hcju/isabelleps huggingface dataset:
maxsimpchain_Cons_reduce: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X (x # xs)" shows "SimplicialComplex.maxsimpchain X xs"
fps_unit_factor_mult_fps_X_2: fixes f :: "'a fps" shows "unit_factor (f * fps_X) = unit_factor f"
ubs_ex_n_maxTr: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "a + n \<le> nasc_seq A a n"
connection_axioms: shows "connection (G \<circ>\<^sub>g F)"
0: shows "mat (row_length A1) (length A2) (A1 \<circ> A2)"
abs_eq: fixes xa :: "int"   and x :: "int" shows "fb.dist (fbint xa) (fbint x) = fb.N (xa - x)"
assms_1: fixes x :: "'a" assumes "eval A x" shows "eval B x"
sdrop_suntil: fixes q :: "'b stream \<Rightarrow> bool"   and j :: "nat"   and \<omega> :: "'b stream"   and p :: "'b stream \<Rightarrow> bool" assumes "q (sdrop j \<omega>)"   and "\<forall>k<j. p (sdrop k \<omega>)" shows "(p suntil q) \<omega>"
Basic: fixes a :: "'c set"   and M :: "'c set set"   and \<Omega> :: "'c set" assumes "a \<in> M" shows "a \<in> smallest_ccdi_sets \<Omega> M"
dist_eqs_1: fixes a' :: "'b"   and list' :: "['b]" shows "[] \<noteq> a' : list'"
imp_for_imp_for': fixes i :: "nat"   and u :: "nat" shows "imp_for i u (\<lambda>_. return True) = imp_for' i u"
steps_preserves_termination: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and c' :: "('d, 'e, 'f) com"   and cs' :: "('d, 'e, 'f) com list"   and css' :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and t :: "('d, 'f) xstate" assumes "\<Gamma>\<turnstile> ([c], [], s) \<rightarrow>\<^sup>* (c' # cs', css', t)"   and "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>c' \<down> t"
dlex_pm_refl: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "dlex_pm s s"
deriv': fixes z :: "complex"   and x :: "int"   and t :: "real" assumes "z \<in> U"   and "a \<le> x"   and "t \<in> {real_of_int a..real_of_int x}" shows "((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"
ordinal_times_0: fixes x :: "ordinal" shows "x * 0 = 0"
graph_homomorphism_on_graph: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'd) labeled_graph"   and R :: "('c \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "graph_homomorphism A B R" shows "graph_homomorphism A (map_graph_fn B f) (R O on_graph B f)"
HEndPhase2_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv6 s"   and "Inv2b s"   and "Inv2c s"   and "HInv3 s"   and "HInv5_inner s p" shows "HInv6 s'"
CFG_SSA_Transformed_notriv_linorder_code_axioms: shows "CFG_SSA_Transformed_notriv_linorder_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var chooseNext_all"
lub_below_iff: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "class.cpo below"   and "po.chain below S" shows "below (po.lub below (range S)) x = (\<forall>i. below (S i) x)"
prems_1: shows "\<forall>a|\<in>|e. case a of (a, b) \<Rightarrow> (case a of (from, to) \<Rightarrow> \<lambda>t. to \<noteq> 0) b"
chain_rel_prodD1: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "('b \<times> 'c) set" assumes "Complete_Partial_Order.chain (rel_prod orda ordb) Y" shows "Complete_Partial_Order.chain orda (fst ` Y)"
Semi: fixes c0 :: "com"   and s :: "loc \<Rightarrow> nat"   and s2 :: "loc \<Rightarrow> nat"   and eval :: "((exp \<times> (loc \<Rightarrow> nat)) \<times> nat \<times> (loc \<Rightarrow> nat)) set"   and c1 :: "com"   and s1 :: "loc \<Rightarrow> nat" assumes "(c0, s) -[eval]-> s2"   and "(c1, s2) -[eval]-> s1" shows "(c0;;c1, s) -[eval]-> s1"
semigroup_ow_axioms: shows "semigroup_ow U (+\<^sub>o\<^sub>w)"
IS_BELOW_ID_def: fixes R :: "('a \<times> 'a) set" shows "IS_BELOW_ID R \<equiv> R \<subseteq> Id"
