hcju/isabelleps huggingface dataset:
intros_37: fixes "is" :: "nat list"   and c :: "i32"   and i :: "nat" assumes "length is \<le> nat_of_int c" shows "\<lparr>[$C ConstInt32 c, $Br_table is i]\<rparr> \<leadsto> \<lparr>[$Br i]\<rparr>"
assms_6: fixes j :: "'b" assumes "j \<in> B" shows "\<exists>x\<in>A. g j \<le>\<^sub>o\<^sub>w f x"
mat_gt_arc_compat2: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "mat_gt_arc A B"   and "B \<ge>\<^sub>m C"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "mat_gt_arc A C"
cmult_commute: fixes i :: "V"   and j :: "V" shows "i \<otimes> j = j \<otimes> i"
satisfies_T0: shows "is_left_adjoint s\<^sub>0"
DmdImplE: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and G :: "'a \<Rightarrow> bool" assumes "sigma \<Turnstile> \<diamond>F"   and "\<turnstile> F \<longrightarrow> G" shows "sigma \<Turnstile> \<diamond>G"
l2_inv6'_trans: shows "{l2_inv6'} TS.trans l2 {> l2_inv6'}"
fSum_def: fixes A :: "('d, 'e) aGroup_scheme"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat"   and m :: "nat" shows "\<Sigma>\<^sub>f A f n m = (if n \<le> m then \<Sigma>\<^sub>e A cmp f (slide n) (m - n) else \<zero>\<^bsub>A\<^esub>)"
hyps_3: shows "dgrad_p_set_le d {y} (insert p F)"
rev_image_eqI: fixes x :: "'d"   and A :: "'d set"   and b :: "'e"   and f :: "'d \<Rightarrow> 'e" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
cond_action_ok_init_fin_descend_stateI: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ((status \<times> '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" assumes "final_thread.cond_action_ok init_fin_final s t ct" shows "cond_action_ok (init_fin_descend_state s) t ct"
translation_galois: fixes T :: "'a set"   and a :: "'a"   and S :: "'a set" shows "(T = (+) a ` S) = (S = (+) (- a) ` T)"
reads_bind_pure: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a"   and S :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "pure f h'"   and "reads S f h h'"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> reads S (g x) h h'" shows "reads S (f \<bind> g) h h'"
perm2: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f bs y c = f (p \<bullet> bs) (p \<bullet> y) c"
RN_deriv_def: fixes M :: "'b measure"   and N :: "'b measure" shows "RN_deriv M N = (if \<exists>f. f \<in> borel_measurable M \<and> density M f = N then SOME f. f \<in> borel_measurable M \<and> density M f = N else (\<lambda>_. 0))"
l7_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A A" shows "M = A"
Qinter_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'b option"   and u :: "'a"   and v' :: "'a" shows "Prim2.Qinter w g r Q \<pi> u v' = (if Prim2.upd_cond w g r Q \<pi> u v' then enat (w {v', u}) else Q v')"
map_leI: fixes m1 :: "'g \<Rightarrow> 'h option"   and m2 :: "'g \<Rightarrow> 'h option" assumes "\<And>x v. m1 x = Some v \<Longrightarrow> m2 x = Some v" shows "m1 \<subseteq>\<^sub>m m2"
join_fmap_join: fixes xsss :: "'d\<cdot>'e\<cdot>'e\<cdot>'e" shows "join\<cdot>(fmap\<cdot>join\<cdot>xsss) = join\<cdot>(join\<cdot>xsss)"
valid_join_msaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'msaux"   and auxlist :: "(nat \<times> event_data option list set) list"   and rel1 :: "event_data option list set" assumes "valid_msaux args cur aux auxlist"   and "table (args_n args) (args_L args) rel1" shows "valid_msaux args cur (join_msaux args rel1 aux) (map (\<lambda>(t, rel). (t, join rel (args_pos args) rel1)) auxlist)"
divisors_int_pos_1: fixes x :: "int" assumes "x \<noteq> 0" shows "set (divisors_int_pos x) = {i. i dvd x \<and> 0 < i}"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (\<lambda>f. T (f, \<I>))"
replication_numbers_def: shows "target.replication_numbers \<equiv> {\<B>' rep x |x. x \<in> \<V>'}"
sorted_append2: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and ys :: "('b, 'c) Combinators list" assumes "all_in_list xs l"   and "singleCombinators xs"   and "all_in_list ys l"   and "singleCombinators ys"   and "FWNormalisationCore.sorted xs l \<and> FWNormalisationCore.sorted ys l \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. smaller x y l)" shows "FWNormalisationCore.sorted (xs @ ys) l"
