hcju/isabelleps huggingface dataset:
of_nat_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (numeral n) = numeral.numeral one plus n"
simps_3: fixes d :: "nat"   and v :: "complex vec"   and va :: "complex vec"   and vb :: "complex vec list" shows "mk_projector_orthog d (v # va # vb) = (let norm2 = v \<bullet>c v in 1 / norm2 \<cdot>\<^sub>m (mat_of_cols d [v] * mat_of_rows d [conjugate v]) + mk_projector_orthog d (va # vb))"
assms_2: fixes L :: "'a language"   and K :: "'a language" assumes "R L K" shows "\<oo> L \<le> \<oo> K \<and> (\<forall>x. R (\<dd> L x) (\<dd> K x))"
finite_reach: fixes \<Sigma> :: "'b set"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" assumes "finite (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0)" shows "finite (reach \<Sigma> \<delta> q\<^sub>0)"
distinct_71: fixes x31 :: "char list"   and x32 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "\<lparr>x31\<rparr>x32 \<noteq> x121;; x122"
retract_inverse: fixes x :: "'c" assumes "x \<in> carrier \<X>\<^bsub>F\<^esub>" shows "\<pi>\<^sub>*\<^bsub>F\<^esub> (\<pi>\<^sup>*\<^bsub>F\<^esub> x) = x"
subgroup_incl: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I (G\<lparr>carrier := P\<rparr>)"   and "subgroup J (G\<lparr>carrier := P\<rparr>)"   and "I \<subseteq> J" shows "subgroup I (G\<lparr>carrier := P, carrier := J\<rparr>)"
poly_ext: fixes p :: "'c poly"   and q :: "'c poly" assumes "\<And>x. poly p x = poly q x" shows "p = q"
MkIde_Cod: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr a" shows "J_C.MkIde (J_C.Cod a) = J_C.cod a"
Parallel_length_post_P1: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and s :: "'b"   and R' :: "'b com"   and t :: "'b" assumes "(Parallel Ts, s) -P1\<rightarrow> (R', t)" shows "\<exists>Rs. R' = Parallel Rs \<and> length Rs = length Ts \<and> (\<forall>i<length Ts. OG_Hoare.post (Rs ! i) = OG_Hoare.post (Ts ! i))"
compact_scaling: fixes s :: "'a set"   and c :: "real" assumes "compact s" shows "compact ((*\<^sub>R) c ` s)"
Pi_pmf_geometric_filter: fixes A :: "'a set"   and p :: "real"   and dflt :: "bool" assumes "finite A"   and "p \<in> {0<..1}" shows "Pi_pmf A 0 (\<lambda>_. geometric_pmf p) = Pi_pmf A dflt (\<lambda>_. bernoulli_pmf p) \<bind> (\<lambda>fb. Pi_pmf {x \<in> A. \<not> fb x} 0 (\<lambda>_. map_pmf Suc (geometric_pmf p)))"
unit_closed: shows "stfx.class_of V (zero_str\<^sub>Y V) \<in> stfx.carrier_direct_lim"
rbl_shiftl: fixes w :: "'a word"   and n :: "nat" shows "rev (to_bl (w << n)) = takefill False (size w) (replicate n False @ rev (to_bl w))"
sigma_sets_subset: fixes a :: "'c set set"   and M :: "'c measure" assumes "a \<subseteq> sets M" shows "sigma_sets (space M) a \<subseteq> sets M"
Dube_indets: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite F"   and "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> Dube (card (\<Union> (indets ` F))) (maxdeg F)"
U0_2: shows "U0 \<inter> W \<subseteq> W0"
finite_measure_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<in> sets MX" shows "Sigma_Algebra.measure MX A \<le> Sigma_Algebra.measure MX B"
assms_2: shows "c2 \<sqsubseteq> c1"
assms_1: shows "i \<in>\<^sub>\<circ> I"
two_comparisons_into_case_order_7: fixes x :: "'b"   and y :: "'b"   and Q :: "'d"   and P :: "'d"   and R :: "'d" shows "(if c1.lt x y then Q else if x = y then P else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
