hcju/isabelleps huggingface dataset:
awalk_def: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b" shows "pre_digraph.awalk G u p v \<equiv> u \<in> verts G \<and> set p \<subseteq> arcs G \<and> pre_digraph.cas G u p v"
measure_distr: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and S :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "S \<in> sets N" shows "Sigma_Algebra.measure (distr M N f) S = Sigma_Algebra.measure M (f -` S \<inter> space M)"
foundation7': fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> not (\<upsilon> x)) = (\<tau> |\<noteq> \<upsilon> x)"
upper_dim_implies_not_one_side_two_sides: assumes "UpperDimAxiom" shows "\<forall>A B X Y. \<not> Col X A B \<and> \<not> Col Y A B \<and> \<not> A B OS X Y \<longrightarrow> A B TS X Y"
equivE2_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<longleftrightarrow>. q" shows "A \<turnstile> q .\<longrightarrow>. p"
group_hom_def: fixes G :: "('f, 'g) monoid_scheme"   and H :: "('h, 'i) monoid_scheme"   and h :: "'f \<Rightarrow> 'h" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
hn_refine_post_other: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>'' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and a :: "'b nres"   and \<Gamma>' :: "assn" assumes "hn_refine \<Gamma> c \<Gamma>'' R a" shows "hn_refine \<Gamma> c (\<Gamma>' \<or>\<^sub>A \<Gamma>'') R a"
consistence_error: fixes j :: "nat" assumes "t (Suc j) \<le> T" shows "dist (x t (Suc j)) (discrete_evolution \<psi> (t (Suc j)) (t j) (x t j)) \<le> B * stepsize j ^ (p + 1)"
simps_1: fixes C :: "String.literal" shows "class_type_of' (Class C) = \<lfloor>C\<rfloor>"
compl_unique: fixes x :: "'a"   and y :: "'a" assumes "x AND y = (0::'a)"   and "x OR y = - (1::'a)" shows "NOT x = y"
col_out2_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and AA :: "'p"   and CC :: "'p" assumes "Col A B C"   and "B Out A AA"   and "B Out C CC" shows "Col AA B CC"
nat_not_finite: fixes R :: "bool" assumes "finite UNIV" shows "R"
usubstappp_choice_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<beta> :: "game" assumes "snd (usubstappp \<sigma> U (\<alpha> \<union>\<union> \<beta>)) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg \<and> snd (usubstappp \<sigma> U \<beta>) \<noteq> undefg"
test_mult: fixes x :: "'a"   and y :: "'a" shows "at (at (at (at x) \<cdot> at (at y))) = at (at y) \<cdot> at (at x)"
that: assumes "typeof_addr h (addr_of_sys_xcpt C) = \<lfloor>Class_type C\<rfloor>"   and "P \<turnstile> C \<preceq>\<^sup>* Throwable" shows "thesis"
trapped_forward_le: fixes V :: "'a \<Rightarrow> real"   and x :: "'a"   and g :: "'a \<Rightarrow> real"   and V' :: "'a \<Rightarrow> 'a \<Rightarrow> real" assumes "V x \<le> 0"   and "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"   and "\<And>x. (V has_derivative V' x) (at x)"   and "\<And>s. s \<in> existence_ivl0 x \<inter> {0..} \<Longrightarrow> V' (flow0 x s) (f (flow0 x s)) \<le> g (flow0 x s) * V (flow0 x s)" shows "trapped_forward x {x. V x \<le> 0}"
simps_5: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "push_forall (Or \<phi> \<psi>) = or (push_forall \<phi>) (push_forall \<psi>)"
nn_integral_less: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integral\<^sup>N M f \<noteq> \<infinity>"   and "AE x in M. f x \<le> g x"   and "\<not> (AE x in M. g x \<le> f x)" shows "integral\<^sup>N M f < integral\<^sup>N M g"
last_message_conv_if: fixes xs :: "'b message_af list"   and m :: "'b message_af" shows "(last_message xs = m) = (if m = NoMsg then \<forall>i<length xs. xs ! i = NoMsg else \<exists>i<length xs. xs ! i = m \<and> (\<forall>j<length xs. i < j \<longrightarrow> xs ! j = NoMsg))"
conj_residuatedI1: fixes f :: "'a \<Rightarrow> 'a" assumes "\<exists>g. conjugation_pair f g" shows "residuated f"
inv_image_def: fixes r :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "inv_image r f = _Collect (x, y) ((f x, f y) \<in> r)"