finite_Union: fixes X :: "'b set set"   and M :: "'b measure" assumes "finite X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
Vagree_univ: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and d :: "(real, 'a) vec" assumes "Vagree (a, b) (c, d) UNIV" shows "a = c \<and> b = d"
normalize_alt_simps_6: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "String.literal"   and as :: "'a expr list"   and b :: "'a expr" shows "(\<Gamma> \<turnstile> StaticOperationCall \<tau> op as \<Rrightarrow> b) = (\<exists>params\<^sub>2. b = StaticOperationCall \<tau> op params\<^sub>2 \<and> \<Gamma> \<turnstile>\<^sub>L as \<Rrightarrow> params\<^sub>2)"
prv_imp_triv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv \<psi>" shows "prv (imp \<phi> \<psi>)"
ChiStar_def: fixes A :: "('a, 'b, 'c) hierauto" shows "ChiStar A \<equiv> (ChiRel A)\<^sup>*"
pairwiseI: fixes S :: "'c set"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; x \<noteq> y\<rbrakk> \<Longrightarrow> R x y" shows "pairwise R S"
distinct_eqvt: fixes pi :: "(name \<times> name) list"   and xs :: "name list" shows "pi \<bullet> distinct xs = distinct (pi \<bullet> xs)"
fresh_freshAbs_substAbs: fixes Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "goodAbs A"   and "fresh ys y Y" shows "freshAbs ys y (A $[Y / y]_ys)"
index_finsum_vec: fixes F :: "'b set"   and i :: "nat"   and n :: "nat"   and vs :: "'b \<Rightarrow> 'c vec" assumes "finite F"   and "i < n"   and "vs \<in> F \<rightarrow> carrier_vec n" shows "finsum_vec TYPE('c) n vs F $ i = (\<Sum>f\<in>F. vs f $ i)"
InvariantConsistentAfterBackjump: fixes M :: "(Literal \<times> bool) list"   and p :: "(Literal \<times> bool) list"   and bClause :: "Literal list"   and bLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantConsistent M"   and "isPrefix p M"   and "isUnitClause bClause bLiteral (elements p)"   and "M' = p @ [(bLiteral, False)]" shows "InvariantConsistent M'"
arr_def: fixes f :: "'a" shows "A.arr f \<equiv> A.domains f \<noteq> {} \<or> A.codomains f \<noteq> {}"
sel_2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "stl (siterate f x) = siterate f (f x)"
axioms_4: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
assms_1: shows "\<exists>n'<n''. \<exists>nid'. (n', nid') \<in> his t n nid"
simps: fixes i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q) = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"
prj1_simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.cod C (prj1 f g) = partial_magma.dom C f"
not_mrw: shows "\<not> P,E \<turnstile> r' \<leadsto>mrw ws r'"
symetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<approx> Q" shows "Q \<approx> P"
shift_eq: fixes i :: "nat"   and j :: "nat"   and e :: "nat \<Rightarrow> 'a"   and T :: "'a" assumes "i = j" shows "(e\<langle>i:T\<rangle>) j = T"
comp_char: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f = (if A1.seq (fst g) (fst f) \<and> A2.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>A\<^sub>1 fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f) else (A1.null, A2.null))"
filterlim_at_top_ge: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and c :: "'c" shows "filterlim f at_top F = (\<forall>Z\<ge>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
axioms_3: fixes X :: "'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 \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
unit_vecs_last_carrier: fixes n :: "nat"   and i :: "nat" shows "set (unit_vecs_last n i) \<subseteq> carrier_vec n"
borel_measurable_norm: shows "norm \<in> borel_measurable borel"