sem_Scon: fixes \<phi>s :: "('U, 'C, 'Out) Noninterference.aprop dfmla set"   and env :: "pvar \<Rightarrow> nat" assumes "finite \<phi>s" shows "sem (Scon \<phi>s) env = scon ((\<lambda>\<phi>. sem \<phi> env) ` \<phi>s)"
leg0_simps_5: shows "cod p\<^sub>0 = p\<^sub>0"
diff_add_assoc2: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "b + c - a = b - a + c"
cases: fixes a1 :: "'g set"   and a2 :: "('h, 'i, 'g, 'j, 'k) system_state"   and a3 :: "('h, 'i, 'g, 'j, 'k) system_state" assumes "system_step a1 a2 a3" obtains sh :: "('h, 'i, 'g, 'j, 'k) system_state" and p :: "'g" and ls' :: "('h, 'i, 'j, 'k) com list \<times> 'i option \<times> 'k" and sh' :: "('h, 'i, 'g, 'j, 'k) system_state" where "a1 = {p__}" and "a2 = sh'__" and "a3 = sh__" and "GST sh__ p__ \<rightarrow>\<^bsub>\<tau>\<^esub> ls'" and "GST sh'__ = (GST sh__)(p__ := ls')" and "HST sh'__ = HST sh__"   | sh :: "('h, 'i, 'g, 'j, 'k) system_state" and p :: "'g" and \<alpha> :: "'j" and \<beta> :: "'h" and ls1' :: "('h, 'i, 'j, 'k) com list \<times> 'i option \<times> 'k" and q :: "'g" and ls2' :: "('h, 'i, 'j, 'k) com list \<times> 'i option \<times> 'k" and sh' :: "('h, 'i, 'g, 'j, 'k) system_state" where "a1 = {p, q}" and "a2 = sh'" and "a3 = sh" and "GST sh p \<rightarrow>\<^bsub>\<guillemotleft>\<alpha>, \<beta>\<guillemotright>\<^esub> ls1'" and "GST sh q \<rightarrow>\<^bsub>\<guillemotright>\<alpha>, \<beta>\<guillemotleft>\<^esub> ls2'" and "p \<noteq> q" and "GST sh' = (GST sh)(p := ls1', q := ls2')" and "HST sh' = HST sh @ [(\<alpha>, \<beta>)]"
closedin_subtopology: fixes U :: "'c topology"   and V :: "'c set"   and S :: "'c set" shows "closedin (subtopology U V) S = (\<exists>T. closedin U T \<and> S = T \<inter> V)"
downset_def: fixes xss :: "'b list set" shows "downset xss = (\<forall>x n. x \<in> xss \<longrightarrow> take n x \<in> xss)"
subgroup_def: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
add_zeror: fixes x :: "'a" shows "x + (0::'a) = x"
nn_integral_cmult: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
isomorphicI: fixes f :: "'a" assumes "local.iso f" shows "isomorphic (local.dom f) (cod f)"
Poly_coeffs: fixes p :: "'a poly" shows "Poly (coeffs p) = p"
Exit_successor_of_Low: fixes a :: "'edge" assumes "valid_edge a"   and "sourcenode a = (_Low_)" shows "targetnode a = (_Exit_)"
deutsch_transform_is_gate: shows "gate 2 U\<^sub>f"
size_heapify: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "size (fst (heapify n xs)) = min (length xs) n"
get_owner_document_owner_document_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document" shows "owner_document |\<in>| document_ptr_kinds h"
unchanged_i: shows "\<forall>x\<le>l. \<forall>ts\<^sub>x \<S>\<^sub>x m\<^sub>x ts\<^sub>x' \<S>\<^sub>x' m\<^sub>x'. c (n + x) = (ts\<^sub>x, m\<^sub>x, \<S>\<^sub>x) \<longrightarrow> c' (n + x) = (ts\<^sub>x', m\<^sub>x', \<S>\<^sub>x') \<longrightarrow> ts\<^sub>x' ! i = u_ts ! i \<and> (\<forall>a\<in>u_owns. \<S>\<^sub>x' a = u_shared a) \<and> (\<forall>a\<in>u_owns. \<S>\<^sub>x a = \<S> a) \<and> (\<forall>a\<in>u_owns. m\<^sub>x' a = u_m a) \<and> (\<forall>a\<in>u_owns. m\<^sub>x a = m a)"
mono: fixes xs :: "'a llist" shows "lfp.mono_body (\<lambda>llength. F llength xs)"
4: shows "v * v\<^sup>T \<le> (v * v\<^sup>T \<sqinter> g)\<^sup>\<star>"
continuous_map_id_subt: fixes X :: "'c topology"   and S :: "'c set" shows "continuous_map (subtopology X S) X id"
infdist_triangle: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" shows "infdist x A \<le> infdist y A + dist x y"
