hcju/isabelleps huggingface dataset:
simps: fixes C :: "'a literal multiset"   and i :: "nat" shows "weight (C, i) = size_factor * size_multiset (size_literal size_atm) C + timestamp_factor * i"
Scan_mk_regular1: fixes T :: "(symbol \<times> character list) set"   and k :: "nat" shows "\<exists>P F. Scan T k = mk_regular1 P F"
terminationLessState1_def: fixes Vbl :: "nat set" shows "terminationLessState1 Vbl = {(state1, state2). (state1, state2) \<in> satFlagLessState \<or> (state1, state2) \<in> lexLessState1 Vbl}"
map_simp: fixes f :: "'d" assumes "D.arr f" shows "D.\<rho>' f = D.inv (D.\<rho> (D.cod f)) \<cdot>\<^sub>D D.map f"
elims_2: fixes x :: "'e list"   and xa :: "'e list set"   and xb :: "'e list" assumes "mcp x xa xb" obtains z :: "'e list" and W :: "'e list set" and p :: "'e list" where "x = z" and "xa = W" and "xb = p" and "prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p)"
Diff_partition: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
trivial_phi: fixes g :: "'g"   and v :: "'val" assumes "uninst_code.ssa.trivial uses phis g v" shows "uninst_code.ssa.phi phis g v \<noteq> None"
m_s_h: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and S :: "'a st" assumes "Measure1 m h"   and "finite X" shows "Measure1.m_s m S X \<le> h * card X"
extremum_strict: fixes a :: "'c" shows "\<not> \<top> < a"
has_int': shows "((\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})) has_integral integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)) {0..1}"
span_list_def: fixes vs :: "'c list" shows "span_list vs \<equiv> {lincomb_list c vs |c. c \<in> {0..<length vs} \<rightarrow> carrier R}"
k_op_included: shows "(k, op) \<in> {0..<t} \<times> set (\<Pi>\<^sub>\<O>)"
rep_eq: fixes x :: "('c \<times> 'd) list" shows "mapping.rep (AList_Mapping.Mapping x) = map_of x"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "natural_transformation A B F G \<tau>"   and "natural_transformation A B G F \<tau>'"   and "inverse_transformations_axioms A B \<tau> \<tau>'" shows "inverse_transformations A B F G \<tau> \<tau>'"
sba_in_L: fixes xa :: "'b" assumes "xa \<in> segment G.face_cycle_succ b a" shows "xa \<in> H.face_cycle_set b"
symmetric_point_uniqueness: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and P1 :: "'a"   and P2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong P A P1"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong P A P2" shows "P1 = P2"
retract_cancel1_aux: fixes ys :: "'d list"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" assumes "cancel1 ys (map f xs)" shows "\<exists>zs. cancel1 zs xs \<and> ys = map f zs"
1: shows "u \<le>\<^sub>F\<^sub>I v"
mr_not_Conc: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators" assumes "singleCombinators p" shows "applied_rule_rev C x p \<noteq> \<lfloor>a \<oplus> b\<rfloor>"
flowpipe_def: fixes X0 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and hl :: "real"   and hu :: "real"   and CX :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and X1 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set" shows "flowpipe X0 hl hu CX X1 = (0 \<le> hl \<and> hl \<le> hu \<and> fst ` X0 \<subseteq> Csafe \<and> fst ` CX \<subseteq> Csafe \<and> fst ` X1 \<subseteq> Csafe \<and> (\<forall>(x0, d0)\<in>X0. \<forall>h\<in>{hl..hu}. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0..h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX)))"
distinct_39: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> Stop"
max_bot2: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order_bot bot less_eq less" shows "ord.max less_eq x bot = x"
ext_surjective: fixes x :: "'a state_scheme" shows "x = \<lparr>heap = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_ext_Tuple_Iso \<circ> id)) x, locals = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_ext_Tuple_Iso \<circ> id)) x, \<dots> = (Record.iso_tuple_snd state_ext_Tuple_Iso \<circ> id) x\<rparr>"
