \<subseteq>\<^sub>g c') \<or> (\<exists>c''. c = Guard f g c'' \<and> (c'' \<subseteq>\<^sub>g c'))"
unique_CryptKey: fixes B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "msg"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent"   and T' :: "msg" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key SesKey, T\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B') \<lbrace>Agent A', Agent B', Key SesKey, T'\<rbrace> \<in> parts (knows Spy evs)"   and "Key SesKey \<notin> analz (knows Spy evs)"   and "evs \<in> kerbV" shows "A = A' \<and> B = B' \<and> T = T'"
bdd_belowI: fixes A :: "'c multiset set"   and m :: "'c multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># x" shows "subset_mset.bdd_below A"
g: assumes "x \<in> t \<union> closure s \<inter> closure t" shows "(g \<longlongrightarrow> l x) (at x within t \<union> closure s \<inter> closure t)"
add_ideals: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set"   and J :: "'e set" assumes "ring R"   and "ideal I R"   and "ideal J R" shows "ideal (I <+>\<^bsub>R\<^esub> J) R"
simps_1: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and E :: "'b equation" shows "validate_ground_lit I (Pos E) = validate_ground_eq I E"
assms_2: shows "R \<in> rcosets stabilizer G \<phi> x"
first_el_smallest: fixes x :: "'a" shows "first_el \<le> x"
wf_type_imp_typ_ok_sig: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ" assumes "wf_type \<Sigma> T" shows "typ_ok_sig \<Sigma> T"
axioms_2: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "eq_linorder_on D cmp" shows "eq_linorder_on_axioms D cmp"
bit_a: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "T\<^sub>p_on_rand' BIT s qs \<le> 175 / 10\<^sup>2 * real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) \<and> BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 15 / 10"
bintrunc_sbintrunc: fixes n :: "nat"   and w :: "int" shows "take_bit (Suc n) (signed_take_bit n w) = take_bit (Suc n) w"
pr_naturality_1: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "\<pp>\<^sub>0[c, d] \<cdot> (f \<otimes> g) = g \<cdot> \<pp>\<^sub>0[a, b]"
sets_completionI: fixes A :: "'b set"   and S :: "'b set"   and N :: "'b set"   and N' :: "'b set"   and M :: "'b measure" assumes "A = S \<union> N"   and "N \<subseteq> N'"   and "N' \<in> null_sets M"   and "S \<in> sets M" shows "A \<in> sets (completion M)"
of_int_divide_less_eq: shows "(of_int a / of_int b < of_int c / of_int d) = (of_int a * of_int d < of_int c * of_int b)"
endofunctor_axioms: shows "endofunctor (\<bullet>) src"
simps_4: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and msg :: "'c"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Send i p q s s' msg) c' = (distributed_system.can_occur channel trans send recv (Send i p q s s' msg) c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c' i = msgs c i @ [Msg msg] \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> no_snapshot_change c c')"
left_mult_one_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "1\<^sub>m nr * A = A"
is_zero_null: fixes p :: "'a poly" shows "is_zero p = (p = 0)"
check_eqv_def: fixes r :: "'t1"   and s :: "'t2" shows "check_eqv r s = (wf1 r \<and> wf2 s \<and> (case closure (init1 r, init2 s) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
closed_map_eq: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "closed_map X X' f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "closed_map X X' g"
in_set_selsort_eq: fixes x :: "real \<times> real"   and xa :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "(x \<in> set (ccw.selsort xa xs)) = (x \<in> set xs)"
simps_2: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and i :: "nat" shows "sublist_at xs (y # ys) (Suc i) = sublist_at xs ys i"
Pair_inject: fixes a :: "'l"   and b :: "'m"   and a' :: "'l"   and b' :: "'m" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
lower_le_mid_2: fixes x :: "float interval" shows "real_of_float (lower x) \<le> real_of_float (mid x)"