bnd_def: shows "bnd = (\<lambda>((l1, r1, sr1), l2, r2, sr2). (l1 * l2, r1 * r2))"
rel_simps_50: fixes n :: "num" shows "\<not> numeral n < (1::'a)"
apply_sequence: fixes f :: "'b perm"   and g :: "'b perm"   and a :: "'b" shows "f \<langle>$\<rangle> (g \<langle>$\<rangle> a) = (f * g) \<langle>$\<rangle> a"
simps_3: fixes a :: "Ferrack.num" shows "bound0 (Lt a) = numbound0 a"
assms_1: fixes r :: "'state"   and a :: "'action"   and b :: "'action" assumes "r \<in> nodes"   and "a \<in> psen p s - {a. en a s}"   and "b \<in> {a. en a r} - pac p"   and "en a (ex b r)" shows "en a r"
meet_assoc_lemma: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot>\<^sub>C f \<noteq> C.null" shows "C.seq g f"
op_ASSUME_bind_def: fixes I :: "bool"   and m :: "'a nres" shows "op_ASSUME_bind I m \<equiv> ASSUME I \<bind> (\<lambda>_. m)"
qelim_ci: fixes qe :: "fm \<Rightarrow> fm"   and bbs :: "bool list"   and p :: "fm"   and bs :: "int list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)" shows "qfree (qelim p qe) \<and> Ifm bbs bs (qelim p qe) = Ifm bbs bs p"
fcard_seteq: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| B"   and "fcard B \<le> fcard A" shows "A = B"
add: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "linearform V f"   and "x \<in> V"   and "y \<in> V" shows "f (x + y) = f x + f y"
invertible_functor_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b" shows "invertible_functor_axioms A B G \<equiv> \<exists>F. inverse_functors A B F G"
pairwise_rel_prime: shows "\<forall>q. List.member qs q \<longrightarrow> coprime p q"
less_eq_Measurement_System_ext_def: fixes x :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>"   and y :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>" shows "(x \<le> y) = (mmore x \<le> mmore y)"
acquired_reads_pending_write_mono: fixes xs :: "'b memref list"   and A :: "nat set" shows "acquired_reads False xs A \<subseteq> acquired_reads True xs A"
Zp_nat_inc_zero: fixes x :: "nat \<Rightarrow> int" shows "[0] \<cdot> x = \<zero>"
higher_deriv_transform_within_open: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and z :: "complex"   and i :: "nat" assumes "f holomorphic_on S"   and "g holomorphic_on S"   and "open S"   and "z \<in> S"   and "\<And>w. w \<in> S \<Longrightarrow> f w = g w" shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
poly_mult_comm: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" shows "poly_mult p q =p poly_mult q p"
SC_sym: fixes X :: "'p set"   and Y :: "'p set" assumes "SC X Y" shows "SC Y X"
mono_Field: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s" shows "Field r \<subseteq> Field s"
lc: shows "NC.lincomb a U = 0\<^sub>v nc"
i_Streams_Connected_nth_conv: fixes outS :: "nat \<Rightarrow> 'd message_af"   and inS :: "nat \<Rightarrow> 'd message_af" shows "i_Streams_Connected outS inS = (\<forall>i. inS i = (case i of 0 \<Rightarrow> NoMsg | Suc k \<Rightarrow> outS k))"
imag_unit_circle_cmat_def: shows "imag_unit_circle_cmat = (1, 0, 0, 1)"
distinct_sorted_wrt_irrefl: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "irreflp rel"   and "transp rel"   and "sorted_wrt rel xs" shows "distinct xs"
set_intros_2: fixes y :: "'a"   and x2 :: "'a stream"   and x1 :: "'a" assumes "y \<in> i.R x2" shows "y \<in> i.R (x1 ## x2)"
eq_divide_eq_numeral_2: fixes w :: "num"   and b :: "'a"   and c :: "'a" shows "(- numeral w = b / c) = (if c \<noteq> (0::'a) then - numeral w * c = b else - numeral w = (0::'a))"
