hcju/isabelleps huggingface dataset:
Res: fixes P :: "pi"   and Q :: "pi"   and a :: "name" assumes "(P, Q) \<in> Rel" shows "(<\<nu>a>P, <\<nu>a>Q) \<in> Rel"
qfree_subst\<^sub>2: fixes \<phi> :: "atom fm"   and l :: "nat"   and u :: "nat" assumes "nqfree \<phi>" shows "qfree (subst\<^sub>2 l u \<phi>)"
octo_trad: fixes x :: "real"   and y :: "real"   and z :: "real"   and w :: "real"   and u :: "real"   and v :: "real"   and q :: "real"   and g :: "real" shows "Octo x y z w u v q g = x *\<^sub>R octo_e0 + y *\<^sub>R octo_e1 + z *\<^sub>R octo_e2 + w *\<^sub>R octo_e3 + u *\<^sub>R octo_e4 + v *\<^sub>R octo_e5 + q *\<^sub>R octo_e6 + g *\<^sub>R octo_e7"
ab_diff_conv_add_uminus: fixes a :: "'q"   and b :: "'q" shows "a - b = a + - b"
lappend_simps_1: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lhd (lappend xs ys) = (if lnull xs then lhd ys else lhd xs)"
t2_exist: shows "t2 \<in> existence_ivl0 x"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "cod (HoVH f) = HoVH (VVV.cod f)"
eq_general: fixes B :: "'b set"   and A :: "'c set"   and h :: "'c \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> 'd"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "prod \<phi> A = prod \<gamma> B"
vec_coeff_append1: fixes i :: "nat"   and a :: "rat vec"   and b :: "rat vec" assumes "i \<in> {0..<dim_vec a}" shows "Abstract_Linear_Poly.coeff (vec_to_lpoly (a @\<^sub>v b)) i = a $ i"
dominating_chain: shows "landau_dominating_chain F (map get_fun gs)"
conversep_le_swap: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "(r \<le> s\<inverse>\<inverse>) = (r\<inverse>\<inverse> \<le> s)"
rel_spmf_restrict_relpI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "rel_spmf R p q"   and "pred_spmf P p"   and "pred_spmf Q q" shows "rel_spmf (R \<upharpoonleft> P \<otimes> Q) p q"
simps_2: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and ct :: "'f conditional_action"   and cts :: "'f conditional_action list" shows "cond_action_oks' s t (ct # cts) = (cond_action_ok' s t ct \<and> cond_action_oks' s t cts)"
sound_isTrue: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>" shows "isTrue \<phi>"
Gcd_gauss_int_def: shows "Gcd \<equiv> normalization_euclidean_semiring_class.Gcd"
lunit_char_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<l>[f] : trg f \<star> f \<Rightarrow> f\<guillemotright>"
edges_graph_def: fixes g :: "graph" shows "\<E> g \<equiv> \<Union> (\<E> ` \<F> g)"
assms_2: shows "coprime e ((P - 1) * (Q - 1))"
ts: shows "\<forall>i<length ts\<^sub>s\<^sub>b. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>s\<^sub>b ! i in ts ! i = (p, is, \<theta>, (), \<D>, \<O>, \<R>)"
signed_take_bit_eq_take_bit_minus: fixes n :: "nat"   and k :: "int" shows "signed_take_bit n k = take_bit (Suc n) k - 2 ^ Suc n * of_bool (bit k n)"
listset_ConsD: fixes x :: "'b"   and xs :: "'b list"   and A :: "'b set"   and As :: "'b set list" assumes "x # xs \<in> listset (A # As)" shows "x \<in> A \<and> xs \<in> listset As"
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n"   and "n \<le> p" shows "sum g {m..<n} + sum g {n..<p} = sum g {m..<p}"
f_Pi: shows "f \<in> A \<rightarrow> carrier G"
assms_1: fixes x :: "real" assumes "x \<in> snd ` set xs" shows "0 < x"
comp_fun_commute_restrict: shows "\<forall>y\<in>A. \<forall>x\<in>A. \<forall>z\<in>A. f y (f x z) = f x (f y z)"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs H" shows "rev_H' a \<in> arcs H"
get_owner_document_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes"   and "known_ptrs h"   and "type_wf h" shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr) \<rightarrow>\<^sub>r document_ptr"
fmpredD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and x :: "'c"   and y :: "'d" assumes "fmpred P m"   and "m $$ x = Some y" shows "P x y"
