hcju/isabelleps huggingface dataset:
semiring_simprules_12: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z) = y \<oplus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> z)"
simps_3: fixes b :: "'b"   and ts :: "'a FOL_Fitting.term list" shows "params (Pred b ts) = paramsts ts"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "dual_wellorder.Max A \<le> a"
that: fixes m :: "nat" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "m cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "thesis"
SMP_D'': fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "t \<in> SMP (set M)"   and "finite_SMP_representation arity Ana \<Gamma> M" shows "\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta>"
simps_2: fixes T :: "Unary_PCF.type"   and U :: "Unary_PCF.type" shows "constant0 (T \<rightarrow> U) = \<Lambda>\<langle>T\<rangle> constant0 U"
prefix_fininfI_empty: fixes w :: "'a stream" shows "[] \<le>\<^sub>F\<^sub>I w"
ub'_def: shows "ub' = sum_upto (\<lambda>n. real n powr - \<alpha>)"
max_Var_floatariths_ode_d_expr: fixes d :: "nat"   and z :: "nat"   and m :: "nat" assumes "max_Var_floatariths ode_e \<le> d"   and "0 < d"   and "3 * d \<le> z" shows "max_Var_floatariths (ode_d_expr d m) \<le> z"
Sup_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Squnion> (insert a A) = a \<squnion> \<Squnion> A"
return_edge_determines_call_and_sum_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f" obtains a' :: "'edge" and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r' :: "'ret" and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" where "a \<in> get_return_edges a'" and "valid_edge a'" and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'" and "CFG_node (sourcenode a') s-p\<rightarrow>\<^bsub>sum\<^esub> CFG_node (targetnode a)"
card_UNIV_finite_4_def: shows "card_UNIV_class.card_UNIV = Phantom(Enum.finite_4) 4"
divergent_def: fixes \<omega> :: "('c \<Rightarrow> real) stream" shows "divergent \<omega> \<equiv> \<forall>t. \<exists>n. t < dur \<omega> n"
thetaIfZO_ZOretr: shows "thetaIfZO \<subseteq> ZOretr (thetaIfZO \<union> ZObis)"
assms_3: shows "formulaFalse F0 (elements (getM state))"
emsapss_decode_help8_def: fixes mHash :: "bit list"   and DB_zero :: "bit list"   and H :: "bit list" shows "emsapss_decode_help8 mHash DB_zero H = emsapss_decode_help9 mHash (generate_salt DB_zero) H"
monic_imp_div_mod_int_poly_degree2: fixes u :: "'a poly"   and p :: "'a poly" assumes "monic u"   and "0 < degree u" shows "\<exists>q r. p = q * u + r \<and> degree r < degree u"
dom_map_of_conv_domA: fixes \<Gamma> :: "('a \<times> 'b) list" shows "dom (map_of \<Gamma>) = domA \<Gamma>"
poly_eq_poly_eval: fixes x :: "'d"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and a :: "'c" shows "poly (poly_of_pm x p) a = poly_eval (\<lambda>y. a when y = x) p"
multiplicity_times_unit_right: fixes c :: "'a"   and p :: "'a"   and x :: "'a" assumes "is_unit c" shows "multiplicity p (c * x) = multiplicity p x"
ladder_stepdown_last_j: fixes L :: "(nat \<times> nat \<times> nat) list" assumes "1 < length L" shows "ladder_last_j (ladder_stepdown L) = ladder_last_j L"
tr_add_context_s: fixes c :: "'f list multiset"   and e :: "('g, 'h) label"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and ce :: "'f list multiset" assumes "(c, e, c') \<in> tr fg"   and "mon_e fg e \<inter> mon_c fg ce = {}" shows "(c + ce, e, c' + ce) \<in> tr fg"
strip_step': fixes S :: "'av Abs_State.st option"   and c :: "'av Abs_State.st option acom" shows "strip (step' S c) = strip c"
inv4: shows "access root path user\<^sub>1 {Writable} = None"
Nil_is_append_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(\<epsilon> = xs \<cdot> ys) = (xs = \<epsilon> \<and> ys = \<epsilon>)"
Meet_Un: fixes A :: "'b set"   and B :: "'b set" shows "\<Sqinter>(A \<union> B) = \<Sqinter>A \<sqinter> \<Sqinter>B"
