hcju/isabelleps huggingface dataset:
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "connex A less_eq"   and "antisymmetric A less_eq" shows "total_pseudo_ordered_set A less_eq"
prems_8: fixes a :: "nat list"   and b :: "nat list" assumes "a \<in> list.set as"   and "b \<in> list.set bs" shows "a < b \<or> b < a"
surj_uni_samp: fixes y :: "nat"   and q :: "nat" assumes "inj_on (\<lambda>b. (y + b) mod q) {..<q}" shows "(\<lambda>b. (y + b) mod q) ` {..<q} = {..<q}"
tau_om1: fixes x :: "'a" shows "\<tau> x \<le> \<tau> (x\<^sup>\<omega>)"
le_right_comm: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong A B D C"
carr_dsum_prod: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'h) aGroup_scheme" shows "carr_dsumag I M \<subseteq> carr_prodag I M"
volumeI: fixes f :: "'b set \<Rightarrow> ennreal"   and M :: "'b set set" assumes "f {} = 0"   and "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> f a"   and "\<And>C. \<lbrakk>C \<subseteq> M; disjoint C; finite C; \<Union> C \<in> M\<rbrakk> \<Longrightarrow> f (\<Union> C) = sum f C" shows "volume M f"
polyhedron_eq_finite_faces: fixes S :: "'b set" shows "polyhedron S = (closed S \<and> convex S \<and> finite {F. F face_of S})"
bigo_const3: fixes c :: "'c" assumes "c \<noteq> (0::'c)" shows "(\<lambda>x. 1::'c) \<in> O(\<lambda>x. c)"
at_within_interior: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> interior S" shows "at x within S = at x"
iNodeOf_tree_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 fc :: "('b, 'd) graph_node \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'b fset \<times> 'b \<Rightarrow> bool"   and t :: "('b, 'd, 'c, 'a) itnode rose_tree"   and ent :: "'b fset \<times> 'b"   and "is" :: "nat list" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "Abstract_Task.iwf freshenLC renameLCs subst anyP antecedent consequent rules assumptions conclusions fc t ent"   and "is \<in> it_paths t" shows "iNodeOf (tree_at t is) \<in> i.R (Abstract_Task.nodes rules assumptions conclusions)"
ccInf_lower: fixes A :: "'a set"   and x :: "'a" assumes "countable A"   and "x \<in> A" shows "Inf A \<le> x"
sup_left_commute: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x"   and y :: "'x"   and z :: "'x" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
mapping_of_inverse: fixes x :: "'b mpoly" shows "MPoly (mapping_of x) = x"
fbox_temp_dyn: fixes a :: "real"   and U :: "(real, 4) vec \<Rightarrow> real set"   and L :: "real"   and G :: "(real, 4) vec \<Rightarrow> bool"   and Q :: "(real, 4) vec \<Rightarrow> bool" assumes "0 < a"   and "\<And>s. s2p UNIV s \<Longrightarrow> s2p (U s) 0 \<and> is_interval (U s) \<and> U s \<subseteq> UNIV" shows "wp x\<acute>= \<lambda>t. temp_vec_field a L & G on U UNIV @ 0 \<lceil>Q\<rceil> = \<lceil>\<lambda>s. s2p UNIV s \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (temp_flow a L \<tau> s)) \<longrightarrow> Q (temp_flow a L t s))\<rceil>"
simps_3: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and z :: "'b"   and zs :: "'b list" shows "x # xs \<simeq> {[], z # zs, P} = (x = z \<and> xs \<simeq> {[], zs, P})"
list_all2_Nil2: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list" shows "list_all2 P xs [] = (xs = [])"
IS_RIGHT_TOTAL_def: fixes R :: "('a \<times> 'b) set" shows "IS_RIGHT_TOTAL R \<equiv> Range R = UNIV"
H_def: shows "H \<equiv> \<Prod>\<^sub># (factors_of_factor_tree r)"
semiring_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "semiring_axioms R \<equiv> ((\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z) \<and> (\<forall>x y z. x \<in> carrier R \<longrightarrow> y \<in> carrier R \<longrightarrow> z \<in> carrier R \<longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y)) \<and> (\<forall>x. x \<in> carrier R \<longrightarrow> \<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>) \<and> (\<forall>x. x \<in> carrier R \<longrightarrow> x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>)"
