hcju/isabelleps huggingface dataset:
sqn_invalidate_not_in_dests: fixes dip :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<notin> dom dests" shows "sqn (invalidate rt dests) dip = sqn rt dip"
Greatest_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.Greatest less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq y x))"
2: shows "deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g) = deg R f * deg R g"
cUNION_cUn: fixes f :: "'e \<Rightarrow> 'd cset"   and A :: "'e cset"   and B :: "'e cset" shows "cUnion (cimage f (cUn A B)) = cUn (cUnion (cimage f A)) (cUnion (cimage f B))"
finsert_fsubset: fixes x :: "'a"   and A :: "'a fset"   and B :: "'a fset" shows "(finsert x A |\<subseteq>| B) = (x |\<in>| B \<and> A |\<subseteq>| B)"
fmdom'_notD: fixes x :: "'e"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "x \<notin> fmdom' m" shows "fmlookup m x = None"
simps_4: fixes \<rho> :: "(nat \<times> exp) list"   and e1 :: "exp"   and e2 :: "exp" shows "psubst \<rho> (EApp e1 e2) = EApp (psubst \<rho> e1) (psubst \<rho> e2)"
Resolution_weaken: fixes S :: "'b literal set set"   and D :: "'b literal set"   and T :: "'b literal set set" assumes "S \<turnstile> D" shows "T \<union> S \<turnstile> D"
not_less_le: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "(\<not> a \<prec> b) = b \<preceq> a"
normal_invE_1: fixes N :: "'a set" assumes "N \<lhd> G\<lparr>carrier := S\<rparr>" shows "subgroup N (G\<lparr>carrier := S\<rparr>)"
linear_subspace_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "clinear f"   and "csubspace S" shows "csubspace (f ` S)"
lc_f': shows "lincomb f' A' = 0\<^sub>v n"
perm_elem: fixes M2 :: "('e, 'f, 'g) FSM"   and V :: "'e list set"   and M1 :: "('e, 'h, 'i) FSM"   and vs :: "'e list"   and vs' :: "('e \<times> 'h) list" assumes "is_det_state_cover M2 V"   and "OFSM M1"   and "OFSM M2"   and "inputs M1 = inputs M2"   and "vs \<in> V"   and "vs' \<in> language_state_for_input M1 (initial M1) vs" obtains V'' :: "('e \<times> 'h) list set" where "V'' \<in> Perm V M1" and "vs' \<in> V''"
wt_term_code: fixes \<Sigma> :: "exesignature"   and t :: "Core.term" assumes "exe_osig_conds (exesorts \<Sigma>)" shows "term_ok' (translate_signature \<Sigma>) t = exeterm_ok' \<Sigma> t"
BS2': 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 f :: "'b"   and g :: "'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g"   and "\<exists>ga. bicategory.adjoint_pair V H \<a> \<i> src trg g ga"   and "src f = src g"   and "category.isomorphic V (H g (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f)) r" shows "\<exists>\<rho>. tabulation V H \<a> \<i> src trg r \<rho> f g"
finite_not_eq_zero_sumI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite {a. f a \<noteq> (0::'b)}"   and "finite {b. g b \<noteq> (0::'b)}" shows "finite {a + b |a b. f a \<noteq> (0::'b) \<and> g b \<noteq> (0::'b)}"
deduct_hom_if_synth: fixes m :: "('fun, 'var) Term.term"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set" assumes "homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec"   and "m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "M \<turnstile>\<^sub>c m" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m m"
nprv_insertShiftI: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "nprv (insert \<phi>1 (insert \<phi>2 F)) \<psi>" shows "nprv (insert \<phi>2 (insert \<phi>1 F)) \<psi>"