semiring_normalization_rules_4: fixes m :: "'a" shows "m + m = ((1::'a) + (1::'a)) * m"
right_idem: fixes a :: "'b"   and b :: "'b" shows "a + b + b = a + b"
grey_protects_white_induct: fixes g :: "'e"   and w :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "(g grey_protects_white w) s"   and "\<And>x. grey x s \<Longrightarrow> P x x"   and "\<And>x y z. \<lbrakk>(x has_white_path_to y) s; P x y; (y points_to z) s; white z s\<rbrakk> \<Longrightarrow> P x z" shows "P g w"
insert_5: shows "sum ((^) 2) (insert x X) = sum ((^) 2) Y"
lt_\<omega>_imp_times_minus_left: fixes m :: "hmultiset"   and n :: "hmultiset"   and l :: "hmultiset" assumes "m < \<omega>"   and "n < \<omega>" shows "(m - n) * l = m * l - n * l"
mignotte_helper_coeff_int': fixes lst :: "complex list"   and i :: "nat" shows "cmod (coeff_int (\<Prod>a\<leftarrow>lst. [:- a, 1:]) (int i)) \<le> real (length lst - 1 choose i) * (\<Prod>a\<leftarrow>lst. max 1 (cmod a)) + real (min i 1 * (length lst - 1 choose nat (int i - 1)))"
interchange_columns_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and n :: "'c"   and m :: "'c" shows "interchange_columns A n m = (\<chi>i j. if j = n then A $ i $ m else if j = m then A $ i $ n else A $ i $ j)"
indpt2: shows "\<not> module.lin_dep class_ring (module_vec TYPE('a) (card {i. i < n \<and> i \<in> I})) (set (cols (submatrix (submatrix A UNIV J) I UNIV)))"
Suc_0_eq_Suc_0_mod_iff: fixes n :: "nat" shows "(Suc 0 = Suc 0 mod n) = (n \<noteq> Suc 0)"
conditional_mutual_information_eq_conditional_entropy: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "\<I>( X ; X | Y ) = \<H>(X | Y)"
leg_simps_3: shows "C.arr Cod\<mu>_Cod\<nu>.leg1"
OrdNotEqP_E: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and A :: "fm" assumes "H \<turnstile> x EQ y" shows "insert (x NEQ y) H \<turnstile> A"
abs_nchotomy: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A" shows "\<exists>xs x X. good X \<and> A = Abs xs x X"
us_faithful: fixes x :: "'a"   and y :: "'a" assumes "us x \<le> us y" shows "x \<le> y"
dim_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "dim S \<le> dim T"
simps_2: fixes x :: "'c" shows "remdups_adj [x] = [x]"
Red_fv: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "fv e' \<subseteq> fv e"
simps_2: fixes x21 :: "'i"   and x22 :: "'j"   and y21 :: "'i"   and y22 :: "'j" shows "(IO x21 x22 = IO y21 y22) = (x21 = y21 \<and> x22 = y22)"
open_closed_invariant: fixes A :: "(('e \<Rightarrow> 'f) \<times> 'g, 'h) automaton"   and I :: "'h \<Rightarrow> bool"   and P :: "('e \<Rightarrow> 'f) \<times> 'g \<Rightarrow> bool"   and U :: "('e \<Rightarrow> 'f) \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and J :: "'e set" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "subreachable A U J"   and "\<And>\<sigma> \<sigma>' s. \<lbrakk>\<forall>j\<in>J. \<sigma>' j = \<sigma> j; P (\<sigma>', s)\<rbrakk> \<Longrightarrow> P (\<sigma>, s)" shows "A \<Turnstile> (act I, U \<rightarrow>) P"
iwf_length_inPorts: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and "is" :: "nat list" assumes "iwf fc t ent"   and "is \<in> it_paths t" shows "length (iAnts (tree_at t is)) \<le> length (inPorts' (iNodeOf (tree_at t is)))"
esize_strict_mono: fixes x :: "'a"   and y :: "'a" assumes "esize x \<noteq> \<infinity>"   and "x < y" shows "esize x < esize y"
circle_y_def: fixes t :: "real" shows "circle_y t = sqrt (1 / 4 - t * t)"
multLess_def: fixes M :: "(Literal \<times> bool) list" shows "multLess M = inv_image (mult (precedesOrder (elements M))) (\<lambda>x. mset (remdups (oppositeLiteralList x)))"
assms_13: shows "isUIP (opposite (getCl state)) (getC state) (getM state)"
rel_set_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "rel_set (BNF_Def.Grp A f) = BNF_Def.Grp {X. X \<subseteq> A} ((`) f)"
act_distrib: fixes p :: "'b"   and q :: "'b"   and x :: "'a" assumes "D p q \<and> Dla x (p \<oplus> q)" shows "\<alpha> x (p \<oplus> q) = \<alpha> x p \<oplus> \<alpha> x q"