cos_ang_circ_simp: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" assumes "E \<noteq> \<mu>1"   and "E \<noteq> \<mu>2" shows "cos (ang_circ E \<mu>1 \<mu>2 p1 p2) = sgn_bool (p1 = p2) * cos (arg (E - \<mu>2) - arg (E - \<mu>1))"
vars_subst_var: fixes x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and lp :: "linear_poly" shows "vars (subst_var x\<^sub>j lp' lp) \<subseteq> vars lp - {x\<^sub>j} \<union> vars lp'"
integrable_on_empty: fixes f :: "'c \<Rightarrow> 'd" shows "f integrable_on {}"
\<II>_altdef: shows "\<II> = items_le (length Doc) (\<I> (length Doc))"
Abs_freelist_in_FreeGroup: fixes ss :: "'a list"   and S :: "'a set" assumes "ss \<in> lists S" shows "Abs_freelist ss \<in> FreeGroup S"
s'_def: shows "s' = (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)"
word_lessThan_Suc_atMost: fixes k :: "'b word" assumes "k \<noteq> - 1" shows "{..<k + 1} = {..k}"
inf_greatest: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> y \<cdot> z"
flush_append_Read\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat"   and m :: "nat \<Rightarrow> nat" shows "flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Read\<^sub>s\<^sub>b volatile a t v])) m = flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) m"
inorder_inv_insert: fixes t :: "'t"   and x :: "'a" assumes "inv t \<and> sorted (inorder t)" shows "inv (insert x t)"
those_transfer: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (list_all2 (rel_option P)) (rel_option (list_all2 P)) those those"
43: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> x = y \<sqinter> x"
view_slens_compl: fixes a :: "<'f, 'd> \<Longleftrightarrow> 'e" shows "\<V>\<^bsub>-\<^sub>L a\<^esub> = \<C>\<^bsub>a\<^esub>"
commutes_with_composition: fixes x :: "'b"   and y :: "'b" assumes "x \<in> \<FF> z'"   and "y \<in> \<FF> z'" shows "\<rho> z' w (+\<^bsub>z'\<^esub> x y) = +\<^bsub>w\<^esub> (\<rho> z' w x) (\<rho> z' w y)"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "LynSuf x = y" obtains "x = \<epsilon>" and "y = \<epsilon>"   | a :: "'a" and w :: "'a list" where "x = a # w" and "y = (if Lyndon (a # w) then a # w else LynSuf w)"
continuous_axioms_def: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real" shows "continuous_axioms V f norm \<equiv> \<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * norm x"
funcset_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
sym_transitive: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semiattractive A less_eq" shows "transitive A (sympartp less_eq)"
sum_fun_add: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite X"   and "f (0::'b) = (0::'c)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (\<Sum> X) = \<Sum> (f ` X)"
fps_Gcd: fixes A :: "'b fps set" assumes "A - {0} \<noteq> {}" shows "Gcd A = fps_X ^ Inf (subdegree ` (A - {0}))"
minus_Assertion_def: fixes p :: "'c Assertion"   and q :: "'c Assertion" shows "p - q = p \<sqinter> - q"
rel_\<I>_eq: shows "rel_\<I> (=) (=) = (=)"
partition: fixes X :: "'b set"   and r :: "'b \<times> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "equiv X (rel_ext r)"   and "set xs \<subseteq> X" shows "set ` set (List_local.partition r xs) = set xs // Restr (rel_ext r) (set xs)"
mst_subgraph_inv: fixes e :: "'a"   and v :: "'a"   and g :: "'a"   and t :: "'a"   and r :: "'a" assumes "e \<le> v \<sqinter> - id v \<sqinter> g"   and "t \<le> g"   and "id v = id r \<sqinter> top" shows "e \<le> id (id r \<sqinter> top) \<sqinter> (id r \<sqinter> top) \<sqinter> g"
preferred_alts_altdef: fixes x :: "'a" shows "preferred_alts le x = {y \<in> carrier. le x y}"
blaschke_unit_circle_fix: fixes a :: "complex" assumes "cmod a \<noteq> 1" shows "unit_circle_fix (blaschke a)"