simps_1: fixes c :: "('Var, 'AExp, 'BExp) Stmt" shows "c \<otimes> [] = c"
abs_eq: fixes x :: "'c" shows "return_pmf x = Abs_pmf (return (count_space UNIV) x)"
comp_null_1: fixes f :: "'a" shows "Left_a.comp Left_a.null f = Left_a.null"
ilc_def: shows "ilc \<equiv> inverse_mod (lead_coeff f) (p ^ n)"
prems: shows "f \<in>\<^sub>\<circ> vproduct I A"
eventually_mono: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter"   and Q :: "'e \<Rightarrow> bool" assumes "eventually P F"   and "\<And>x. P x \<Longrightarrow> Q x" shows "eventually Q F"
assms_1: shows "i \<in>\<^sub>\<circ> j"
greatest_cong: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and x' :: "'c"   and A :: "'c set" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed\<^bsub>L\<^esub> A" shows "greatest L x A = greatest L x' A"
atom_map_atoms: fixes 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.bounded_lattice inf less_eq less sup bot top" shows "\<Union> (range (bounded_lattice.atom_map less_eq less bot)) = Collect (bounded_lattice.atom less bot)"
3: fixes x :: "brick list"   and y :: "brick list"   and ys :: "wall" shows "basic x \<otimes> y * ys = (if codomain_block x = 0 then (x \<otimes> y) * ys else (x \<otimes> y) * (basic (make_vert_block (nat (codomain_block x))) \<otimes> ys))"
simps_3: fixes uu :: "mreal" shows "mreal_plus uu MInfty = MInfty"
separation_by_component_closed: fixes S :: "'a set" assumes "closed S"   and "\<not> connected (- S)" obtains C :: "'a set" where "C \<in> components S" and "\<not> connected (- C)"
word_of_int_via_signed: fixes mask :: "int"   and shift :: "int"   and index :: "nat"   and overflow :: "int"   and least :: "int"   and i :: "int"   and arbitrary1 :: "int \<Rightarrow> 'a word"   and arbitrary2 :: "int \<Rightarrow> 'a word" assumes "mask = mask LENGTH('a)"   and "shift = 1 << LENGTH('a)"   and "index = LENGTH('a) - 1"   and "overflow = 1 << LENGTH('a) - 1"   and "least = - overflow" shows "word_of_int i = (let i' = i AND mask in if i' !! index then if i' - shift < least \<or> overflow \<le> i' - shift then arbitrary1 i' else word_of_int (i' - shift) else if i' < least \<or> overflow \<le> i' then arbitrary2 i' else word_of_int i')"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Inl x1 = Inl y1) = (x1 = y1)"
mantissa_float: fixes m :: "int"   and e :: "int" assumes "odd m" shows "mantissa (float_of (real_of_int m * 2 powr real_of_int e)) = m"
on_equator_rep: fixes a :: "proj2" assumes "z_non_zero a"   and "proj2_incident a equator" shows "\<exists>x. a = proj2_abs (vector [x, 0, 1])"
Under_decr: fixes A :: "'a set"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "A \<subseteq> B" shows "Under r B \<subseteq> Under r A"
simps_3: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_letter prop\<^sub>n(a) \<nu> = (if a \<in> \<nu> then true\<^sub>n else false\<^sub>n)"
nth_oo_def: fixes v :: "'b"   and a :: "'b array"   and i :: "nat" shows "nth_oo v a i \<equiv> Array.len a \<bind> (\<lambda>l. if i < l then Array.nth a i else return v)"
listgcd_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "listgcd (x # xs) = gcd x (listgcd xs)"
se_mono_for_sub: fixes c1 :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and c1' :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf"   and c2' :: "('d, 'e) conf" assumes "se c1 l c1'"   and "se c2 l c2'"   and "c2 \<sqsubseteq> c1" shows "c2' \<sqsubseteq> c1'"
norm_1_bounded: shows "bdd_above {\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> = 1}"
sorted_butlast: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "xs \<noteq> []"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (butlast xs)"
absorb2: fixes a :: "'b"   and b :: "'b" assumes "(a::'b) \<le> (b::'b)" shows "sup (a::'b) (b::'b) = b"
fin: shows "(\<Sum>i. emeasure M (A i)) \<noteq> \<infinity>"