simps_2: fixes n :: "nat"   and s :: "'b stream" shows "sdrop (Suc n) s = sdrop n (stl s)"
Rep_lam_cases: fixes y :: "lam_Rep" assumes "y \<in> Collect lam_Rep_set" obtains x :: "lam" where "y = Rep_lam x"
stable_pair_on_def: fixes ds :: "'d set" shows "stable_pair_on ds = (\<lambda>(XD, XH). XD = - Rf CH XH \<and> XH = - Rf (CD_on ds) XD)"
S5nI: fixes M :: "('e, 'g, 'f) KripkeStructure" assumes "\<And>a. equiv (worlds M) (relations M a)" shows "S5n M"
2: assumes "punit.lt (rep_list p) = punit.lt (rep_list q)"   and "punit.lc (rep_list p) + punit.lc (rep_list q) = (0::'b)" shows "thesis"
\<Gamma>_consts_simps_3: fixes b :: "'sets set" shows "\<Gamma> (Fun (Abs b) []) = Var Value"
rep_eq: fixes x :: "uint32" shows "Rep_uint64 (uint64_of_uint32 x) = ucast (Rep_uint32 x)"
oid_def: fixes v :: "Value" shows "oid v = (case v of objV C a \<Rightarrow> a)"
bit_mod_2_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (modulo a (numeral.numeral one plus (num.Bit0 num.One))) n = (n = 0 \<and> \<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
a1: shows "\<forall>x. (\<Sum>m | m \<in> monomials mpa \<and> lookup m va = x. monomial (MPoly_Type.coeff mpa m) (Poly_Mapping.update va 0 m)) = 0"
finite_LB: fixes xs :: "real list"   and f :: "atom fm" shows "finite {(r - \<langle>cs,xs\<rangle>) / c |r c cs. Less r (c # cs) \<in> set (atoms\<^sub>0 f) \<and> 0 < c}"
freshEnv_swapEnv_exchange2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "goodEnv rho" shows "freshEnv xs x2 (rho &[x2 \<and> x1]_xs) = freshEnv xs x1 rho"
small_nstep_SucD: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" assumes "\<sigma>' \<in> small_nstep P \<sigma> (Suc n)" shows "\<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>' \<in> small_nstep P \<sigma>1 n"
apply_commute_commute: fixes f :: "('c, 'b) commute"   and x :: "'c"   and y :: "'c" shows "apply_commute f x y = apply_commute f y x"
enum_alt_inj: fixes x :: "nat"   and y :: "nat" assumes "enum_alt x = enum_alt y" shows "x = y \<or> enum_alt x = None"
success_heapI: fixes f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap" shows "success (heap f) h"
g2_g1_identical: fixes k :: "nat" assumes "k < m"   and "k \<noteq> i"   and "k \<noteq> i - 1" shows "gs2.gso k = fs.gs.gso k"
op_set_isEmpty_info_rel_set: fixes empty_i :: "'e \<Rightarrow> bool"   and A :: "('e \<times> 'f set) set"   and I :: "('g \<times> 'h) set" assumes "GEN_OP empty_i is_empty (A \<rightarrow> bool_rel)" shows "(\<lambda>x. empty_i (snd x), op_info_is_empty) \<in> \<langle>I, A\<rangle>info_rel \<rightarrow> bool_rel"
l1_refines_init_s0: shows "init l1 \<subseteq> R01s `` init s0"
lookup_append: fixes e :: "name list"   and i :: "name"   and n :: "nat"   and j :: "name" shows "lookup (e @ [i]) n j = abst_dbtm i (length e + n) (lookup e n j)"
l_new_shadow_root_get_child_nodes_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_child_nodes_axioms get_child_nodes get_child_nodes_locs \<equiv> (\<forall>ptr' new_shadow_root_ptr h h' r. ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_shadow_root_ptr h'. h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr) \<rightarrow>\<^sub>r [])"