extremum_strict: fixes a :: "'k word" shows "\<not> a < 0"
dom_fun_upd: fixes f :: "'f \<Rightarrow> 'g option"   and x :: "'f"   and y :: "'g option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
is_aligned_to_bl: fixes w :: "'b word"   and n :: "nat" shows "is_aligned w n = (True \<notin> set (drop (size w - n) (to_bl w)))"
IT: shows "RETURN tsl' \<le> it_to_sorted_list (key_rel ordR) (map_to_set m')"
matrix_kauff_mat: fixes ws :: "wall" assumes "is_tangle_diagram ws" shows "rat_poly.row_length (kauff_mat ws) = 2 ^ nat (domain_wall ws) \<and> length (kauff_mat ws) = 2 ^ nat (codomain_wall ws) \<and> mat (rat_poly.row_length (kauff_mat ws)) (length (kauff_mat ws)) (kauff_mat ws)"
st_2: shows "g C1_differentiable_on {c..b} - T"
valuations_equiv: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "\<forall>x\<in>carrier K. 0 \<le> v x \<longrightarrow> 0 \<le> v' x" shows "v_equiv K v v'"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_diff a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "wt t \<ge>\<^sub>p wt s" and "head t >\<^sub>h\<^sub>d head s"
le_diff_conv2: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b" shows "(c \<subseteq># b - a) = (c + a \<subseteq># b)"
comp_null_2: fixes f :: "'j" shows "J f J.null = J.null"
col2_cop__cop: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D"   and "C \<noteq> D"   and "Tarski_neutral_dimensionless.Col Bet C D E"   and "Tarski_neutral_dimensionless.Col Bet C D F" shows "Tarski_neutral_dimensionless.Coplanar Bet A B E F"
left_neutral: fixes a :: "'a" shows "(1::'a) * a = a"
subst_lit_def: fixes L :: "'a literal"   and \<sigma> :: "'s" shows "L \<cdot>l \<sigma> = map_literal (\<lambda>A. A \<cdot>a \<sigma>) L"
bigo_bounded_alt: fixes f :: "'c \<Rightarrow> 'd"   and c :: "'d"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. (0::'d) \<le> f x"   and "\<forall>x. f x \<le> c * g x" shows "f \<in> O(g)"
assms_2: shows "a \<in>\<^sub>\<circ> B"
Sup_upper2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and u :: "'d"   and A :: "'d set"   and v :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq v u" shows "less_eq v (Sup A)"
interp_ins_append_non_memb: fixes pre :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and suf :: "('a \<times> 'a option) list" assumes "insert_ops (pre @ [(oid, Some ref)] @ suf)"   and "ref \<notin> set (interp_ins pre)" shows "ref \<notin> set (interp_ins (pre @ [(oid, Some ref)] @ suf))"
swap_pair_fun'': fixes C :: "'b \<times> 'a \<Rightarrow> real \<times> real" shows "(\<lambda>y. prod.swap (C (0::'b, y))) = (\<lambda>x. (C (0::'b, x) \<bullet> j, C (0::'b, x) \<bullet> i))"
convex_hull_finite_step: fixes S :: "'c set"   and a :: "'c"   and w :: "real"   and y :: "'c" assumes "finite S" shows "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y) = (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a)"
subterms_of_irred_trms_are_irred: fixes t :: "'a trm"   and S :: "'a eclause set" shows "trm_rep t S \<noteq> t \<longrightarrow> st_red S t"
indep_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "X_insert.indep_in \<E> X" shows "X \<subseteq> \<E>"
poly_x_minus_y_pCons: fixes a :: "'a"   and p :: "'a poly" shows "poly_x_minus_y (pCons a p) = [:[:a:]:] + poly_x_minus_y p * x_y"
knows2knowPQ_k: fixes P :: "specID"   and m :: "Keys"   and Q :: "specID" assumes "knows P [kE m] \<or> knows Q [kE m]" shows "know P (kKS m) \<or> know Q (kKS m)"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "('i, 'j, 'k, 'l) SLin_state_scheme" shows "P13 (s1, s2) = (finite (initVals s2) \<and> finite (abortVals s1) \<and> finite (abortVals s2))"
antisym_conv1: fixes x :: "'b list"   and y :: "'b list" assumes "\<not> x <p y" shows "(x \<le>p y) = (x = y)"