ldfix: shows "LeftDerivationFix (\<alpha> @ \<delta>) i D j (\<beta> @ \<delta>')"
that: fixes as\<^sub>1 :: "('a \<times> bool) list"   and as\<^sub>2 :: "('a \<times> bool) list" assumes "as = as\<^sub>1 @ as\<^sub>2"   and "as\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)"   and "as\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)" shows "thesis"
outside_convex: fixes S :: "'b set" assumes "convex S" shows "outside S = - S"
simps_2: fixes m :: "nat" shows "geod 0 (Suc m) = y0"
subgroups_of_size_def: fixes p :: "nat" shows "subgroups_of_size p = {H. subgroup H G \<and> card H = p}"
cases: fixes a1 :: "('a, 'b) tm"   and a2 :: "('a, 'b) tm" assumes "sub a1 a2" obtains s :: "('a, 'b) tm" where "a1 = s__" and "a2 = s__"   | s :: "('a, 'b) tm" and t :: "('a, 'b) tm" and u :: "('a, 'b) tm" where "a1 = sa__" and "a2 = App u__ t__" and "sub sa__ t__"   | s :: "('a, 'b) tm" and t :: "('a, 'b) tm" and u :: "('a, 'b) tm" where "a1 = s" and "a2 = App t u" and "sub s t"
pre3: shows "r|\<sigma>3| -s (r \<down>l \<sigma>2 \<union> r \<down>l \<sigma>1 \<union> r \<down>m {#\<beta>#} \<union> r \<down>m {#\<alpha>#}) = {#}"
m_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier M1"   and "y \<in> carrier M1" shows "x \<oplus>\<^bsub>M1\<^esub> y = y \<oplus>\<^bsub>M1\<^esub> x"
in_real_itvl_get_bounds_tighten: fixes x :: "real_alg_3"   and n :: "nat" shows "real_of_3 x \<in>\<^sub>i get_itvl_3 ((tighten_bounds_3 ^^ n) x)"
seminorm_def: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" shows "seminorm V norm \<equiv> (\<forall>x. x \<in> V \<longrightarrow> 0 \<le> norm x) \<and> (\<forall>x a. x \<in> V \<longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> norm (x + y) \<le> norm x + norm y)"
Nmlize_Trg_2: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Trg t\<^bold>\<rfloor> = Trg t"
Nml_Nmlize_4: fixes t :: "'a local.term" assumes "Arr t" shows "Dom \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom t\<^bold>\<rfloor>"
IS_PURE_def: fixes P :: "('b \<times> 'd) set \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'b \<Rightarrow> assn" shows "IS_PURE P R \<equiv> is_pure R \<and> P (the_pure R)"
simps_12: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v2 :: "'b" shows "AExp_Lexorder.less_aexp_aux (Plus e1 e2) (V v2) = False"
case_prod_o_map_prod: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> 'd"   and g2 :: "'b \<Rightarrow> 'e" shows "(\<lambda>(x, y). f x y) \<circ> map_prod g1 g2 = (\<lambda>(l, r). f (g1 l) (g2 r))"
InvariantsVarsAfterApplyUnitPropagate: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantQCharacterization False (getQ state) (getF state) (getM state)"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl" shows "let state' = applyUnitPropagate state in InvariantVarsM (getM state') F0 Vbl \<and> InvariantVarsQ (getQ state') F0 Vbl"
iril: shows "0 \<le> real_of_rat (ir - il)"
legal_zpassign_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A = (\<forall>x. zhmset_of (tkbo_coefs.min_passign ground_heads_var \<delta> arity_sym wt_sym x) \<le> A x)"
rel_I1: fixes s :: "'d"   and U :: "'c set"   and x :: "'c set \<times> 'd" assumes "s \<in> \<O>\<^sub>Y U"   and "x \<in> stfx.class_of U s"   and "U \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" shows "stfx.rel (U, s) x"
front_tickFree_implies_tickFree: fixes t :: "'b event list"   and a :: "'b event" assumes "front_tickFree (t @ [a])" shows "tickFree t"
no_head_hd: fixes \<C> :: "'b list set"   and u :: "'b list"   and b :: "'b list" assumes "code \<C>"   and "u \<in> no_head \<C> b"   and "u \<noteq> \<epsilon>" shows "hd (Dec \<C> u) \<noteq> b"
Zero_fract_def: shows "0 = Fraction_Field.Fract (0::'b) (1::'b)"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