assms_9: shows "r\<^sup>\<circ> * q \<le> q * r\<^sup>\<circ>"
delay_bisimulation_measure_flip: shows "delay_bisimulation_measure trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1 \<mu>2 \<mu>1"
rel_of_def: fixes m :: "'a \<Rightarrow> 'b option"   and P :: "'a \<times> 'b \<Rightarrow> bool" shows "rel_of m P \<equiv> {(k, v). m k = Some v \<and> P (k, v)}"
add_carrier_mat: fixes B :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'c mat" assumes "B \<in> carrier_mat nr nc" shows "A + B \<in> carrier_mat nr nc"
induct: fixes xa :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "xa \<in> orbit f x"   and "P (f x)"   and "\<And>y. \<lbrakk>y \<in> orbit f x; P y\<rbrakk> \<Longrightarrow> P (f y)" shows "P xa"
Cons_4: shows "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"
ineq_dfa_def: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "ineq_dfa n ks l = (let (is, js) = dioph_ineq_dfs n ks l in (map (\<lambda>j. make_bdd (\<lambda>xs. the (is ! int_encode ((j - eval_dioph ks xs) div 2))) n []) js, map ((\<le>) 0) js))"
success_cond_effect: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and h :: "heap" assumes "success (shiftr_p a key i) h"   and "cond (shiftr_p a key i) h" shows "effect (shiftr_p a key i) h h True"
that: fixes e :: "real" assumes "0 < e"   and "{x. infdist x A \<le> e} \<subseteq> B" shows "thesis"
step': fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and l\<^sub>i :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "P s (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)"
frameChainEq': fixes xvec :: "name list"   and \<Psi> :: "'d"   and yvec :: "name list"   and \<Psi>' :: "'d" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" and "distinctPerm p" and "yvec = p \<bullet> xvec" and "\<Psi>' = p \<bullet> \<Psi>"
span_clauses_2: fixes S :: "'a set" shows "(0::'a) \<in> More_Modules.ideal S"
set_append: fixes xs :: "'b list"   and ys :: "'b list" shows "set (xs @ ys) = set xs \<union> set ys"
prems_1: shows "A2 = 0\<^sub>m n (Suc m)"
Collect_mono_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(Collect P \<subseteq> Collect Q) = (\<forall>x. P x \<longrightarrow> Q x)"
WTrts_hext_mono: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,E,h \<turnstile> es [:] Ts"   and "h \<unlhd> h'" shows "P,E,h' \<turnstile> es [:] Ts"
path_first_loc: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and loc :: "'a"   and loc' :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list" assumes "graph weights"   and "graph.path weights loc loc' xs"   and "xs \<noteq> []" shows "(case xs ! 0 of (s, l, t) \<Rightarrow> s) = loc"
pairs_substI': fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "subst_domain \<theta> \<inter> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = {}" shows "F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> = F"
lift_Suc_antimono_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f (Suc n)) (f n)"   and "n \<le> n'" shows "less_eq (f n') (f n)"
phiDefNodesE: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> phiDefNodes g v" obtains ns :: "'node list" and m :: "'node" where "g \<turnstile> n-ns\<rightarrow>m" and "\<forall>n\<in>set ns. v \<notin> defs g n" and "v \<in> uses g m"
map_filter_map_of: fixes P :: "'a \<Rightarrow> bool"   and m :: "('a \<times> 'b) list" shows "map_filter P (map_of m) = map_of (filter (\<lambda>(k, uu_). P k) m)"