connected_root_iff1: fixes r :: "'a"   and x :: "'a" assumes "point r" shows "connected_root r x = (1 ; x \<le> r\<^sup>T ; x\<^sup>+)"
F_le_T: fixes x :: "real" assumes "a \<le> x" shows "F x \<le> T"
fps_lr_inverse_nth_0_1: fixes f :: "'a fps"   and x :: "'a" shows "fps_left_inverse f x $ 0 = x"
hyps: shows "strip_while ((=) (0::'r)) as \<bullet>\<cdot> ms = as \<bullet>\<cdot> ms"
assms: fixes c' :: "real" assumes "c < ereal c'" shows "\<exists>s. s \<bullet> (1::'a) = c' \<and> fds_converges f s"
bin_nth_of_bl: fixes bl :: "bool list"   and n :: "nat" shows "bit (bl_to_bin bl) n = (n < length bl \<and> rev bl ! n)"
simps: fixes v :: "name"   and p :: "tm"   and sx :: "name"   and y :: "name"   and ax :: "name"   and x :: "name" assumes "atom v \<sharp> (p, sx, y, ax, x)"   and "atom x \<sharp> (p, sx, y, ax)"   and "atom ax \<sharp> (p, sx, y)"   and "atom y \<sharp> (p, sx)"   and "atom sx \<sharp> p" shows "Special_axP p = SyntaxN.Ex v (SyntaxN.Ex x (SyntaxN.Ex ax (SyntaxN.Ex y (SyntaxN.Ex sx (FormP (Var x) AND VarP (Var v) AND TermP (Var y) AND AbstFormP (Var v) Zero (Var x) (Var ax) AND SubstFormP (Var v) (Var y) (Var x) (Var sx) AND p EQ Q_Imp (Var sx) (Q_Ex (Var ax)))))))"
mdecls_visible: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list" assumes "wf_prog wf_md P"   and "is_class P C"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" shows "\<exists>Mm. P \<turnstile> C sees_methods Mm \<and> (\<forall>(M, b, Ts, T, m)\<in>set ms. Mm M = \<lfloor>((b, Ts, T, m), C)\<rfloor>)"
C_restr_C_restr: fixes r :: "C"   and r' :: "C"   and v :: "C \<rightarrow> 'a" shows "(v|\<^bsub>r'\<^esub>)|\<^bsub>r\<^esub> = v|\<^bsub>(r' \<sqinter> r)\<^esub>"
P_Suc: fixes a :: "'a"   and b :: "'b" assumes "P a b"   and "Q a b" shows "P (f\<cdot>a) b"
Un_iff: fixes c :: "'x"   and A :: "'x set"   and B :: "'x set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
lookup_proj_poly_syz_alt: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a"   and i :: "nat" shows "lookup (proj_poly_syz n p) (term_of_pair (t, i)) = lookup p (term_of_pair (t, i + n))"
freshTauTransition: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
getEnv_idEnv: fixes xs :: "'e"   and x :: "'f" shows "idEnv xs x = None"
Thread_def: shows "Thread \<equiv> STR ''java/lang/Thread''"
Limsup_compose_continuous_antimono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "antimono f"   and "F \<noteq> bot" shows "Limsup F (\<lambda>n. f (g n)) = f (Liminf F g)"
unintersect2_def: fixes X :: "'d set" shows "unintersect2 X = SPEC ((\<subseteq>) X)"
demonic_def: fixes Q :: "'e \<Rightarrow> 'f"   and p :: "'f" shows "[: Q :] p = {s. Q s \<sqsubseteq> p}"
simps_2: fixes n :: "nat"   and I :: "float interval" shows "subdivide_interval (Suc n) I = (let m = mid I in subdivide_interval n (Ivl (lower I) m) @ subdivide_interval n (Ivl m (upper I)))"
intros: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and cnv :: "('a, 'b, 'c, 'd) converter"   and f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
assms_2: shows "dim_vec b = dim_col A\<^sup>T"
pred_mono_strong: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b stream"   and Pa :: "'b \<Rightarrow> bool" assumes "pred_stream P x"   and "\<And>z. \<lbrakk>z \<in> sset x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_stream Pa x"
abs_conv_abscissa_add_le: fixes f :: "'a fds"   and g :: "'a fds" shows "abs_conv_abscissa (f + g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
hyps: assumes "\<forall>x\<in>set xs. ChamberComplexEndomorphism X (f x)" shows "ChamberComplexEndomorphism X (fold f xs)"