undefined_in_PiM_empty: fixes M :: "'b \<Rightarrow> 'c measure" shows "(\<lambda>x. undefined) \<in> space (Pi\<^sub>M {} M)"
from_nat_ia_not_i: shows "mod_type_class.from_nat ia \<noteq> i"
fMax_in: fixes A :: "'b fset" assumes "A \<noteq> {||}" shows "fMax A |\<in>| A"
dim_eq_card_independent: fixes B :: "'b set" assumes "vs1.independent B" shows "vs1.dim B = card B"
hom: fixes r :: "real \<Rightarrow> 'c" assumes "path r"   and "path_image r \<subseteq> U"   and "pathstart r = z"   and "pathfinish r = z" shows "\<exists>q. path q \<and> path_image q \<subseteq> C \<and> pathstart q = a \<and> pathfinish q = a \<and> homotopic_paths S (f \<circ> r) (p \<circ> q)"
r_update_aux_recfn: shows "recfn 4 r_update_aux"
inv_def: shows "local.inv \<equiv> THE F. inverse_functors (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G"
prems_1: shows "\<Gamma>\<turnstile>sequence Seq (x # xs) \<down> Normal s"
radical_sqrt_quadratic_equation: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "a \<in> radical_sqrt"   and "b \<in> radical_sqrt"   and "c \<in> radical_sqrt"   and "a * x\<^sup>2 + b * x + c = 0"   and "\<not> (a = 0 \<and> b = 0 \<and> c = 0)" shows "x \<in> radical_sqrt"
cf_assn_def: fixes N :: "nat" shows "??.Network_Impl.cf_assn N \<equiv> asmtx_assn N cap_assn"
upper: fixes a :: "ereal" assumes "l < a" shows "\<forall>\<^sub>F n in sequentially. v n < a"
simps_1: shows "\<^bold>\<I>\<^bold>\<down> = \<^bold>\<I>"
not_min_maxsimpchainI_betw: fixes x :: "'a set"   and ys :: "'a set list"   and y :: "'a set"   and xs :: "'a set list" assumes "gallery (x # ys @ [y])"   and "order.greater (length xs) (length ys)" shows "\<not> min_gallery (x # xs @ [y])"
cINF_greatest: 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 A :: "'c set"   and m :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m (f x)" shows "less_eq m (Inf (f ` A))"
gfun_elim: fixes w :: "bval"   and f :: "(val \<times> val) fset" assumes "w \<in> good (VFun f)" shows "\<exists>x e \<rho>. w = BClos x e \<rho> \<and> good_fun f x e \<rho>"
differentiable_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and t :: "'a set"   and s :: "'a set" assumes "f differentiable_on t"   and "s \<subseteq> t" shows "f differentiable_on s"
rcos_module_rev: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and x' :: "'c" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "x' \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H" shows "x' \<in> H #>\<^bsub>G\<^esub> x"
step_inf: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "\<And>x y z. \<lbrakk>r x y; r x z\<rbrakk> \<Longrightarrow> y = z"   and "r x y"   and "Inf.inf r x" shows "Inf.inf r y"
grounding_of_clss_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and CC :: "'b literal multiset set" shows "substitution_ops.grounding_of_clss subst_atm CC = \<Union> (substitution_ops.grounding_of_cls subst_atm ` CC)"
repeat_mset_subset_in: fixes A :: "'b set multiset"   and B :: "'b set"   and X :: "'b set"   and n :: "nat"   and x :: "'b" assumes "\<And>a. a \<in># A \<Longrightarrow> a \<subseteq> B"   and "X \<in># repeat_mset n A"   and "x \<in> X" shows "x \<in> B"
imply_weaken: fixes ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm"   and ps' :: "('a, 'b) fm list" assumes "ps \<turnstile> q"   and "set ps \<subseteq> set ps'" shows "ps' \<turnstile> q"
remdups_append1_drop: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> set ys" shows "remdups (xs @ ys) = remdups ys"