endofunctor_L: shows "endofunctor (\<cdot>) L"
N: shows "(\<Sum>i<m. llength (lnth (LCons xs' xss'') i)) = enat N"
simps: fixes k :: "int" shows "bits_of k = odd k # (if k = 0 \<or> k = - 1 then [] else bits_of (k div 2))"
IH: fixes X :: "((real \<times> real pdevs) \<times> real) list" assumes "approx_floatariths_aformerr p ras VS = Some X" shows "interpret_floatariths ras vs \<in> aforms_err e X"
check_xcpt_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and n :: "nat"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and a :: "'addr" shows "check_xcpt P h n pc xt a = (\<exists>C. typeof_addr h a = \<lfloor>Class_type C\<rfloor> \<and> (case match_ex_table P C pc xt of None \<Rightarrow> True | \<lfloor>(pc', d')\<rfloor> \<Rightarrow> d' \<le> n))"
signed_drop_bit_beyond: fixes n :: "nat"   and w :: "'a word" assumes "LENGTH('a) \<le> n" shows "signed_drop_bit n w = (if bit w (LENGTH('a) - Suc 0) then - 1 else 0)"
fold_graph_fold: fixes A :: "'d set"   and z :: "'e" assumes "finite A" shows "fold_graph g z A (Finite_Set.fold g z A)"
field_sum_of_halves: fixes x :: "'a" shows "x / (2::'a) + x / (2::'a) = x"
plus_comm: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
item_dot_of_inc_item: fixes x :: "item"   and k :: "nat" shows "item_dot (inc_item x k) = item_dot x + 1"
hintikka_Extend_S: fixes C :: "(nat, 'b) form set set"   and S :: "(nat, 'b) form set" assumes "consistency C"   and "S \<in> C"   and "infinite (- \<Union> (params ` S))" shows "hintikka (Extend S (mk_finite_char (mk_alt_consistency (close C))) from_nat)"
subst_drop_fset: fixes S :: "name fset"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "fdisjnt S (frees t)" shows "subst t (fmdrop_fset S env) = subst t env"
Seq_reds: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>c\<^sub>1;; c\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>" obtains s'' :: "char list \<Rightarrow> val option" where "\<langle>c\<^sub>1,s\<rangle> \<rightarrow>* \<langle>Skip,s''\<rangle>" and "\<langle>c\<^sub>2,s''\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"
br_while_algo: fixes \<delta> :: "('d, 'e) ta_rule set" assumes "finite \<delta>" shows "while_algo (br_algo \<delta>)"
subset_image_iff: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "order.greater_eq (f ` A) B = (\<exists>AA. order.greater_eq A AA \<and> B = f ` AA)"
nextss_def: fixes g :: "(node \<times> node) list"   and xs :: "node list" shows "nextss g xs = set g `` set xs"
get_nth_i: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list" assumes "list_repr i ba xs"   and "i < length xs" shows "get_nth_i ba = xs ! i"
runit_simps_5: fixes f :: "'a" assumes "ide f" shows "cod \<r>[f] = f"
csmset: shows "fmset G cs = fmset G bs - fmset G as"
cases: fixes a :: "'c list"   and A :: "'c set"   and n :: "nat" assumes "a \<in> nlists A n" obtains xs :: "'c list" where "a = xs" and "set xs \<subseteq> A" and "length xs = n"
dens': fixes M :: "(real \<times> real) measure"   and \<delta> :: "real \<times> real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel \<delta>" shows "has_density (distr M borel (\<lambda>(x, y). f x y)) lborel (g \<delta>)"
that: assumes "h' = h"   and "r = get h a" shows "thesis"
map_matrix_def: fixes f :: "'e \<Rightarrow> 'b"   and x :: "(('e, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.map_matrix f x = (\<chi>i j. f (x $h i $h j))"