Strong_dlHPP_Bisimulation_def: fixes d :: "'d"   and PP :: "nat set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" shows "SdlHPPB d PP R \<equiv> sym R \<and> trans R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))) \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 \<alpha> p. SRabbr (V ! i) m1 \<alpha> p m2 \<and> m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1' \<longrightarrow> (\<exists>p' \<alpha>' m2'. SRabbr (V' ! i) m1' \<alpha>' p' m2' \<and> stepResultsinR p p' R \<and> (\<alpha>, \<alpha>') \<in> R \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'))"
l2: shows "\<forall>t p. cotrace init t \<longrightarrow> (\<exists>i. enat i \<le> llength t \<and> ps (coS t i) p \<noteq> None)"
p\<^sub>1_simps: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and h :: "'b"   and k :: "'b" assumes "cospan_of_maps_in_bicategory_of_spans V H \<a> \<i> src trg h k" shows "trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg k) h)) = src k"
seq_extract_Cons: fixes A :: "nat set"   and x :: "'b"   and xs :: "'b list" shows "A \<upharpoonleft>\<^sub>l (x # xs) = (if 0 \<in> A then [x] else []) @ {j. Suc j \<in> A} \<upharpoonleft>\<^sub>l xs"
mon_c_mono: fixes c :: "'a list multiset"   and c' :: "'a list multiset"   and fg :: "('a, 'c, 'd, 'b, 'e) flowgraph_rec_scheme" assumes "c \<subseteq># c'" shows "mon_c fg c \<subseteq> mon_c fg c'"
fold_remove1_mset: fixes ws :: "'a list"   and vs :: "'a list" assumes "mset ws \<subseteq># mset vs" shows "mset (fold remove1 ws vs) + mset ws = mset vs"
weakly_preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (G (src\<^sub>C \<mu>)) (src\<^sub>D (G \<mu>))"
col_2cycle: fixes P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "col P Q R" shows "col P R Q"
cosh_def: fixes x :: "'b" shows "cosh x = (exp x + exp (- x)) /\<^sub>R 2"
minus_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus> y \<in> carrier R"
hyp: fixes cs :: "('a, 'b, 'c) com list" shows "\<Gamma>\<turnstile>cs,css \<Down> Fault f"
dist_fun_def: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" shows "dist x y = (\<Sum>n. (1 / 2) ^ n * min (dist (x (from_nat n)) (y (from_nat n))) 1)"
cases_scheme: fixes r :: "('b, 'c, 'd, 'e) spr_simWorld_scheme" obtains sprFst :: "('b, 'c, 'd) BEState" and sprLst :: "('b, 'c, 'd) BEState" and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set" and more :: "'e" where "r = \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr>"
finish_simps_4: fixes u :: "'v"   and s :: "'v tarjan_state" shows "stack (finish u s) = tl (stack s)"
CTXT_def: fixes inp :: "nat"   and ct :: "(char list \<times> nat \<times> unit list) list"   and outp :: "nat"   and P :: "'a" shows "CTXT inp ct outp P \<equiv> P"
abs_eq: fixes x :: "circline_mat" shows "of_ocircline (abs_ocircline x) = abs_circline x"
null_space_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "null_space A = {x. A *v x = 0}"
exhaust: fixes y :: "ChildAct" obtains "y = SayIKnow"   | "y = SayNothing"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size"   and "ProofStepInduct_NonTrivial_P_k_pre_axioms G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
AQ_D: fixes v :: "('a \<times> bool list) list"   and n :: "nat"   and m :: "nat"   and a :: "'a" assumes "v \<in> lang n (Atom (AQ m a))"   and "m < n"   and "a \<in> set \<Sigma>" shows "\<exists>x. v = [x] \<and> fst x = a \<and> snd x ! m"
Max_insert: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "signed.Max (insert x A) = signed.max x (signed.Max A)"
WT: shows "\<I>1, \<I>2 \<turnstile>\<^sub>C conv \<surd>"
assms_5: shows "snd a \<otimes> snd h [^] e' = g' [^] z'"
step_Normal_elim_cases_15: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (DynCom c # cs, css, Normal s) \<rightarrow> u" obtains "u = (c s # cs, css, Normal s)"
sin_arccos_abs: fixes y :: "real" assumes "\<bar>y\<bar> \<le> 1" shows "sin (arccos y) = sqrt (1 - y\<^sup>2)"
full_normalize_poly_def: fixes p :: "(char list list \<times> int) list" shows "full_normalize_poly p = sort_all_coeffs p \<bind> (\<lambda>p. sort_poly_spec p \<bind> (\<lambda>p. RETURN (merge_coeffs0 p)))"
K3_msg_in_parts_spies: fixes Kas' :: "agent"   and A :: "agent"   and KeyA :: "nat"   and authK :: "msg"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas' A (Crypt KeyA \<lbrace>authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs" shows "authTicket \<in> parts (knows Spy evs)"