xor_one_right: fixes x :: "'a" shows "x XOR - (1::'a) = NOT x"
cobounded1: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.min a b \<preceq>\<^sub>t a"
msetext_dersh_wf: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. msetext_dersh gt ys xs)"
le_iff_sup: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<preceq> y) = (x \<oplus> y = y)"
vsubst_ident: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X" shows "X #[z // z]_zs = X"
right_diff_distrib': 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times a (minus b c) = minus (times a b) (times a c)"
pivot_positions_def: fixes zer :: "'c"   and A :: "'c mat" shows "pivot_positions_gen zer A \<equiv> pivot_positions_main_gen zer A (dim_row A) (dim_col A) 0 0"
poly_card_roots_less_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "card {x. a < x \<and> x < b \<and> poly p x = 0} = count_roots_between p a b - (if poly p b = 0 \<and> a < b \<and> p \<noteq> 0 then 1 else 0)"
identify_blocks_iff: fixes k :: "nat"   and A :: "'b mat" assumes "k < n" shows "set (identify_blocks A k) = {(i, j) |i j. i \<le> j \<and> j < k \<and> (\<forall>l\<ge>i. l < j \<longrightarrow> A $$ (l, Suc l) = (1::'b)) \<and> (Suc j \<noteq> k \<longrightarrow> A $$ (j, Suc j) \<noteq> (1::'b)) \<and> (0 < i \<longrightarrow> A $$ (i - 1, i) \<noteq> (1::'b))}"
right_diff_distrib': fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times a (minus b c) = minus (times a b) (times a c)"
monad_prob_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and sample :: "'f pmf \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e" shows "monad_prob return bind sample \<equiv> monad return bind \<and> monad_prob_axioms bind sample"
pivot_eq_satisfies_eq: fixes y :: "nat"   and e :: "nat \<times> linear_poly"   and v :: "nat \<Rightarrow> 'a" assumes "y \<in> rvars_eq e" shows "v \<Turnstile>\<^sub>e e = v \<Turnstile>\<^sub>e pivot_eq e y"
internal_induct_lemma: fixes X :: "nat set star" assumes "0 \<in> Iset X"   and "\<forall>n. n \<in> Iset X \<longrightarrow> n + 1 \<in> Iset X" shows "Iset X = UNIV"
continuous_within_upperhalf_Arg2pi: fixes z :: "complex" assumes "z \<noteq> 0" shows "continuous (at z within {z. 0 \<le> Im z}) Arg2pi"
frameResComm: fixes x :: "name"   and y :: "name"   and F :: "'b frame" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>F \<simeq>\<^sub>F \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>F"
fold_set_fold_remdups: fixes y :: "'b"   and xs :: "'a list" shows "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
simps_9: fixes f :: "'f \<Rightarrow> 'e"   and x21 :: "'f"   and x22 :: "'f list" shows "map f (x21 # x22) = f x21 # map f x22"
inv_left: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bijection f" shows "inv f (f a) = a"
IH: shows "det (\<chi>k. if k = i then row i A + y else row k A) = det A"
DERIV_subset: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "(f has_field_derivative f') (at x within s)"   and "t \<subseteq> s" shows "(f has_field_derivative f') (at x within t)"
types_conf_Wtrt_conf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts" shows "\<exists>Ts'. P,E,h \<turnstile> es [:] Ts' \<and> conformable Ts' Ts"
subs_to_same_BL: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat" assumes "RedBlack prb"   and "sub \<in> subs prb" shows "fst (subsumee sub) = fst (subsumer sub)"
shadow_root_ptr_casts_commute2: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^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 (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>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr) = Some shadow_root_ptr"