rationalizable_binary_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "rationalizable_binary_on A f = (\<forall>B\<subseteq>A. f B = {y \<in> B. \<forall>x\<in>B. y \<in> f {x, y}})"
properties_loops_ge_mu: fixes i :: "nat"   and j :: "nat" assumes "seq (i + j) = seq i"   and "0 < j" shows "mu \<le> i"
sum_upto_ln_conv_sum_upto_mangoldt: fixes x :: "real" shows "sum_upto (\<lambda>n. ln (real n)) x = sum_upto (\<lambda>n. mangoldt n * real (nat \<lfloor>x / real n\<rfloor>)) x"
ipurge_tr_aux_cons: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and x :: "'e"   and xs :: "'e list" shows "ipurge_tr_aux I D U (x # xs) = (if \<exists>u\<in>U. (u, D x) \<in> I then ipurge_tr_aux I D (insert (D x) U) xs else x # ipurge_tr_aux I D U xs)"
vcn: shows "vecsqnorm \<equiv> \<lambda>A. (fst A)\<^sup>2 + (snd A)\<^sup>2"
normal_subgroup_intersect: fixes M :: "'a set"   and N :: "'a set" assumes "M \<lhd> G"   and "N \<lhd> G" shows "M \<inter> N \<lhd> G"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "float" shows "real_of_float (float_plus_down x xa xb) = plus_down x (real_of_float xa) (real_of_float xb)"
one_step: fixes g :: "nat \<Rightarrow> nat" assumes "g \<in> local.F_strict" shows "Determinant.det (submatrix A UNIV (g ` {0..<n})) * Determinant.det (submatrix B (g ` {0..<n}) UNIV) = (\<Sum>(x, y)\<in>local.Z_good g. local.weight x y)"
dim_eq: fixes S :: "'c set"   and X :: "'c set" assumes "real_vector_space_on S"   and "X \<subseteq> S" shows "vector_space_on.dim S (*\<^sub>R) X = dim X"
inv_left: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "bijection f" shows "inv f (f a) = a"
in_set_butlastD: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
perm_setP: fixes as :: "'c list"   and bs :: "'c list"   and P :: "'c set \<Rightarrow> bool" assumes "as <~~> bs"   and "P (set as)" shows "P (set bs)"
mayDiverge_Seq_R: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>*t s'"   and "mayDiverge c2 s'" shows "mayDiverge (c1 ;; c2) s"
pochhammer_of_int: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "int"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "comm_semiring_1.pochhammer times one plus zero (ring_1.of_int one plus zero minus x) n = ring_1.of_int one plus zero minus (pochhammer x n)"
in_hom_def: fixes f :: "'b"   and a :: "'b"   and b :: "'b" shows "C'.in_hom f a b \<equiv> a \<in> C'.domains f \<and> b \<in> C'.codomains f"
vars_subset_if_in_strand_ik2: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
entailment_and_implicates: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" assumes "entails_formula S1 S2" shows "implicates S2 \<subseteq> implicates S1"
collapse: fixes prod :: "'k \<times> 'l" shows "(fst prod, snd prod) = prod"
diff_eq1: fixes V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x - y = x + - y"
not_all: fixes P :: "'a \<Rightarrow> bool" shows "(\<not> (\<forall>x. P x)) = (\<exists>x. \<not> P x)"
denormal: fixes x :: "('e, 'f) IEEE.float" assumes "is_denormal x"   and "valof x = real_of_int (denormal_mantissa x) * 2 powr real_of_int (denormal_exponent TYPE(('e, 'f) IEEE.float))" shows "P x"
lem: shows "\<forall>P A. (\<exists>x\<in>A. \<exists>y. P x y) = (\<exists>y. \<exists>x\<in>A. P x y)"
at_within_closed_interval_finite: fixes a :: "real"   and x :: "real"   and b :: "real"   and S :: "real set" assumes "a < x"   and "x < b"   and "x \<notin> S"   and "finite S" shows "at x within {a..b} - S = at x"
intros_2: fixes xs :: "'c llist"   and x :: "'c" assumes "lfinite xs" shows "lfinite (LCons x xs)"
continuous_on_mult_const: fixes s :: "'a set"   and c :: "'a" shows "continuous_on s ((*) c)"
exists_snapshot_for_all_p: fixes t :: "('a, 'b, 'c) event list"   and p :: "'a" assumes "trace init t final" shows "\<exists>i. \<not> ps (S t i) p \<noteq> None \<and> ps (S t (Suc i)) p \<noteq> None"