tight_imp_convergent_subsubsequence: fixes \<mu> :: "nat \<Rightarrow> real measure"   and s :: "nat \<Rightarrow> nat" assumes "tight \<mu>"   and "strict_mono s" shows "\<exists>r M. strict_mono r \<and> real_distribution M \<and> weak_conv_m (\<mu> \<circ> s \<circ> r) M"
ratfps_subdegree_altdef: fixes x :: "'b ratfps" shows "ratfps_subdegree x = subdegree (fps_of_ratfps x)"
set_val_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_val_locs character_data_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
heapify_heap_is_heap: fixes t :: "'c Tree" shows "is_heap (heapify t)"
llist_all2_all_lnthI: fixes xs :: "'g llist"   and ys :: "'h llist"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "llength xs = llength ys"   and "\<And>n. enat n < llength xs \<Longrightarrow> P (lnth xs n) (lnth ys n)" shows "llist_all2 P xs ys"
abs_eq: fixes xa :: "rat"   and x :: "nat" shows "lp_monom xa x = Abs_linear_poly (\<lambda>y. if x = y then xa else 0)"
coneE: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "p \<in> cone (h, U)" obtains a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "a \<in> P[U]" and "p = a * h"
frac_eq_id: fixes x :: "'a" assumes "x \<in> {0::'a..<1::'a}" shows "frac x = x"
Complex_mult_complex_of_real: fixes x :: "real"   and y :: "real"   and r :: "real" shows "Complex x y * cor r = Complex (x * r) (y * r)"
weakCongruenceE_1: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<cong> Q" shows "P \<leadsto><weakBisimulation> Q"
empty_comp2: fixes \<sigma> :: "char list \<Rightarrow> fterm" shows "\<epsilon> \<cdot> \<sigma> = \<sigma>"
aff: fixes C :: "'a set" assumes "C \<in> \<M>" shows "aff_dim C = d"
scalar_coeff: fixes a :: "'a"   and p :: "'a list" assumes "a \<in> carrier R" shows "local.coeff (map ((\<otimes>) a) p) = (\<lambda>i. a \<otimes> local.coeff p i)"
seq_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<times> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "partial_magma.arr (product_category.comp C1 C2) (product_category.comp C1 C2 g f) = (partial_magma.arr C1 (C1 (fst g) (fst f)) \<and> partial_magma.arr C2 (C2 (snd g) (snd f)))"
R_def: fixes p :: "real"   and A :: "'a set"   and u :: "nat"   and l :: "nat" shows "random_skip_list.R p A u l = map_pmf (\<lambda>f. steps A f 0 u l) (random_skip_list.SL p A)"
awalk_decomp_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "cas u p v"   and "awalk_verts u p = xs @ y # ys" obtains q :: "'b list" and r :: "'b list" where "cas u q y" and "cas y r v" and "p = q @ r" and "awalk_verts u q = xs @ [y]" and "awalk_verts y r = y # ys"
map_key_PP_Macaulay_list: fixes fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "map (Poly_Mapping.map_key PP) (punit.Macaulay_list fs) = pp_pm.punit.Macaulay_list (map (Poly_Mapping.map_key PP) fs)"
k': shows "clock_simAbs k' \<in> (\<Union>a. Clock.sim_equiv_class a ` Clock.jkbpC)"
assms_3: shows "set lss = valid_states (prob_proj PROB vs)"
ground_inference_saturated_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.ground_inference_saturated trm_ord sel pos_ord filter_trms S = (\<forall>C P \<sigma> C'. basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> grounding_set P \<sigma> \<longrightarrow> basic_superposition.redundant_inference trm_ord C S P \<sigma>)"
lem': fixes tp :: "bool list"   and s :: "bool list"   and A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(tp, s) \<in> (eps (RegExp2NAe.star A))\<^sup>*" shows "\<forall>p. tp = True # p \<longrightarrow> (\<exists>r. ((p, r) \<in> (eps A)\<^sup>* \<or> (\<exists>q. (p, q) \<in> (eps A)\<^sup>* \<and> fin A q \<and> (start A, r) \<in> (eps A)\<^sup>*)) \<and> s = True # r)"