last_conv_nth: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "last xs = xs ! (\<^bold>|xs\<^bold>| - 1)"
fax_def: fixes f :: "'fsym" shows "fax f \<equiv> [Pos (cOfFax f)]"
feasible_sound: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "feasible t"   and "sound P" shows "sound (t P)"
simps_46: fixes x31 :: "'b local.term"   and x32 :: "'b local.term"   and x5 :: "'b local.term" shows "x31 \<^bold>\<otimes> x32 \<noteq> \<^bold>\<l>\<^bold>[x5\<^bold>]"
rbt_lookup_keys: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "dom (ord.rbt_lookup less t) = set (RBT_Impl.keys t)"
upper_triangular_upt_imp_upper_triangular: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "upper_triangular_upt_k A (nrows A)" shows "upper_triangular A"
rep_eq: shows "zhmsetmset 0 = {#}\<^sub>z"
hfun_hlam: fixes A :: "hf"   and B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun (A * B) C F" shows "hfun A (hexp B C) (hlam A B C F)"
encode_frame_axioms_is_cnf: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "is_cnf (encode_all_frame_axioms \<Pi> t)"
groupWIs3_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs3 c rs = (if has_default_policy rs then groupWIs3_default_policy c rs else groupWIs2 c rs)"
compare_Unit_def: shows "compare = (\<Lambda> \<langle>\<rangle> \<langle>\<rangle>. EQ)"
ring_homomorphism_induced_morphism: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e"   and x :: "'d"   and V :: "'f set" assumes "ind_mor_btw_stalks X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f x"   and "is_open\<^sub>Y V"   and "f x \<in> V" shows "ring_homomorphism \<phi>\<^bsub>X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y f \<phi>\<^sub>f x\<^esub> (stalk.carrier_stalk is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y (f x)) (stalk.add_stalk \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U}) (stalk.mult_stalk \<O>\<^sub>Y \<rho>\<^sub>Y mult_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U}) (stalk.zero_stalk \<O>\<^sub>Y \<rho>\<^sub>Y zero_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U} V) (stalk.one_stalk \<O>\<^sub>Y \<rho>\<^sub>Y one_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U} V) (stalk.carrier_stalk is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X x) (stalk.add_stalk \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U}) (stalk.mult_stalk \<O>\<^sub>X \<rho>\<^sub>X mult_str\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U}) (stalk.zero_stalk \<O>\<^sub>X \<rho>\<^sub>X zero_str\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} (f \<^sup>\<inverse> X V)) (stalk.one_stalk \<O>\<^sub>X \<rho>\<^sub>X one_str\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} (f \<^sup>\<inverse> X V))"
IH: shows "(monom (UP Zp) \<one> na__ \<bullet> a) k = (monom (UP Zp) \<one> na__ \<bullet> b) k"
max: fixes \<A> :: "'a set set" assumes "\<A> \<in> Field r"   and "(\<P>, \<A>) \<in> r" shows "\<A> = \<P>"
of_nat_less_imp_less: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)" shows "m < n"
lexordp_eq_refl: fixes less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" shows "ord.lexordp_eq less xs xs"
lesub_def: fixes x :: "'a"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = r x y"
fls_of_int_nth: fixes n :: "int"   and k :: "int" shows "fls_nth (of_int n) k = (if k = 0 then of_int n else (0::'a))"
congBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s\<^sub>l Q" shows "P \<sim>\<^sub>l Q"
distinct_14: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "Cn x41 x42 x43 \<noteq> S"
simps_2: fixes p :: "'d"   and Tl :: "'c trm list" shows "nvA (Pr p Tl) = {}"
fps_expansion_cong: fixes g :: "complex \<Rightarrow> complex"   and h :: "complex \<Rightarrow> complex"   and x :: "complex" assumes "\<forall>\<^sub>F x in nhds x. g x = h x" shows "fps_expansion g x = fps_expansion h x"