psi_convex1: fixes c :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and d :: "real \<times> real"   and u :: "real"   and v :: "real" assumes "psi c a b"   and "psi d a b"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "psi (u *\<^sub>R c + v *\<^sub>R d) a b"
coeffs_reflect_poly: fixes p :: "'b poly" shows "coeffs (reflect_poly p) = rev (dropWhile ((=) (0::'b)) (coeffs p))"
maximalD2: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set"   and y :: "'c" assumes "x \<in> maximal f S"   and "y \<in> S" shows "f y \<le> f x"
ev_holds_sset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "ev (holds P) xs = (\<exists>x\<in>sset xs. P x)"
addrow_carrier: fixes ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat"   and n :: "nat"   and nc :: "nat" shows "(mat_addrow_gen ad mul a k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
stream_enc_unique: fixes \<Sigma> :: "'b list"   and i :: "nat"   and I :: "(nat + nat set) list"   and w :: "'b list" assumes "formula \<Sigma>"   and "i < length I"   and "\<exists>p. I ! i = Inl p" shows "\<exists>!p. snd (formula.stream_enc \<Sigma> (w, I) !! p) ! i"
simps_6: fixes x1 :: "ct list" shows "Bot \<noteq> TopType x1"
yun_factorization_hom: fixes hom :: "'a \<Rightarrow> 'b"   and f :: "'a poly" assumes "field_hom_0' hom" shows "yun_factorization gcd (map_poly hom f) = map_prod hom (map (\<lambda>(f, i). (map_poly hom f, i))) (yun_factorization gcd f)"
Collect_imp_eq: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
simps_3: fixes a :: "MIR.num"   and b :: "MIR.num" shows "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
holomorphic_on_divide: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "g holomorphic_on s"   and "\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0" shows "(\<lambda>z. f z / g z) holomorphic_on s"
IfTrue: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c1 :: "com"   and c2 :: "com"   and cs :: "com list" assumes "b s" shows "((IF b THEN c1 ELSE c2) # cs, s) \<rightarrow> (c1 # cs, s)"
simps_6: shows "normalize_match (MatchNot MatchAny) = []"
rank_of_eq_card_basis_in: fixes B :: "'a set" assumes "basis_in \<E> B" shows "rank_of \<E> = card B"
idle_squareI: fixes v :: "state \<Rightarrow> 'b"   and s :: "state"   and t :: "state"   and A :: "state \<times> state \<Rightarrow> bool" assumes "(s, t) \<Turnstile> unchanged v" shows "(s, t) \<Turnstile> [A]_v"
mat_mult_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and i :: "'b"   and j :: "'d" shows "(f \<otimes> g) i j \<equiv> \<Sum> {f i k \<cdot> g k j |k. k \<in> UNIV}"
cong: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "comm_monoid_add.sum plus zero \<equiv> comm_monoid_add.sum plus zero"
induct: fixes P :: "'g list \<Rightarrow> 'g list \<Rightarrow> bool"   and a0 :: "'g list"   and a1 :: "'g list" assumes "\<And>ys. P [] ys"   and "\<And>x xs ys. P ys xs \<Longrightarrow> P (x # xs) ys" shows "P a0 a1"
app_iso_inv_subgraph: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and H :: "('e, 'f) pre_digraph" assumes "pre_digraph.digraph_isomorphism G hom"   and "subgraph H G" shows "app_iso (pre_digraph.inv_iso G hom) (app_iso hom H) = H"
mult_commute: fixes x :: "nat discr Box"   and y :: "nat discr Box" shows "x * y = y * x"
inc_inc_dot: fixes a :: "nat"   and b :: "nat"   and x :: "item" shows "inc_dot a (inc_dot b x) = inc_dot (a + b) x"
pow_generator_eq_iff_cong: fixes G :: "('c, 'd) cyclic_group_scheme"   and x :: "nat"   and y :: "nat" assumes "cyclic_group G"   and "finite (carrier G)" shows "(\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> x = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> y) = [x = y] (mod order G)"
cases: fixes x :: "(Vars.var \<times> exp) list" obtains "x = []"   | v :: "Vars.var" and e :: "exp" and \<Gamma> :: "(Vars.var \<times> exp) list" where "x = (v, e) # \<Gamma>"
to_list_correct: fixes e :: "'env" shows "map_of (to_list e) = get e"