sv01_1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> V\<^sub>0\<^sub>1" shows "s\<^bsub>01\<^esub> (f \<triangleright> 0) = auxhyp (f \<triangleright> 0)"
rep_eq: fixes x :: "('d, 'c) sq_matrix" shows "to_fun (cofactor x) = (\<lambda>i j. Square_Matrix.det (minor x i j))"
23: shows "irreflexive (prim_P w v e\<^sup>T * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * e * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
hyps_3: shows "eval_poly hom (sum p A) x = (\<Sum>k\<in>A. eval_poly hom (p k) x)"
hyps: fixes i :: "nat"   and j :: "nat" shows "(i < length v * row_length M \<and> j < length M) \<and> mat (row_length M) (length M) M \<longrightarrow> vec_mat_Tensor v M ! j ! i = v ! (i div row_length M) * M ! j ! (i mod row_length M)"
bot_least: fixes a :: "'a" shows "(zero_class.zero::'a) \<le> a"
SkipMonotone: fixes G :: "Var \<Rightarrow> L"   and p :: "L" shows "Monotone (\<lambda>T (s, t). EQ G p s t)"
comparator_lex_comp_aux: shows "comparator lex_comp_aux"
inj: shows "inj_on (\<lambda>g. g - {x}) \<G>"
WP1Conca: fixes p :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "DenyAll \<notin> set p" shows "wellformed_policy1 (a # p)"
card_eq_bij: fixes A :: "'b set"   and B :: "'c set" assumes "card A = card B"   and "finite A"   and "finite B" obtains f :: "'b \<Rightarrow> 'c" where "bij_betw f A B"
labeled_tiebreaker_lifting_axioms_def: fixes Inf_F :: "'a inference set"   and Inf_FL :: "('a \<times> 'b) inference set" shows "labeled_tiebreaker_lifting_axioms Inf_F Inf_FL \<equiv> (\<forall>\<iota>\<^sub>F Ll. \<iota>\<^sub>F \<in> Inf_F \<longrightarrow> length Ll = length (prems_of \<iota>\<^sub>F) \<longrightarrow> (\<exists>L0. Infer (zip (prems_of \<iota>\<^sub>F) Ll) (concl_of \<iota>\<^sub>F, L0) \<in> Inf_FL)) \<and> (\<forall>\<iota>\<^sub>F\<^sub>L. \<iota>\<^sub>F\<^sub>L \<in> Inf_FL \<longrightarrow> Infer (map fst (prems_of \<iota>\<^sub>F\<^sub>L)) (fst (concl_of \<iota>\<^sub>F\<^sub>L)) \<in> Inf_F)"
lfp_Siml: fixes S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and P :: "('c, 'd) action \<Rightarrow> ('c, 'd) action" assumes "\<And>X. X \<preceq>S Q \<Longrightarrow> P X \<preceq>S Q"   and "mono P" shows "lfp P \<preceq>S Q"
full_abstraction_and_enc_respects_binary_pred_impl_SRel_iff_TRel_reflects: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "enc_respects_binary_pred Pred" shows "rel_reflects_binary_pred (_Collect (P, Q) (\<exists>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<and> (SP, SQ) \<in> SRel)) Pred = rel_reflects_binary_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
pw_eeq_iff: fixes m :: "('c + 'd) nres"   and n :: "('c + 'd) nres" shows "(m = n) = (nofail m = nofail n \<and> (\<forall>e. inres m (Inl e) = inres n (Inl e)) \<and> (\<forall>x. inres m (Inr x) = inres n (Inr x)))"
zip_range_id: fixes A :: "'b list"   and B :: "'c list" assumes "length A = length B" shows "snd ` set (zip A B) = set B"
simps_8: fixes f :: "'f \<Rightarrow> 'e" shows "map f [] = []"
prems_15: shows "getSATFlag state' = UNDEF \<longrightarrow> InvariantEquivalentZL (getF state') (getM state') F0'"
bij_betw_implies_surj_on_and_card_eq: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "finite A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "bij_betw f A B = (f ` A = B \<and> card A = card B)"