execn_Stuck_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "s = Stuck" shows "t = Stuck"
no_\<tau>moves2_def: shows "no_\<tau>moves2 \<equiv> \<lambda>s2 t. wset s2 t = None \<and> (\<exists>x. thr s2 t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> r2.silent_move t (x, shr s2) (x', m')))"
center_on_geodesic_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "center x y z \<in> {y--z}"
opp_closed: fixes a :: "'a"   and b :: "'a"   and p :: "'a point" assumes "on_curve a b p" shows "on_curve a b (opp p)"
premise_3: shows "\<^bold>\<forall>x xa. E x \<and> \<not> E xa \<longrightarrow> x \<^bold>= (\<^bold>\<tau>x. \<Phi> x) \<or> xa = (\<^bold>\<tau>x. \<Phi> x) \<longrightarrow> x = (\<^bold>\<tau>x. \<Phi> x)"
foldr_transfer: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
A: shows "ipurge_tr_aux I D U (x # xs) = x # ipurge_tr_aux I D U xs"
Quotient_endofun: shows "Quotient (=) abs_endofun rep_endofun cr_endofun"
purgeIdle_map_Idle: fixes s :: "'a list" shows "GM_sec_model.purgeIdle (map Idle s) = []"
flow_image_eq_rev: fixes x :: "'a"   and S :: "real set" shows "flow0 x ` S = rev.flow0 x ` uminus ` S"
rep_eq: fixes x :: "'a regular"   and xa :: "'a regular" shows "(x < xa) = (Rep_regular x < Rep_regular xa)"
path_connected_imp_connected: fixes S :: "'a set" assumes "path_connected S" shows "connected S"
iT_cut_ge1_1: fixes t :: "nat"   and n :: "nat" assumes "t \<in> [\<dots>n]" shows "[\<dots>n] \<down>\<ge> t = [t\<dots>,n - t]"
case_two_psi_longest_prefix: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and y :: "nat" assumes "total_cons s"   and "\<exists>t. eval (total_cons.r_prefixes s) [t, j] \<up>"   and "y = total_cons.longest_prefix s j" shows "(\<forall>x. 0 < x \<and> x < y \<longrightarrow> eval (total_cons.r_psi s) [j, x] \<down>= 0 \<or> eval (total_cons.r_psi s) [j, x] \<down>= 1) \<and> (\<forall>x\<ge>y. eval (total_cons.r_psi s) [j, x] \<up>)"
image_mset_mset_set: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "image_mset f (mset_set A) = mset_set (f ` A)"
comp_eq_id_dest: fixes a :: "'h \<Rightarrow> 'g"   and b :: "'f \<Rightarrow> 'h"   and c :: "'f \<Rightarrow> 'g"   and v :: "'f" assumes "a \<circ> b = id \<circ> c" shows "a (b v) = c v"
final: shows "final (c\<^sub>f, s\<^sub>f)"
r: fixes x :: "'a" assumes "x \<in> S" shows "0 < r x \<and> (\<exists>C\<in>\<C>. ball x (2 * r x) \<subseteq> C)"
midpoint_existence: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<exists>X. Tarski_neutral_dimensionless.Midpoint Bet Cong X A B"
mult_time_constant_self_financing: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1"   and "self_financing Mkt pf1"   and "\<forall>n w. qty n w = qty (Suc n) w" shows "self_financing Mkt (qty_mult_comp pf1 qty)"
has_bochner_integral_FTC_Icc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "has_bochner_integral lborel (\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) (F b - F a)"
poly_eval_focus: fixes a :: "'b \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (focus X p) = poly_subst (\<lambda>x. if x \<in> X then a x else monomial (1::'c) (monomial 1 x)) p"
strict_implies_order: fixes a :: "'b"   and b :: "'b" assumes "codomain.greater b a" shows "codomain.greater_eq b a"
pt3: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and " pi1 \<triangleq> pi2 " shows "pi1 \<bullet> x = pi2 \<bullet> x"
assms_2: fixes n :: "nat" assumes "0 < n" shows "A \<inter> (T ^^ n) -` A = {}"
IH: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> set T"   and "decomp_closure M M'"   and "intruder_synth M' t"   and "intruder_deduct M' t"   and "t \<sqsubseteq> t" shows "intruder_synth M' t"
hypnat_add_is_0: fixes m :: "nat star"   and n :: "nat star" shows "(m + n = 0) = (m = 0 \<and> n = 0)"