one_time_pad: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "map_spmf (([\<oplus>]) xs) (spmf_of_set (nlists UNIV n)) = spmf_of_set (nlists UNIV n)"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
pinduct: fixes a0 :: "nat \<times> nat \<times> 'd"   and a1 :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and a2 :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and P :: "nat \<times> nat \<times> 'd \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> bool" assumes "gb_schema_aux_dom (a0, a1, a2)"   and "\<And>data bs ps. \<lbrakk>gb_schema_aux_dom (data, bs, ps); \<And>x xa xb xc xd xe y. \<lbrakk>ps \<noteq> []; x = sel gs bs ps (snd data); xa = ps -- x; xb = compl gs bs xa x (snd data); xc = fst data - count_const_lt_components (fst xb); xc \<noteq> 0; xd = add_indices xb (snd data); (xe, y) = xd\<rbrakk> \<Longrightarrow> P (xc, y) (ab gs bs xe y) (ap gs bs xa xe y)\<rbrakk> \<Longrightarrow> P data bs ps" shows "P a0 a1 a2"
WT_converter_map_converter: fixes f :: "'e \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'f"   and \<I> :: "('e, 'f) \<I>"   and f' :: "'c \<Rightarrow> 'g"   and g' :: "'h \<Rightarrow> 'd"   and \<I>' :: "('g, 'h) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "map_\<I> (inv f) (inv g) \<I>, map_\<I> f' g' \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "inj f"   and "surj g" shows "\<I>, \<I>' \<turnstile>\<^sub>C map_converter f g f' g' conv \<surd>"
int_neq_iff: fixes w :: "int"   and z :: "int" shows "(w \<noteq> z) = (w < z) \<or> z < w"
lfinite_lfilter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "lfinite (lfilter P xs) = (lfinite xs \<or> finite {n. enat n < llength xs \<and> P (lnth xs n)})"
isSurjAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'c, 'd, 'i) Recursion.model_scheme" shows "isSurjAll h hA MOD \<equiv> isSurj h MOD \<and> isSurjAbs hA MOD"
f: shows "f \<in> [UU]\<^bsup>2\<^esup> \<rightarrow> {..<Suc (Suc 0)}"
addO_Z: fixes n :: "Ord" shows "addO Z n = n"
circ_loop_fixpoint: fixes y :: "'a"   and z :: "'a" shows "y \<sqinter> (top \<sqinter> z) \<squnion> z = top \<sqinter> z"
triangle: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src g = trg f" shows "(g \<star> \<l>[f]) \<cdot> \<a>[g, src g, f] = \<r>[g] \<star> f"
sorted_a_le: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and x :: "'a extended"   and l :: "nat"   and y :: "'a extended"   and u :: "nat" assumes "integrate_insert_commute M a s"   and "idx s x = Inr l"   and "idx s y = Inr u" shows "(l < u) = (a x < a y)"
map_optionT_def: shows "map_optionT = map_optionT' undefined"
app_strict: fixes x :: "'d" shows "\<bottom> x = \<bottom>"
order_sym_trans: fixes y :: "'a set"   and x :: "'a set"   and z :: "'a set" assumes "y \<sqsubseteq>\<^sup>s x"   and "Q.sym y z"   and "x \<in> Q"   and "y \<in> Q"   and "z \<in> Q" shows "z \<sqsubseteq>\<^sup>s x"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(PWt x1 = PWt y1) = (x1 = y1)"
coboundedI: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "a \<in> A" shows "ordered_powerprod_lin.Min A \<preceq> a"
listToFun_futz_rev: fixes M :: "'a \<Rightarrow> 'b"   and xs :: "('a \<times> 'b list) list" assumes "\<And>x. M x \<in> (if x \<in> fst ` set xs then {uu_. \<exists>y ys. uu_ = y \<and> (x, ys) \<in> set xs \<and> y \<in> set ys} else {undefined})"   and "distinct (map fst xs)" shows "M \<in> set (listToFun xs)"
integral_Dirichlet_kernel_half: fixes n :: "nat" shows "integral\<^sup>L (lebesgue_on {0..pi}) (Dirichlet_kernel n) = pi / 2"
Subset_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> t SUBS u IFF t' SUBS u'"
subring_props_2: fixes K :: "'a set" assumes "subfield K R" shows "\<zero> \<in> K"
floor_eq_iff: fixes x :: "'d"   and a :: "int" shows "(\<lfloor>x\<rfloor> = a) = (of_int a \<le> x \<and> x < of_int a + (1::'d))"