distinct_42: fixes x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Await x71 x72 \<noteq> x41 ;; x42"
types_wellformed_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.types_wellformed bexp_vars \<C>_vars \<Gamma> \<equiv> \<forall>x\<in>dom \<Gamma>. sifum_types.type_wellformed bexp_vars \<C>_vars (the (\<Gamma> x))"
mf0smcle: fixes X :: "complex mat"   and Y :: "complex mat" assumes "X \<le>\<^sub>L Y"   and "X \<in> carrier_mat d d"   and "Y \<in> carrier_mat d d" shows "c \<cdot>\<^sub>m (X - f 0) \<le>\<^sub>L c \<cdot>\<^sub>m (Y - f 0)"
lt_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "tail p \<noteq> 0" shows "lt (tail p) \<prec>\<^sub>t lt p"
D: shows "\<D>s (e # es) \<lfloor>dom l\<rfloor>"
sv: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "sound (v P)"
poly_altdef: fixes p :: "'a poly"   and x :: "'a" shows "poly p x = (\<Sum>i\<le>degree p. poly.coeff p i * x ^ i)"
dom_minus: fixes f :: "'h \<Rightarrow> 'g option"   and x :: "'h"   and A :: "'h set" assumes "f x = None" shows "dom f - insert x A = dom f - A"
eval_merge_primfun: fixes f :: "primfun \<times> real"   and g :: "primfun \<times> real"   and x :: "real" assumes "fst f = fst g" shows "eval_primfun (merge_primfun f g) x = eval_primfun f x * eval_primfun g x"
complete_bipartite_digraph_pair_def: fixes m :: "nat"   and n :: "nat"   and G :: "'b pair_pre_digraph" shows "K\<^bsub>m,n\<^esub> (with_proj G) = (finite (pverts G) \<and> (\<exists>U V. pverts G = U \<union> V \<and> U \<inter> V = {} \<and> card U = m \<and> card V = n \<and> parcs G = U \<times> V \<union> V \<times> U))"
\<psi>\<^sub>1\<^sub>0_tensor_\<psi>\<^sub>1\<^sub>1_is_\<psi>\<^sub>1: fixes n :: "nat" assumes "1 \<le> n" shows "Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)) \<Otimes> Matrix.mat 2 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2)) = Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1)))"
hom_dvd: fixes p :: "'b"   and q :: "'b" assumes "p dvd q" shows "[:p:] dvd [:q:]"
n_rw2: shows "ncols_iarray (IArray (map (\<lambda>x. IArray (map (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) [0..<CARD('b)])) [0..<CARD('c)])) = CARD('b)"
prime_elems_mult_not: fixes P :: "'a set"   and x :: "'a"   and y :: "'a" assumes "prime_ideal R P"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<notin> P"   and "y \<notin> P" shows "x \<cdot>\<^sub>r y \<notin> P"
induct: fixes P :: "'b heap \<Rightarrow> 'b heap \<Rightarrow> bool"   and a0 :: "'b heap"   and a1 :: "'b heap" assumes "\<And>h. P h Empty"   and "\<And>v va. P Empty (Hp v va)"   and "\<And>x hsx y hsy. P (Hp x hsx) (Hp y hsy)" shows "P a0 a1"
ord_1: fixes k :: "nat" assumes "k < length \<alpha>s" shows "Ord (\<alpha>s ! k)"
f_expand_take_mod: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" assumes "n mod k = 0" shows "xs \<odot> k \<down> n = xs \<down> (n div k) \<odot> k"
2: shows "Cring_Poly.compose (R\<lparr>carrier := S\<rparr>) (monom (UP (R\<lparr>carrier := S\<rparr>)) a n) g = a \<odot>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n"
support_wpt: fixes c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (wp\<^sub>1 c Q) \<subseteq> support Q"
D_reset: shows "[reset' A1 n r v (0::'c)]\<^bsub>v,n\<^esub> = ([D]\<^bsub>v,n\<^esub> \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub>"