lunit_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and tensor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unity :: "'a"   and lunit :: "'a \<Rightarrow> 'a"   and runit :: "'a \<Rightarrow> 'a"   and assoc :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.ide C a" shows "tensor unity (lunit a) = C (tensor (elementary_monoidal_category.\<iota> unity lunit) a) (category.inv C (elementary_monoidal_category.\<alpha> C tensor assoc (monoidal_category.unity C (elementary_monoidal_category.\<iota> unity lunit), monoidal_category.unity C (elementary_monoidal_category.\<iota> unity lunit), a)))"
P1D1Q1: shows "(P1, D1, Q1) = Smith_2xn A2"
meet_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
a: shows "a + (1::'a) \<noteq> (0::'a)"
dns2: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
cong: fixes is_node :: "'d \<Rightarrow> bool"   and memb :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "DFS.rel is_node memb \<equiv> DFS.rel is_node memb"
merkle_interface_aux: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" shows "merkle_interface h bo m = merge_on UNIV h bo m"
le_minus_one_simps_4: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq one (uminus one)"
\<E>p_\<E>_eq: fixes final :: "'c \<Rightarrow> bool"   and r :: "('e, 'a, 'c, 'd, 'f, 'b) semantics"   and convert_RA :: "'e \<Rightarrow>f nat \<Rightarrow> 'b list"   and \<sigma> :: "('e, 'a) locks \<times> (('a \<Rightarrow> ('c \<times> 'e \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'f wait_set_status option) \<times> 'a set" shows "multithreaded_base.\<E>p final r convert_RA \<sigma> = (\<lambda>x. x \<in> multithreaded_base.\<E> final r convert_RA \<sigma>)"
elims: fixes x :: "op"   and y :: "nat" assumes "ar x = y" obtains "x = OpNeg" and "y = 1"   | "x = OpAdd" and "y = 2"   | "x = OpMul" and "y = 2"
induct: fixes P :: "real \<Rightarrow> bool"   and x1 :: "real"   and x2 :: "real"   and Pa :: "real \<Rightarrow> real \<Rightarrow> bool" assumes "finite_Psegments P x1 x2"   and "\<And>a b. b \<le> a \<Longrightarrow> Pa a b"   and "\<And>s a b. \<lbrakk>s \<in> {a..<b}; s = a \<or> P s; \<forall>t\<in>{s<..<b}. P t; finite_Psegments P a s; Pa a s\<rbrakk> \<Longrightarrow> Pa a b"   and "\<And>s a b. \<lbrakk>s \<in> {a..<b}; s = a \<or> P s; \<forall>t\<in>{s<..<b}. \<not> P t; finite_Psegments P a s; Pa a s\<rbrakk> \<Longrightarrow> Pa a b" shows "Pa x1 x2"
compl_le_swap1: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<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 y :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq y (uminus x)" shows "less_eq x (uminus y)"
tuple_in_hom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "\<guillemotleft>tuple f g : local.dom f \<rightarrow> local.prod (cod f) (cod g)\<guillemotright>"
S_def: fixes n :: "nat" shows "S n \<equiv> \<lambda>x. \<Sum>i<n. X i x"
Class_widen2: fixes P :: "'a prog"   and C :: "String.literal"   and T :: "ty" shows "P \<turnstile> Class C \<le> T = (\<exists>D. T = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D)"
nat_exists_least_iff': fixes P :: "nat \<Rightarrow> bool" shows "(\<exists>n. P n) = (P (Least P) \<and> (\<forall>m<Least P. \<not> P m))"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P \<alpha> P' A\<^sub>P \<Psi>\<^sub>P"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "x \<sharp> A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) \<alpha> (\<lparr>\<nu>x\<rparr>P') (x # A\<^sub>P) \<Psi>\<^sub>P"
vp_def: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" shows "n -as\<rightarrow>\<^sub>\<surd>* n' \<equiv> n -as\<rightarrow>* n' \<and> valid_path as"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'c \<Rightarrow> 'a"   and x :: "'c list"   and y :: "'b list" shows "list_all2 Sb (map i x) y = list_all2 (\<lambda>x. Sb (i x)) x y"
subspace_tangent_space: fixes p :: "'a" shows "subspace (sub.tangent_space p)"
hyps_3: assumes "f \<noteq> {}"   and "\<forall>s\<in>f. \<exists>p. p division_of s" shows "\<exists>p. p division_of \<Inter> f"