keep_drop_sels_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "snd (R\<^sup>k) = R"
b'_Basis: fixes i :: "'a" assumes "i \<in> Basis" shows "b' i \<in> {..<n}"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "\<And>a. partial_magma.ide A a \<Longrightarrow> category.inverse_arrows B (\<tau> a) (\<tau>' a)" shows "inverse_transformations_axioms A B \<tau> \<tau>'"
reciprocal_degree: fixes P :: "'a poly" shows "reciprocal_poly (degree P) P = reflect_poly P"
dual_complete_lattice: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.complete_lattice Sup Inf sup (\<lambda>x y. less_eq y x) (\<lambda>x y. less y x) inf top bot"
mul_ext_stri_imp_nstri: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and as :: "'b list"   and bs :: "'b list" assumes "fst (mul_ext f as bs)" shows "snd (mul_ext f as bs)"
wf_TC_less: shows "wf {(x, y). x \<sqsubset> y}"
splitAt_no_ram: fixes ram :: "'a"   and vs :: "'a list" assumes "ram \<notin> set vs" shows "splitAt ram vs = (vs, [])"
set_eq_iff_mset_eq_distinct: fixes x :: "'a list"   and y :: "'a list" assumes "distinct x"   and "distinct y" shows "(set x = set y) = (mset x = mset y)"
WhileFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and n :: "nat" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> =n\<Rightarrow> Normal s"
a1: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>'. p = ( \<Phi>' \<Rightarrow>* \<Psi>' \<oplus> F \<nabla> [x].A)"
scale_sum_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "module scale" shows "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))"
IH: assumes "\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> f x" shows "(\<Sum>x\<leftarrow>xs. ennreal (f x)) = ennreal (sum_list (map f xs))"
\<tau>move1ThrowNull: fixes P :: "'a prog"   and h :: "'heap" shows "\<tau>move1 P h (throw null)"
pull_negpos_def: fixes x :: "wall"   and y :: "wall" shows "pull_negpos x y \<equiv> x = basic [brick.under] \<circ> basic [over] \<and> y = basic [vert, vert] \<circ> basic [vert, vert]"
FNC: fixes l1 :: "'a list"   and l2 :: "'a list"   and \<sigma> :: "'b" assumes "l0 = l1 @ l2"   and "I l1 l2 \<sigma>"   and "\<not> c \<sigma>" shows "P \<sigma>"
intros_17: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es1 :: "b_e list"   and es2 :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es1 : tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es2 : tn _> tm" shows "\<C> \<turnstile> [b_e.If tf es1 es2] : tn @ [T_i32] _> tm"
zer: fixes b :: "complex"   and c :: "complex" assumes "closed_segment b c \<subseteq> S" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
assms_5: shows "t\<^sub>i \<in> set T"
cut_ge_Max_empty: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I < t" shows "I \<down>\<ge> t = {}"
fls_left_inverse_eq_inverse: fixes f :: "'a fls" shows "fls_left_inverse f (inverse (fls_nth f (fls_subdegree f))) = inverse f"
ccw_half_segments_of_aform_2: fixes xi :: "real \<times> real"   and xj :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and c :: "real \<times> real"   and d :: "real \<times> real" assumes "(xi, xj) \<in> set (half_segments_of_aform X)"   and "(c, d) \<in> set (half_segments_of_aform X)" shows "ccw xi xj d"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv t) S"   and "wfrestrictedvars\<^sub>s\<^sub>t S' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> (V \<union> fv t)" shows "wf\<^sub>s\<^sub>t (V \<union> fv t) (S @ S')"
tr: shows "(s, a, s') \<in> automaton.trans (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o)"
imp_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "imp \<phi> \<psi> \<equiv> dis (neg \<phi>) \<psi>"
insert_lclss_of_state_3: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "insert (C, Old) (lclss_of_state (N, P, Q)) = lclss_of_state (N, P, Q \<union> {C})"
set_count_size_min: fixes n :: "nat"   and A :: "'b multiset"   and a :: "'b" assumes "n \<le> count A a" shows "n \<le> size A"