cases: fixes x :: "'a tree" obtains "x = \<langle>\<rangle>"   | l :: "'a tree" and a :: "'a" and r :: "'a tree" where "x = \<langle>l, a, r\<rangle>"
nth_prime_eqI': fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "card {q. prime q \<and> q \<le> p} = Suc n" shows "nth_prime n = p"
moebius_commute: fixes x :: "'b"   and n :: "nat" shows "x * moebius_mu n = moebius_mu n * x"
interval_eq_iff: fixes a :: "'a interval"   and b :: "'a interval" shows "(a = b) = (lower a = lower b \<and> upper a = upper b)"
closest_pair_bf_eq_val_closest_pair_bf_tm: fixes ps :: "(int \<times> int) list" shows "Time_Monad.val (closest_pair_bf_tm ps) = closest_pair_bf ps"
heq: fixes y :: "'c" assumes "y \<in> U" shows "h (y, 0) = p (f y)"
dom_one: shows "ad (ad (1::'a)) = (1::'a)"
ex_graph''_def: shows "ex_graph'' \<equiv> Lgraph {(home, cloud), (sphone, cloud), (cloud, hospital)} (\<lambda>x. if x = cloud then {''Patient'', ''Eve''} else if x = hospital then {''Doctor''} else {}) ex_creds ex_locs"
mult_1s_2: fixes a :: "'a" shows "a \<cdot> Numeral1 = a"
arrow: shows "C.ide a \<and> \<guillemotleft>x : local.map a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] d\<guillemotright>"
struct_maps_memE_1: fixes f :: "'a \<Rightarrow> 'b"   and T :: "('a, 'c) partial_object_scheme"   and S :: "('b, 'd) partial_object_scheme"   and x :: "'a" assumes "f \<in> struct_maps T S"   and "x \<in> carrier T" shows "f x \<in> carrier S"
atom_map_compl_pres: fixes x :: "'a" shows "atom_map (- x) = Collect atom - atom_map x"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> ns_shared"   and "KK \<subseteq> - range shrK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
sem: fixes I :: "('sf, 'sc, 'sz) interp" shows "prog_sem I (EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid1 vid1)) = {(\<nu>, mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> (sol t)) |\<nu> sol t. 0 \<le> t \<and> (sol solves_ode (\<lambda>_. ODE_sem I (OSing vid1 (f1 fid1 vid1)))) {0..t} {x. mk_v I (OSing vid1 (f1 fid1 vid1)) \<nu> x \<in> fml_sem I (p1 vid1 vid1)} \<and> VSagree (sol 0) (fst \<nu>) {x |x. Inl x \<in> FVP (EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid1 vid1))}}"
hyps: fixes root :: "(att \<times> char list, att, nat) env" assumes "can_exec root (xs @ [y])" shows "\<exists>root' root''. root \<Midarrow>xs\<Rightarrow> root' \<and> root' \<midarrow>y\<rightarrow> root''"
Diff_eq_empty_iff: fixes A :: "'b set"   and B :: "'b set" shows "((A::'b set) - (B::'b set) = {}) = (A \<subseteq> B)"
is_heap_list_simps_2: fixes h :: "'c"   and t :: "('c, 'd) bintree"   and ts :: "('c, 'd) bintree list" shows "is_heap_list h (t # ts) = (is_heap_list h ts \<and> is_heap_list (priority t) (children t) \<and> h \<le> priority t)"
elims: fixes x :: "nat"   and xa :: "real mpoly"   and y :: "real mpoly \<times> real mpoly \<times> real mpoly" assumes "get_coeffs x xa = y" obtains var :: "nat" and x :: "real mpoly" where "xa__ = var" and "xa = x" and "y = (isolate_variable_sparse x var 2, isolate_variable_sparse x var 1, isolate_variable_sparse x var 0)"
intros: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set"   and c :: "'f"   and s :: "('e \<times> 'f) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
assms_2: shows "b \<in> {- 1, 0, 1}"