cases: fixes x :: "(nat \<times> nat \<times> nat) list" obtains "x = []"   | v :: "nat \<times> nat \<times> nat" where "x = [v]"   | n0 :: "nat" and j0 :: "nat" and i0 :: "nat" and n1 :: "nat" and j1 :: "nat" and ix1 :: "nat" and L :: "(nat \<times> nat \<times> nat) list" where "x = (n0, j0, i0) # (n1, j1, ix1) # L"
sees_method_fun: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and TS :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'a option"   and D' :: "String.literal" assumes "P \<turnstile> C sees M: TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M: TS'\<rightarrow>T' = m' in D'" shows "TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
disj: shows "u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
Sup_range_comp: fixes p :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and S :: "'a \<Rightarrow> 'c" shows "\<Squnion>range p \<circ> S = (SUP w. p w \<circ> S)"
homeomorphic_sym: fixes s :: "'b set"   and t :: "'c set" shows "s homeomorphic t = t homeomorphic s"
assms_1: shows "m = insert_weights s (shallow_model Y Z M N) w"
S5: shows "mod7 (c_fst (c_pair 6 (c_pair (sch_to_nat sch1) (sch_to_nat sch2)))) = 6"
zip_Nil: fixes ys :: "'c list" shows "zip [] ys = []"
sams_lea2__sams: fixes A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "SAMS A' B' C' D' E' F'"   and "A B C LeA A' B' C'"   and "D E F LeA D' E' F'" shows "SAMS A B C D E F"
nths_assoc_three_B: fixes A :: "nat set"   and B :: "nat set"   and C :: "nat set"   and xs :: "'b list" assumes "A \<inter> B = {}"   and "(A \<union> B) \<inter> C = {}" shows "nths (nths xs (ind_in_set (A \<union> B \<union> C) ` (A \<union> B))) (ind_in_set (A \<union> B) ` B) = nths (nths xs (ind_in_set (A \<union> B \<union> C) ` (B \<union> C))) (ind_in_set (B \<union> C) ` B)"
asIMOD_igFreshIGOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igFreshIGOp (asIMOD SEM)"
greaterThanLessThan_iff: fixes i :: "'c"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "(i \<in> ord.greaterThanLessThan less l u) = (less l i \<and> less i u)"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'c \<Rightarrow> 'a"   and x :: "'c fset"   and y :: "'b fset" shows "rel_fset Sb (i |`| x) y = rel_fset (\<lambda>x. Sb (i x)) x y"
pair_fun_upd_def: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a \<times> 'b" shows "pair_fun_upd f p \<equiv> case p of (k, v) \<Rightarrow> f(k := v)"
orbit_eq: fixes s :: "'a" assumes "s \<in> S" shows "\<gamma>\<^sup>\<phi> s = {\<phi> t s |t. t \<in> T}"
simps_11: fixes e :: "char list exp"   and M :: "char list"   and es :: "char list exp list" shows "fv (e\<bullet>M(es)) = fv e \<union> fvs es"
by_length_paths_le: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "by_length k (paths_le k P) = by_length k P"
eq_onp_def: fixes R :: "'a \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
inclusion_morphism: fixes Y :: "'a set set" assumes "folding_g.ChamberSubcomplex Y" shows "ChamberComplexMorphism Y X id"
Agent_Hash: fixes X :: "agent"   and X' :: "msg" shows "Messages.Agent X \<noteq> Hash X'"
S18: shows "pr_gr upb = g_pair (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x)"
matches_to_match_list_normalize: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "matches \<gamma> m a p = match_list \<gamma> (normalize_match m) a p"
Diff_Diff_Int: fixes A :: "'h set"   and B :: "'h set" shows "A - (A - B) = A \<inter> B"
eSwapAbs_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and A :: "'d"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "\<forall>us s. \<not> igWlsAbs MOD (us, s) A" shows "eSwapAbs MOD zs z1 z2 (OK A) = ERR"
incid_inters_left: fixes P :: "Points"   and l :: "Lines"   and m :: "Lines" assumes "P \<in> inters l m" shows "incid P l"
c_fst_is_pr: shows "c_fst \<in> PrimRec1"