Iff_E2: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm" assumes "H \<turnstile> A"   and "insert B H \<turnstile> C" shows "insert (B IFF A) H \<turnstile> C"
unit_factor_poly_def: fixes p :: "'a poly" shows "unit_factor p = [:unit_factor (lead_coeff p):]"
consistent_eq_solution_solve: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "consistent A b = is_solution (fst (the (solve A b))) A b"
prop_restrict: fixes x :: "'b"   and Z :: "'b set"   and X :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> Z"   and "Z \<subseteq> {x \<in> X. P x}" shows "P x"
high_proc\<^sub>2_def: fixes s\<^sub>H :: "stmt" shows "high_proc\<^sub>2 s\<^sub>H \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> s\<^sub>H \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma> ''lowIn'') + the (\<sigma> ''highIn'') \<le> the (\<sigma>' ''highOut'')"
principal_dchar_def: fixes n :: "nat" shows "\<chi>\<^sub>0\<^bsub>n\<^esub> = (\<lambda>k. if coprime k n then 1 else 0)"
max_pos_imp_Interp: fixes C :: "'b literal multiset"   and N :: "'b literal multiset set"   and A :: "'b" assumes "C \<in> N"   and "C \<noteq> {#}"   and "Max_mset C = Pos A"   and "{#} = {#}" shows "Interp N C \<Turnstile> C"
less_imp_not_less: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y" shows "(\<not> y \<prec> x) = True"
simps_3: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval' (Plus a\<^sub>1 a\<^sub>2) s = optplus (paval' a\<^sub>1 s) (paval' a\<^sub>2 s)"
valid_eval: fixes eval :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and eval_list :: "v sem_env \<Rightarrow> exp0 list \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v list, v) result"   and eval_match :: "v sem_env \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0" assumes "eval eval eval_list eval_match" shows "evaluate True env s e (eval env e s)"
simps_1: fixes a :: "int"   and b :: "int" shows "Pc a \<langle>*\<rangle> Pc b = Pc (a * b)"
cop_F_range_inv_def: fixes ds :: "'b set"   and fp :: "'a set" shows "cop_F_range_inv ds fp = (\<forall>x\<in>fp. x \<in> Field (Pd (Xd x)) \<and> Xd x \<in> ds)"
assms_3: shows "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"
kD_addpreRT: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and npre :: "nat set" assumes "dip \<in> kD rt" shows "kD (the (addpreRT rt dip npre)) = kD rt"
the: shows "m = (THE m. m \<in> obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)"
inject: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex"   and y31 :: "'a"   and y32 :: "'a ifex"   and y33 :: "'a ifex" shows "(IF x31 x32 x33 = IF y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
qmpt_factorE_3: fixes proj :: "'a \<Rightarrow> 'd"   and M2 :: "'d measure"   and T2 :: "'d \<Rightarrow> 'd" assumes "I.qmpt_factor proj M2 T2" shows "qmpt M2 T2"
split_swapidseq_ext: fixes k :: "nat"   and n :: "nat"   and S :: "'a set"   and p :: "'a \<Rightarrow> 'a" assumes "k \<le> n"   and "swapidseq_ext S n p" obtains q :: "'a \<Rightarrow> 'a" and r :: "'a \<Rightarrow> 'a" and U :: "'a set" and V :: "'a set" where "swapidseq_ext U (n - k) q" and "swapidseq_ext V k r" and "p = q \<circ> r" and "U \<union> V = S"
linear_on_push_forward_inclusion: fixes p :: "'a" shows "linear_on (sub.tangent_space p) (tangent_space p) (*\<^sub>R) (*\<^sub>R) inclusion.push_forward"
UINF_pred_ueq: fixes v :: "('c, 'b) uexpr"   and P :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> x | \<guillemotleft>x\<guillemotright> =\<^sub>u v \<bullet> P x) = msubst P v"
closed_segment_as_isometric_image: fixes x :: "'b"   and y :: "'b" shows "(\<lambda>t. x + (t / dist x y) *\<^sub>R (y - x)) ` {0..dist x y} = closed_segment x y"
ex_un_null: shows "\<exists>!n. \<forall>f. A n f = n \<and> A f n = n"