uer: shows "length (xs @ [x, x]) = length (OPT2 (xs @ [x, x]) R)"
star_inductr_eq: fixes z :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" assumes "z \<union> y O x = y" shows "z O x\<^sup>* \<subseteq> y"
p_eq: fixes ta :: "real" assumes "0 \<le> ta" shows "p s s' ta = (of_bool (s = s') + integral {0..ta} (\<lambda>t. escape_rate s * exp (escape_rate s * t) * measure_pmf.expectation (J s) (\<lambda>s''. p s'' s' t))) * exp (ta *\<^sub>R - escape_rate s)"
in_set_prefixes: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<in> set (prefixes ys)) = prefix xs ys"
eff_def: fixes c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "eff c s i = snd (snd (wt_cont_eff c s i))"
decompose_literal_def: fixes L :: "'b literal"   and t :: "'b trm"   and s :: "'b trm"   and p :: "sign" shows "decompose_literal L t s p = (\<exists>e. p = pos \<and> L = Pos e \<and> decompose_equation e t s \<or> p = neg \<and> L = Neg e \<and> decompose_equation e t s)"
Diag_implies_Arr: fixes t :: "'a local.term" assumes "Diag t" shows "Arr t"
mset_set_Suc_Suc: fixes a :: "nat"   and b :: "nat" shows "mset_set {Suc a..<Suc b} = Suc `# mset_set {a..<b}"
enat_add2_eq: fixes y :: "enat"   and x :: "nat"   and z :: "enat" shows "(y + enat x = z + enat x) = (y = z)"
subfield_m_inv_2: fixes K :: "'a set"   and k :: "'a" assumes "subfield K R"   and "k \<in> K - {\<zero>}" shows "k \<otimes> inv k = \<one>"
vNM_utilityD_2: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "p \<succeq>[relation] q = (measure_pmf.expectation q u \<le> measure_pmf.expectation p u)"
compact_Int_closedin: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "compactin X S"   and "closedin X T" shows "compactin X (S \<inter> T)"
x_def: shows "x \<equiv> THE x. x ~ s' \<and> x \<in> set_pmf (action cfg)"
borel_measurable_count_space: fixes f :: "'e \<Rightarrow> 'f"   and S :: "'e set" shows "f \<in> borel_measurable (count_space S)"
simps_2: fixes Cs :: "char list list"   and a :: "nat" shows "ics_classes (Throwing Cs a) = Cs"
simps: fixes A :: "(('a, 'c) vec, 'b) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_PQ A bezout = diagonal_to_Smith_aux_PQ [0..<min (nrows A) (ncols A) - 1] bezout (mat (1::'a), A, mat (1::'a))"
isDiagonal_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "isDiagonal A = (\<forall>a b. to_nat a \<noteq> to_nat b \<longrightarrow> A $ a $ b = (0::'d))"
compMb2: fixes e :: "'a expr1" shows "compMb2 e = (max_stack e, max_vars e, compE2 e @ [Return], compxE2 e 0 0)"
long_chain_on_path: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord f X"   and "finite X" shows "\<exists>P\<in>\<P>. X \<subseteq> P"
squarefree_factorial_semiring: fixes n :: "'b" assumes "n \<noteq> (0::'b)" shows "squarefree n = (\<forall>p. prime p \<longrightarrow> \<not> p\<^sup>2 dvd n)"
h: shows "bij_betw (\<lambda>(i, j). (j, i)) (Sigma {..n} lessThan) (SIGMA j:{..<n}. {Suc j..n})"
max_reg_list_cons: fixes a :: "vname gexp"   and G :: "vname gexp list" shows "max_reg_list (a # G) = max (GExp.max_reg a) (max_reg_list G)"
integrable_on_cdivide: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and c :: "real" assumes "f integrable_on s" shows "(\<lambda>x. f x / of_real c) integrable_on s"
high_def: fixes p :: "'b ivl" shows "high p = snd (Rep_ivl p)"
rp: shows "restrict_cnf_atom a v S \<turnstile> \<box>"