9520: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> - - (x \<sqinter> z) = bot"
vector_space_ow_axioms_def: fixes U\<^sub>V\<^sub>S :: "'d set"   and plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale \<equiv> ((\<forall>x a. x \<in> U\<^sub>V\<^sub>S \<longrightarrow> scale a x \<in> U\<^sub>V\<^sub>S) \<and> (\<forall>x y a. x \<in> U\<^sub>V\<^sub>S \<longrightarrow> y \<in> U\<^sub>V\<^sub>S \<longrightarrow> scale a (plus\<^sub>V\<^sub>S x y) = plus\<^sub>V\<^sub>S (scale a x) (scale a y))) \<and> (\<forall>x a b. x \<in> U\<^sub>V\<^sub>S \<longrightarrow> scale (a + b) x = plus\<^sub>V\<^sub>S (scale a x) (scale b x)) \<and> (\<forall>x a b. x \<in> U\<^sub>V\<^sub>S \<longrightarrow> scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. x \<in> U\<^sub>V\<^sub>S \<longrightarrow> scale (1::'e) x = x)"
ESem_Let: fixes as :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Terms.Let as body \<rbrakk>\<^bsub>\<rho>\<^esub> = C_case\<cdot>(\<N>\<lbrakk> body \<rbrakk>\<^bsub>\<N>\<lbrace>as\<rbrace>\<rho>\<^esub>)"
is_joinI: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z"   and "\<And>a. \<lbrakk>x \<sqsubseteq> a; y \<sqsubseteq> a\<rbrakk> \<Longrightarrow> z \<sqsubseteq> a" shows "x \<squnion> y = z"
ext: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
simps_2: fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle>\<^sub>s\<^sub>t) = fv t"
is_rbt_rbt_minus: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.is_rbt less t1"   and "ord.is_rbt less t2" shows "ord.is_rbt less (ord.rbt_minus less t1 t2)"
a: shows "T \<in> M \<rightarrow>\<^sub>M N"
cross3_simps_10: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * b * c = a * (b * c)"
dim_eq: shows "dim_vec (col (submatrix A UNIV J) j) = dim_vec (col A (pick J j))"
sortedP_Cons: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "linorder_list0.sortedP le (x # xs)" obtains "\<forall>xa\<in>set xs. le x xa" and "linorder_list0.sortedP le xs"
assms_2: shows "I \<le> |x\<acute>=f & G on U S @ t\<^sub>0] I"
full_Suc_elim: fixes n :: "nat"   and t :: "'b tree234" assumes "full (Suc n) t" obtains l :: "'b tree234" and r :: "'b tree234" and p :: "'b" where "t = \<langle>l__, p__, r__\<rangle>" and "full n l__" and "full n r__"   | l :: "'b tree234" and m :: "'b tree234" and r :: "'b tree234" and p :: "'b" and q :: "'b" where "t = \<langle>la__, pa__, m__, q__, ra__\<rangle>" and "full n la__" and "full n m__" and "full n ra__"   | l :: "'b tree234" and m :: "'b tree234" and m' :: "'b tree234" and r :: "'b tree234" and p :: "'b" and q :: "'b" and q' :: "'b" where "t = \<langle>l, p, m, q, m', q', r\<rangle>" and "full n l" and "full n m" and "full n m'" and "full n r"
uminus1: fixes a :: "'a"   and b :: "'a" shows "ccw (0::'a) (- a) b = ccw (0::'a) b a"
mult: fixes x :: "V"   and y :: "V" shows "x * y = (SUP u\<in>elts y. lift (x * u) x)"
coeff_0_add_fract_nonzero: fixes x :: "'b poly fract"   and y :: "'b poly fract" assumes "constant_term (denominator x) \<noteq> (0::'b)"   and "constant_term (denominator y) \<noteq> (0::'b)" shows "constant_term (denominator (x + y)) \<noteq> (0::'b)"
lex_fun_alt: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "lex_fun s t = (s = t \<or> (\<exists>x. s x < t x \<and> (\<forall>y<x. s y = t y)))"
HEndPhase1_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2b s"   and "HEndPhase1 s s' p" shows "Inv2b s'"
length_getTvars: fixes \<sigma>l :: "'tp list" shows "length (TE.getTvars \<sigma>l) = length \<sigma>l"
simps_3: fixes x :: "quat" shows "quat_proj x (Suc (Suc 0)) = Im2 x"
invariant_weakenE: fixes A :: "('c, 'd) automaton"   and PI :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and QI :: "'d \<Rightarrow> bool" assumes "A \<TTurnstile> (PI \<rightarrow>) P"   and "\<And>s. P s \<Longrightarrow> Q s"   and "\<And>a. QI a \<Longrightarrow> PI a" shows "A \<TTurnstile> (QI \<rightarrow>) Q"