interval_lowerbound: fixes a :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "interval_lowerbound (cbox a b) = a"
Fourier_sum_0_explicit: fixes f :: "real \<Rightarrow> real"   and n :: "nat" shows "(\<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k 0) = (Fourier_coefficient f 0 / sqrt 2 + (\<Sum>k = 1..n div 2. Fourier_coefficient f (2 * k))) / sqrt pi"
prj_chine_assoc_2: shows "\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.chine_assoc = \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0\<^sub>1"
matrix_vmult_column_sum: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "\<exists>f. A *v x = (\<Sum>y\<in>columns A. f y *s y)"
dim_def: fixes V :: "('a \<Rightarrow> real) set" shows "diff_fun_space.m1.dim V = (if \<exists>b\<subseteq>diff_fun_space. \<not> sub.diff_fun_space.m1.dependent b \<and> sub.diff_fun_space.m1.span b = sub.diff_fun_space.m1.span V then card (SOME b. b \<subseteq> diff_fun_space \<and> \<not> sub.diff_fun_space.m1.dependent b \<and> sub.diff_fun_space.m1.span b = sub.diff_fun_space.m1.span V) else 0)"
Nats_le_whn: fixes n :: "nat star" assumes "n \<in> \<nat>" shows "n \<le> whn"
relcompEpair: fixes a :: "'a"   and c :: "'c"   and r :: "('a \<times> 'd) set"   and s :: "('d \<times> 'c) set" assumes "(a, c) \<in> r O s" obtains b :: "'d" where "(a, b) \<in> r" and "(b, c) \<in> s"
ufa_\<alpha>_memI: fixes x :: "nat"   and l :: "nat list"   and y :: "nat" assumes "x < length l"   and "y < length l"   and "rep_of l x = rep_of l y" shows "(x, y) \<in> ufa_\<alpha> l"
UN_E: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" assumes "b \<in> \<Union> (\<P> B A)" obtains x :: "'c" where "x \<in> A" and "b \<in> B x"
m2: assumes "v = v'" shows "verticesFrom f21 v' = v' # ws @ u # snd (splitAt u (verticesFrom f v'))"
greaterThanAtMost_subseteq_atLeastAtMost_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
intros_1: fixes t :: "('b, 'c, 'd, 'e) Tree"   and tset :: "('b, 'c, 'd, 'e) Tree set['b]" assumes "t \<in> set_bset tset" shows "(t, tConj tset) \<in> Tree_wf"
C'_def: shows "C' = (\<lambda>i j. if i \<in> S \<and> j \<in> S \<and> seq (\<phi> i) (\<phi> j) then \<psi> (\<phi> i \<cdot> \<phi> j) else n)"
atLeastLessThan_iff: fixes i :: "'e word"   and l :: "'e word"   and u :: "'e word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" shows "bounded_linear_axioms f"
MINIM: fixes a' :: "'a" assumes "a' \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> (b, a') \<in> r" shows "(a, a') \<in> r"
star_of_neq_HNatInfinite: fixes N :: "nat star"   and n :: "nat" assumes "N \<in> HNatInfinite" shows "hypnat_of_nat n \<noteq> N"
orbit_altdef_bounded: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "(f ^^ n) s = s"   and "0 < n" shows "orbit f s = {(f ^^ m) s |m. m < n}"
LIM_zero_cancel: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" assumes "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'c)) F" shows "(f \<longlongrightarrow> l) F"
has_typ1_imp_typ_of1: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t : ty" shows "typ_of1 Ts t = Some ty"
f7: fixes A :: "'a set"   and Aa :: "'a set" shows "- A - - Aa = Aa - A"
fps_to_fls_eq_zero_iff: fixes f :: "'a fps" shows "(fps_to_fls f = 0) = (f = 0)"
isomorphic_def: fixes F :: "('c, 'd) Group_scheme"   and G :: "('e, 'f) Group_scheme" shows "F \<cong> G = (\<exists>f. gbij\<^bsub>F,G\<^esub> f)"
bprv_\<phi>G_eqv_all_not_PPf: shows "bprv (eqv \<phi>G (all yy (neg (PPf (Var yy) \<langle>\<phi>G\<rangle>))))"
zdvd_period: fixes a :: "int"   and d :: "int"   and x :: "int"   and t :: "int"   and c :: "int" assumes "a dvd d" shows "(a dvd x + t) = (a dvd x + c * d + t)"
multi_nonempty_split: fixes M :: "'c multiset" assumes "M \<noteq> {#}" shows "\<exists>A a. M = add_mset a A"
undiag_diag_fatom: fixes n :: "nat" shows "nat_of_fatom (fatom_of_nat n) = n"
inner2_def: fixes x :: "'b"   and n :: "'b"   and l :: "'b" shows "inner2 x n l = (x \<bullet> n, x \<bullet> l)"