finite_ksimplexes: fixes p :: "nat"   and n :: "nat" shows "finite {s. ksimplex p n s}"
map_pmf_random_bst_eq_rqs_cost: fixes A :: "'a set" assumes "finite A" shows "map_pmf ipl (random_bst A) = rqs_cost (card A)"
Units_l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<otimes> y = x \<otimes> z) = (y = z)"
Inf_trans_lower: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" assumes "t \<in> S"   and "\<forall>u\<in>S. \<forall>P. sound P \<longrightarrow> sound (u P)" shows "le_trans (Inf_trans S) t"
defined: shows "f ` d12.src.carrier \<subseteq> d12.dest.carrier"
set_pmf_not_empty: fixes M :: "'a pmf" shows "set_pmf M \<noteq> {}"
assms: shows "X \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
f_Exec_Stream_expand_shrink_last_map_nth_eq_f_Exec_Comp: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and f :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "0 < k"   and "n < length xs" shows "map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) \<div>\<^sub>l k ! n = f (f_Exec_Comp trans_fun (xs \<down> Suc n \<odot> k) c)"
distinct_sort: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "distinct (sort_key f xs) = distinct xs"
map_ascii_of_idem: fixes str :: "char list" assumes "is_ascii str" shows "map String.ascii_of str = str"
trss_stack_comp_s: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and r :: "'a list" assumes "((s, c), e, s', c') \<in> trss fg" shows "((s @ r, c), e, s' @ r, c') \<in> trss fg"
add_left_imp_eq: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
lookup_AList_delete: fixes k' :: "'a"   and k :: "'a"   and al :: "('a \<times> 'b) list" assumes "k' \<noteq> k" shows "lookup (\<lambda>x. x = k) al = lookup (\<lambda>x. x = k) (AList.delete k' al)"
simps_3: fixes e\<^sub>1 :: "'a allegorical_term"   and e\<^sub>2 :: "'a allegorical_term" shows "translation (A_Cmp e\<^sub>1 e\<^sub>2) = (let G\<^sub>1 = translation e\<^sub>1; G\<^sub>2 = translation e\<^sub>2 in graph_union (map_graph_fn G\<^sub>1 (\<lambda>x. if x = 0 then 0 else x + card (vertices G\<^sub>2) - 1)) (map_graph_fn G\<^sub>2 (\<lambda>x. if x = 0 then card (vertices G\<^sub>2) else x)))"
wellformed_init: fixes t :: "'t1" assumes "wf1 t" shows "wellformed1 (init1 t)"
simps_9: fixes x1 :: "'b"   and x41 :: "('a, 'b) process_term"   and x42 :: "('a, 'b) process_term" shows "VAR x1 \<noteq> CH x41 x42"
dist_decreases_open_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "x \<in> {a<--<b}" shows "dist c x < dist c a \<or> dist c x < dist c b"
injectivity: fixes x :: "'b"   and y :: "'b" assumes "to_fract x = to_fract y" shows "x = y"
inv: shows "2 * g * x - 2 * g * h = v * v"
map\<^sub>0_simps_1: fixes a :: "'d" assumes "D.obj a" shows "C.obj (F.map\<^sub>0 a)"
last_quotient: assumes "1 < length \<GG>" shows "last comp\<GG>.quotients = G Mod \<GG> ! (length \<GG> - 1 - 1)"
simps: fixes s :: "'s" shows "unfoldr s = (if has_next s then let (a, s') = next s in a # unfoldr s' else [])"
parent_child_rel_acyclic: fixes h :: "(_) heap" assumes "heap_is_wellformed h" shows "acyclic (parent_child_rel h)"
rep_eq: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "pmf (pmf_of_multiset M) = (\<lambda>x. real (count M x) / real (size M))"
prime_gt_0_int: fixes p :: "int" assumes "prime p" shows "0 < p"
elims_3: fixes x :: "Runs.var set option"   and xa :: "Runs.var set" assumes "\<not> in_progressS x xa" obtains S :: "Runs.var set" and S' :: "Runs.var set" where "x = Some S" and "xa = S'__" and "\<not> S'__ \<subseteq> S"   | S' :: "Runs.var set" where "x = None" and "xa = S'"
INF_eqI: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and x :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x (f i)"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y (f i)) \<Longrightarrow> less_eq y x" shows "Inf (f ` A) = x"
order_on_defs_5: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"