infE: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('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, 'f) xstate" assumes "AlternativeSmallStep.inf \<Gamma> (c # cs) css s" obtains "AlternativeSmallStep.inf \<Gamma> [c] [] s"   | t :: "('e, 'f) xstate" where "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" and "AlternativeSmallStep.inf \<Gamma> cs css t"
assms_4: shows "fst ` X1' \<subseteq> fst ` X1"
comp_MkArr: fixes A :: "'a E.term"   and B :: "'a E.term"   and F :: "'a"   and C :: "'a E.term"   and G :: "'a" assumes "arr (MkArr A B F)"   and "arr (MkArr B C G)" shows "MkArr B C G \<cdot> MkArr A B F = MkArr A C (G \<cdot>\<^sub>B F)"
Tw_inj: fixes E :: "('e, 'f) Order_scheme" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))" shows "inj_on (Tw\<^bsub>D,E\<^esub>) (carrier D)"
ex_move_tl: fixes r :: "cell list"   and n :: "nat" shows "\<exists>na. tl r = tl (r @ Bk \<up> n) @ Bk \<up> na \<or> tl (r @ Bk \<up> n) = tl r @ Bk \<up> na"
integrable_const_bound: fixes f :: "'a \<Rightarrow> 'b"   and B :: "real"   and M :: "'a pmf" assumes "AE x in measure_pmf M. norm (f x) \<le> B"   and "measure_pmf.random_variable M borel f" shows "integrable (measure_pmf M) f"
cases: fixes x :: "('fsym, 'psym) lit" obtains a :: "('fsym, 'psym) atm" where "x = Pos a__"   | a :: "('fsym, 'psym) atm" where "x = Neg a"
ENF_refl_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ENF_refl P \<equiv> sinvar_all_edges_normal_form P \<and> (\<forall>p1. P p1 p1)"
coboundedI2: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "semilattice_order f less_eq less"   and "less_eq b c" shows "less_eq (f a b) c"
assms: shows "\<forall>PROB as s. P PROB \<and> finite PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
gs': fixes j :: "nat" assumes "j < i - 1" shows "fs.gs.gso j \<in> Rn"
mkPX_ci: fixes l :: "'a list"   and A :: "pol"   and b :: "nat"   and C :: "pol" assumes "in_carrier l" shows "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
pt_swap: fixes a :: "'c"   and b :: "'c"   and x :: "'b" assumes "pt TYPE('b) TYPE('c)"   and "at TYPE('c)" shows "[(a, b)] \<bullet> x = [(b, a)] \<bullet> x"
elims_3: fixes x :: "('a \<times> ('b \<Rightarrow> 'c)) list"   and xa :: "'a \<Rightarrow> 'c"   and xb :: "'b" assumes "\<not> postList x xa xb" obtains xc :: "'a" and e :: "'b \<Rightarrow> 'c" and xs :: "('a \<times> ('b \<Rightarrow> 'c)) list" and l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = (xc, e) # xs" and "xa = l" and "xb = s" and "\<not> (l xc = e s \<and> postList xs l s)"
F_is_locally_essentially_surjective: fixes b :: "'d"   and b' :: "'d"   and f :: "'c" assumes "D.obj b"   and "D.obj b'"   and "\<guillemotleft>f : F.map\<^sub>0 b \<rightarrow>\<^sub>C F.map\<^sub>0 b'\<guillemotright>"   and "C.ide f" shows "\<guillemotleft>\<Phi> (\<Psi> f b b') f (F (\<eta>'.map\<^sub>0 b' \<star>\<^sub>D G f \<star>\<^sub>D \<eta>\<^sub>0 b)) : f \<cong>\<^sub>C F (\<eta>'.map\<^sub>0 b' \<star>\<^sub>D G f \<star>\<^sub>D \<eta>\<^sub>0 b)\<guillemotright>"
gbm_invar_split_4: fixes g :: "'m1" assumes "gbm_invar g" shows "valid_graph (gbm_\<alpha> g)"
psubst_eq_rawpsubst2: fixes \<phi> :: "'c"   and x1 :: "'a"   and x2 :: "'a"   and t1 :: "'b"   and t2 :: "'b" assumes "\<phi> \<in> fmla"   and "x1 \<in> var"   and "x2 \<in> var"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<noteq> x2"   and "x2 \<notin> FvarsT t1" shows "psubst \<phi> [(t1, x1), (t2, x2)] = rawpsubst \<phi> [(t1, x1), (t2, x2)]"