grey_protects_whiteD: fixes g :: "'e"   and w :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "(g grey_protects_white w) s" shows "grey g s \<and> (g = w \<or> white w s)"
trace_add: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" shows "trace (A + B) = trace A + trace B"
type_less_x_Real: fixes \<tau> :: "'a basic_type" assumes "\<tau> < Real" obtains "\<tau> = OclVoid"   | "\<tau> = UnlimitedNatural"   | "\<tau> = Integer"
ASSUMPTION_I: fixes A :: "bool" assumes "A" shows "ASSUMPTION A"
xs: shows "distinct [d1h1, d1h2, d1h3, d2h1, d2h2, d2h3, d3h1, d3h2, d3h3]"
cases: fixes a1 :: "'c list"   and a2 :: "'d list" assumes "same_length a1 a2" obtains "a1 = []" and "a2 = []"   | xs :: "'c list" and ys :: "'d list" and x :: "'c" and y :: "'d" where "a1 = x # xs" and "a2 = y # ys" and "same_length xs ys"
croots2_def: fixes p :: "complex poly" shows "croots2 p = (let a = poly.coeff p 2; b = poly.coeff p 1; c = poly.coeff p 0; b2a = b / (2 * a); bac = b2a\<^sup>2 - c / a; e = csqrt bac in remdups [- b2a + e, - b2a - e])"
non_empty_ith_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and C :: "(nat \<Rightarrow> 'b) set"   and i :: "nat" assumes "minimal_element P A"   and "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "ith (eq_upto C (minimal_element.lexmin P A C) i) i \<subseteq> A"
Can_red2_2: fixes a :: "'a local.term"   and b :: "'a local.term" assumes "Ide a"   and "Diag a"   and "Ide b"   and "Diag b" shows "a \<^bold>\<Down> b \<in> Hom (a \<^bold>\<otimes> b) \<^bold>\<lfloor>a \<^bold>\<otimes> b\<^bold>\<rfloor>"
Inter_ZeroR: fixes r :: "'a language" shows "Inter r Zero = Zero"
iMOD_union: fixes r :: "nat"   and r' :: "nat"   and m :: "nat" assumes "r \<le> r'"   and "r mod m = r' mod m" shows "[ r, mod m ] \<union> [ r', mod m ] = [ r, mod m ]"
sec_id_pubk_more_msg: fixes m :: "nat"   and m' :: "nat"   and n :: "nat" assumes "{m, m'} \<subseteq> bad_sec_prik"   and "(n, m) \<notin> bad_id"   and "(n, m') \<in> bad_id" shows "{ID n (Sec_PubKey m), Hash (ID n (Sec_PubKey m)), \<langle>n, ID n (Sec_PubKey m)\<rangle>} \<subseteq> spied (sec_id_pubk_more n m) \<and> \<langle>n, Hash (ID n (Sec_PubKey m))\<rangle> \<notin> spied (sec_id_pubk_more n m)"
Le_le_LeI: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "Le a \<le> Le b"
finite_UnI: fixes F :: "'a set"   and G :: "'a set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
l_sort_form_aux_all: fixes X :: "'b list \<times> 'b list \<times> 'b list" shows "l_sort_form (l_sort_aux X)"
lprod_2_1: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> R" shows "([c, a], [b, c]) \<in> lprod R"
assert_def: fixes p :: "'c"   and q :: "'c" shows "{. p .} q \<equiv> p \<sqinter> q"
sets: shows "{{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} |i. i < n} = (\<lambda>i. {dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs}) ` {..<n}"
assms_3: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and L :: "char list set"   and u' :: "sterm" assumes "l \<in> dom f"   and "Obj f T = t"   and "Obj f T => Obj f' T"   and "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u' = \<sigma>[s,p] t'')"   and "z = Obj (f'(l \<mapsto> u')) T" shows "Q"
prj1_in_hom': fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.in_hom C (prj1 f g) (partial_magma.dom C (prj0 f g)) (partial_magma.dom C f)"
top: shows "UNIV \<in> sigma_sets UNIV (range greaterThan)"
mult_1: fixes a :: "'a" shows "(1::'a) * a = a"
trace_simps_6: fixes sig :: "'i signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"