ltln_strong_weak_1: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n F\<^sub>n \<psi> and\<^sub>n \<phi> W\<^sub>n \<psi>"
r_sv01_def: shows "r_sv01 \<equiv> let at0 = Cn 1 r_nth [recf.Id 1 0, Z]; at1 = Cn 1 r_nth [recf.Id 1 0, r_const 1]; m = Cn 1 r_amalgamate [at0, at1]; c = Cn 1 r_inconsist [m, recf.Id 1 0]; p = Cn 1 r_pdec1 [Cn 1 r_parallel [at0, at1, c]]; g = Cn 1 r_ifeq [c, r_length, m, Cn 1 r_ifz [p, at1, at0]] in Cn 1 (r_lifz r_auxhyp g) [Cn 1 r_eq [r_length, r_const 1], recf.Id 1 0]"
distinct_1: fixes x2 :: "'b" shows "extB.Bot \<noteq> extB.Val x2"
get_shadow_root_delete_shadow_root: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
enumerate_regs_def: fixes t :: "transition" shows "Transition.enumerate_regs t = \<Union> (set (map GExp.enumerate_regs (Guards t))) \<union> \<Union> (set (map AExp.enumerate_regs (Outputs t))) \<union> \<Union> (set (map (\<lambda>(uu_, u). AExp.enumerate_regs u) (Updates t))) \<union> \<Union> (set (map (\<lambda>(r, uu_). AExp.enumerate_regs (V (R r))) (Updates t)))"
carrier_stalk_def: shows "st.carrier_stalk \<equiv> dl.carrier_direct_lim"
simps_2: fixes k :: "nat"   and x :: "'a" shows "Suc k \<otimes> x = k \<otimes> x \<squnion> x"
A_Issues_B: fixes A :: "agent"   and B :: "agent"   and Ticket :: "msg"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Says A B \<lbrace>Ticket, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "A Issues B with Crypt K \<lbrace>Agent A, Number Ta\<rbrace> on evs"
apply_operations_Broadcast: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and m :: "'a \<times> ('a, 'b) operation" shows "apply_operations (xs @ [Broadcast m]) = apply_operations xs"
not_coprimeE: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "\<not> algebraic_semidom.coprime times one a b" obtains c :: "'b" where "dvd.dvd times c a" and "dvd.dvd times c b" and "\<not> dvd.dvd times c one"
moebius_mu_1: shows "moebius_mu 1 = (1::'a)"
ap_stream_homo: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "Applicative.ap (pure f) (pure x) = pure (f x)"
Cn_imp_bis: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(imp_PL p q \<in> Cn A) = (Cn (A \<union> {q}) \<subseteq> Cn (A \<union> {p}))"
EQ_imp_SUBS: fixes t :: "tm"   and u :: "tm"   and H :: "fm set" shows "insert (t EQ u) H \<turnstile> t SUBS u"
source_out_not_cycle: fixes p :: "'v list"   and x :: "'v" assumes "cycle \<Delta> p" shows "(source \<Delta>, x) \<notin> set (cycle_edges p)"
valid_plan_from_def: fixes P :: "ast_problem"   and M :: "object atom formula set"   and \<pi>s :: "plan_action list" shows "ast_problem.valid_plan_from P M \<pi>s = (\<exists>M'. ast_problem.plan_action_path P M \<pi>s M' \<and> M' \<^sup>c\<TTurnstile>\<^sub>= goal P)"
ipresIGWlsAbs_def: fixes hA :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'c, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'h, 'o, 'p, 'q, 'd, 'r) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
"case": fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(case WriterT x of WriterT x \<Rightarrow> f x) = f x"
to_nat_def: shows "mod_type_class.to_nat = nat \<circ> Rep"
prx: shows "\<forall>n w. prices Mkt2 x n w = cls_val_process Mkt2 pf n w"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>t s. a1 = t \<and> a2 = s \<and> is_App t \<and> (fun t \<ge>\<^sub>t s \<or> arg t \<ge>\<^sub>t s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t >\<^sub>h\<^sub>d head s \<and> chkvar t s \<and> chksubs (>\<^sub>t) t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> chksubs (>\<^sub>t) t s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s))))"
cont_if_else_above: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "cont f"   and "cont g"   and "\<And>x. f x \<sqsubseteq> g x"   and "\<And>x y. \<lbrakk>x \<sqsubseteq> y; P y\<rbrakk> \<Longrightarrow> P x"   and "adm P" shows "cont (\<lambda>x. if P x then f x else g x)"