contf: shows "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of U}) Y f"
encode1_lt: fixes i :: "nat"   and dims :: "nat list"   and vars :: "nat set" assumes "i < state_sig.d dims" shows "partial_state.encode1 dims vars i < partial_state.d1 dims vars"
Diff_Compl: fixes A :: "'a set"   and B :: "'a set" shows "A\<setminus>- B = A \<inter> B"
simps_1: fixes uu :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'd)"   and uv :: "'d \<Rightarrow> bool" shows "normalized_n_primitive uu uv MatchAny = True"
dube_eq_0: shows "(\<lambda>z. z + int (card X) - 1 gchoose (card X - 1)) = (\<lambda>z. z - int (poly_deg f) + int (card X) - 1 gchoose (card X - 1) + Hilbert_poly aa z + Hilbert_poly bb z)"
c1_2: shows "\<forall>\<^sub>F x in F. R (norm (f1 x)) (c1 * norm (g1 x))"
assms_1: shows "e \<in> UNIV \<rightarrow> I"
IH: shows "ord_spmf (=) (resample' A B) (spmf_of_set B)"
indRelLSTPO_impl_SRel_and_TRel_weakly_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs SRel SWB"
\<psi>D: fixes H :: "'a set"   and G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rrsmult :: "('b, 'a) aezfun \<Rightarrow> (('b, 'a) aezfun \<Rightarrow> 'c) \<Rightarrow> ('b, 'a) aezfun \<Rightarrow> 'c"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set"   and T :: "'c \<Rightarrow> 'd" assumes "FrobeniusReciprocity H G smult V rrsmult smult' W"   and "T \<in> FGModuleHomSet G smult V smult' W" shows "FrobeniusReciprocity.\<psi>_condition H G smult V rrsmult smult' T (FrobeniusReciprocity.\<psi> H G smult V rrsmult smult' W T)"
B: shows "Esigma 87633 = 69615 + 87633"
is_cadjoint_def: fixes F :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'a" shows "is_cadjoint F G = (\<forall>x y. F x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y)"
flowsto_source_unionI: fixes X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and A :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and B :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and Z :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" assumes "flowsto X0 T A B"   and "flowsto Z T A B" shows "flowsto (X0 \<union> Z) T A B"
finite_INT: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Inter> (A ` I) \<in> sets M"
run_altc_sample_writer: fixes altc_sample :: "'g \<Rightarrow> ('h \<Rightarrow> 'd) \<Rightarrow> 'd"   and p :: "'g"   and f :: "'h \<Rightarrow> ('e, 'f, 'd) writerT" shows "run_writer (altc_sample_writer altc_sample p f) = altc_sample p (\<lambda>p. run_writer (f p))"
has_limits_def: fixes uu :: "'a" shows "has_limits uu \<equiv> \<forall>J. category J \<longrightarrow> has_limits_of_shape J"
act1: shows "opnet onp p\<^sub>1 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>1) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>1. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>1 \<longrightarrow> \<sigma>' i = \<sigma> i)))"
assms_9: shows "\<I> \<turnstile>g gpv \<surd>"
wens_single_finite_Un_eq: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat" assumes "single_valued act" shows "wens_single_finite act B k \<union> wp act (wens_single_finite act B k) \<in> range (wens_single_finite act B)"
openE: fixes S :: "'a set"   and x :: "'a" assumes "open S"   and "x \<in> S" obtains e :: "real" where "0 < e" and "ball x e \<subseteq> S"
orecvmsg_simps_2: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and ips :: "nat set"   and m :: "'e" shows "orecvmsg P \<sigma> (groupcast ips m) = True"
induct: fixes P :: "char list \<Rightarrow> bool"   and a0 :: "char list" assumes "P []"   and "\<And>c s. (\<And>y ts ya tsa yaa tsb. \<lbrakk>c \<notin> set ''/>''; parse_name (c # s) = Inr (y, ts); exactly ''='' ts = Inr (ya, tsa); parse_attribute_value tsa = Inr (yaa, tsb)\<rbrakk> \<Longrightarrow> P tsb) \<Longrightarrow> P (c # s)" shows "P a0"
soundness_fresh: fixes A :: "'b set"   and n :: "nat"   and p :: "('c, 'b) fm"   and i :: "'b"   and M :: "('d, 'c) model"   and g :: "'b \<Rightarrow> 'd"   and w :: "'d" assumes "A, n \<turnstile> [([\<^bold>\<not> p], i)]"   and "i \<notin> nominals p" shows "M, g, w \<Turnstile> p"
mult_isor: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" assumes "x \<subseteq> y" shows "x ; z \<subseteq> y ; z"