snd_hyp_edge_at: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and "is" :: "nat list"   and n :: "nat"   and s :: "'c" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "snd (Solved_Task.hyp_edge_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions c is n s) = Solved_Task.hyp_edge_to anyP c is"
comp_fun_commute_axioms: shows "comp_fun_commute (add_mset \<circ> f)"
Unwinding_def: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec" shows "Unwinding SES \<V> \<equiv> SES_valid SES \<and> isViewOn \<V> E\<^bsub>SES\<^esub>"
gb_red_aux_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "dickson_grading d" shows "dgrad_p_set_le d (set (gb_red_aux bs ps)) (args_to_set ([], bs, ps))"
sum_list_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ns :: "'a list" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(monoid_add.sum_list plus zero ns = zero) = (\<forall>n\<in>set ns. n = zero)"
simps_3: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and X :: "char list set" shows "bury (c\<^sub>1;; c\<^sub>2) X = bury c\<^sub>1 (L c\<^sub>2 X);; bury c\<^sub>2 X"
incrementI: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and h :: "real star" assumes "f NSdifferentiable x" shows "increment f x h = (*f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
restrict_edges_def: fixes g :: "'b ugraph"   and E :: "('b \<times> 'b) set" shows "restrict_edges g E \<equiv> graph (nodes g) (edges g \<inter> (E \<union> E\<inverse>))"
elements_mat_mat_of_row: fixes v :: "'b vec" shows "elements_mat (mat_of_row v) = set\<^sub>v v"
fls_divide_regpart: fixes f :: "'a fls"   and g :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g" shows "fls_regpart (f / g) = fls_regpart f / fls_regpart g"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F F F"
hoare_wp: fixes x :: "'b"   and q :: "'b Assertion" shows "hoare (wp x q) x q"
valuations_eqTr1: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "Vr K v = Vr K v'"   and "\<forall>x\<in>carrier (Vr K v). v x = v' x" shows "v = v'"
set_mult_commute: fixes N :: "'a set"   and x :: "'a set"   and y :: "'a set" assumes "N \<subseteq> carrier G"   and "x \<in> rcosets N"   and "y \<in> rcosets N" shows "x <#> y = y <#> x"
deg_pm_minus_id_iff: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and s :: "'a \<Rightarrow>\<^sub>0 nat" shows "(deg_pm (t - s) = deg_pm t) = (keys t \<inter> keys s = {})"
set_cases: fixes e :: "'g"   and a :: "'g list" assumes "e \<in> set a" obtains z2 :: "'g list" where "a = e # z2__"   | z1 :: "'g" and z2 :: "'g list" where "a = z1 # z2" and "e \<in> set z2"
markedElementsToNonMemberAreAllMarkedElements: fixes e :: "'b"   and M :: "('b \<times> bool) list" assumes "e \<notin> set (elements M)" shows "markedElementsTo e M = markedElements M"
Rats_add: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a + b \<in> \<rat>"
Mapping_inject: fixes x :: "'b \<Rightarrow> 'c option"   and y :: "'b \<Rightarrow> 'c option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Mapping x = Mapping y) = (x = y)"
ESem_simps_as_defined_3: fixes x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> Var x \<rbrakk>\<^bsub>\<rho>\<^esub> = \<rho> x"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1))))) \<and> t2 = snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2)))))"
region_compatible_suntil: fixes \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and x :: "('s \<times> ('c \<Rightarrow> real)) stream"   and \<psi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool" assumes "pred_stream (\<lambda>s. \<phi> (reps (abss s)) = \<phi> s) x"   and "pred_stream (\<lambda>s. \<psi> (reps (abss s)) = \<psi> s) x" shows "(holds (\<lambda>x. \<phi> (reps x)) suntil holds (\<lambda>x. \<psi> (reps x))) (smap abss x) = (holds \<phi> suntil holds \<psi>) x"
ideal_ele_sumTr1: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<plusminus>\<^bsub>R\<^esub> b \<in> I"   and "a \<in> I" shows "b \<in> I"