n_choose_2_nat: fixes n :: "nat" shows "n choose 2 = n * (n - 1) div 2"
inj_dual_iff: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "(dual_prod x = dual_prod y) = (x = y)"
xcpt_eff_def: fixes i :: "'c instr"   and P :: "'d prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" shows "xcpt_eff i P pc \<tau> et \<equiv> let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>((case C of None \<Rightarrow> Class Throwable | \<lfloor>C'\<rfloor> \<Rightarrow> Class C') # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et)"
u_outside: fixes x :: "'v" assumes "x \<notin> B \<Gamma>" shows "u x = 0"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck_1: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = X"
right_lang_def: fixes q :: "hf" shows "right_lang q \<equiv> {u. nextl q u \<in> dfa.final M}"
det3_zero_translate_plus: fixes a :: "real \<times> real"   and x :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "coll (a + x) (b + x) (c + x) = coll a b c"
freshBinp_def2: fixes xs :: "'varSort"   and x :: "'var"   and binp :: "'d \<Rightarrow> ('e, 'd, 'varSort, 'var, 'f) qAbs set option" shows "freshBinp xs x binp = qFreshBinp xs x (pickBinp binp)"
red_nonzero: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" shows "p \<noteq> 0"
26307: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) \<le> - x"
prod_constant: fixes y :: "'a"   and A :: "'b set" shows "(\<Prod>x\<in>A. y) = y ^ card A"
path_prepend: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set"   and l :: "'a list"   and w :: "'a" assumes "(u, v) \<in> E"   and "path E v l w" shows "path E u (u # l) w"
map_scale_eq_monom_mult: fixes c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "c \<cdot> p = MPoly_Type_Class.punit.monom_mult c (0::'c) p"
image_add_int_atLeastLessThan: fixes l :: "int"   and u :: "int" shows "(\<lambda>x. x + l) ` {0..<u - l} = {l..<u}"
assms: shows "\<forall>x y. (x, y) \<in> {(x, y). R x y}\<^sup>* \<longrightarrow> (y, x) \<in> {(x, y). R x y}\<^sup>*"
leof_RES_rule_2: fixes X :: "'b set"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> X \<Longrightarrow> \<Phi> x" shows "RES X \<le>\<^sub>n SPEC \<Phi>"
run_total_induct: fixes P :: "nat \<Rightarrow> 'state_t \<Rightarrow> ('dom_t \<Rightarrow> 'action_t list list) \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'state_t"   and a2 :: "'dom_t \<Rightarrow> 'action_t list list" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution CISK_control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting"   and "\<And>s execs. P 0 s execs"   and "\<And>n s execs. \<lbrakk>interrupt (Suc n); P n (cswitch (Suc n) s) execs\<rbrakk> \<Longrightarrow> P (Suc n) s execs"   and "\<And>n execs s. \<lbrakk>\<not> interrupt (Suc n); int.thread_empty (execs (current s)); P n s execs\<rbrakk> \<Longrightarrow> P (Suc n) s execs"   and "\<And>n execs s. \<lbrakk>\<not> interrupt (Suc n); \<not> int.thread_empty (execs (current s)); P n (int.step (int.next_state s execs) (int.next_action s execs)) (int.next_execs s execs)\<rbrakk> \<Longrightarrow> P (Suc n) s execs" shows "P a0 a1 a2"
oparboth: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and t :: "'f"   and t' :: "'f"   and T :: "('f \<times> 'e seq_action \<times> 'f) set"   and i :: "nat" assumes "((\<sigma>, s), receive m, \<sigma>', s') \<in> S"   and "(t, send m, t') \<in> T" shows "((\<sigma>, s, t), \<tau>, \<sigma>', s', t') \<in> oparp_sos i S T"
iwlsFSwSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "iwlsFSwSTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSwapAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSwapClsSTR MOD \<and> igAbsCongSSTR MOD"
axiom_distribimp_def: fixes p :: "fol fm"   and q :: "fol fm"   and r :: "fol fm" shows "axiom_distribimp p q r \<equiv> Thm (Imp (Imp p (Imp q r)) (Imp (Imp p q) (Imp p r)))"