that: fixes c1 :: "('a, 'c) chart" assumes "c1 \<in> src.atlas"   and "x \<in> domain c1"   and "k-smooth_on (codomain c1) (f \<circ> inv_chart c1)" shows "thesis"
nonneg_cexprI_shift: fixes t :: "pdf_type"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<And>x \<sigma>. \<lbrakk>x \<in> type_universe t; \<sigma> \<in> space (state_measure V \<Gamma>)\<rbrakk> \<Longrightarrow> 0 \<le> extract_real (cexpr_sem (case_nat x \<sigma>) e)" shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"
measure_pmf: fixes x :: "'b pmf" shows "measure_pmf x \<in> {M. prob_space M \<and> sets M = UNIV \<and> (AE x in M. Sigma_Algebra.measure M {x} \<noteq> 0)}"
distinct_29: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d" shows "x31\<not>x32:arrive(x33) \<noteq> \<tau>\<^sub>n"
StepRecSeq2: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat"   and c\<^sub>1 :: "cmd" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "c\<^sub>1;; prog \<turnstile> \<langle>c,s,l + #:c\<^sub>1\<rangle> \<leadsto> \<langle>c',s',l' + #:c\<^sub>1\<rangle>"
arr_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "J.arr (J.dom f)"
fst_eqD: fixes x :: "'a"   and y :: "'b"   and a :: "'a" assumes "fst (x, y) = a" shows "x = a"
leg_simps_4: shows "C.dom f.leg1 = f.apex"
canI_associator_1_3: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<a>[f, g, src g] = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src g\<^bold>\<rangle>\<^sub>0) ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>src g\<^bold>\<rangle>\<^sub>0)"
eq_onp_same_args: fixes P :: "'h \<Rightarrow> bool"   and x :: "'h" shows "eq_onp P x x = P x"
star_prod_unfold: fixes x :: "'a"   and y :: "'a" shows "(1::'a) + x \<cdot> (y \<cdot> x)\<^sup>\<star> \<cdot> y = (x \<cdot> y)\<^sup>\<star>"
rcos_ring_hom: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R" shows "(+>\<^bsub>R\<^esub>) I \<in> ring_hom R (R Quot I)"
C_def: shows "C = (\<lambda>n. UNIV - A n)"
well_related_set_axioms: shows "well_related_set B (\<sqsubseteq>)"
prems_4: shows "well_valued0 (xs @ as) (Node i j a t1 t2)"
hyps: fixes I :: "nat \<Rightarrow> 'a set" assumes "I \<in> {..Suc n} \<rightarrow> {J. ideal J R}"   and "\<And>i j. \<lbrakk>i \<le> Suc n; j \<le> Suc n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "finprod (ideals_set R) I {..n} <+> I (Suc n) = carrier R"
FL_valid_Act: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and P :: "'a"   and f :: "'d"   and \<alpha> :: "'c"   and x :: "('e, 'b, 'c, 'd) formula" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid satisfies transition effect_apply P (Act f \<alpha> x) = (\<exists>\<alpha>' x' P'. Act f \<alpha> x = Act f \<alpha>' x' \<and> transition (effect_apply f P) \<langle>\<alpha>',P'\<rangle> \<and> effect_nominal_ts.FL_valid satisfies transition effect_apply P' x')"
sams_lea2_suma2__lea: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.LeA Bet Cong D E F D' E' F'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A' B' C' D' E' F'"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A' B' C' D' E' F' G' H' I'" shows "Tarski_neutral_dimensionless.LeA Bet Cong G H I G' H' I'"
add_mset_remove_trivial: fixes x :: "'k"   and M :: "'k multiset" shows "add_mset x M - {#x#} = M"
that: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set \<psi>"   and "map (sasp_op_to_strips \<Psi>) ops = [\<phi>\<^sub>O \<Psi> op]" shows "thesis"
internalize_card_of_ordLeq: fixes A :: "'a set"   and r :: "('b \<times> 'b) set" shows "(|A| \<le>o r) = (\<exists>B\<subseteq>Field r. |A| =o |B| \<and> |B| \<le>o r)"
individually_rational_on_empty: fixes ds :: "'d set" shows "individually_rational_on ds {}"
blinfun_eqI: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
map_Val_eq: fixes vs :: "val list"   and vs' :: "val list" assumes "map Val vs = map Val vs'" shows "vs = vs'"
fundamental_theorem_1: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and \<Gamma>' :: "(name \<times> ty) list"   and \<theta> :: "(name \<times> trm) list"   and \<theta>' :: "(name \<times> trm) list" assumes " \<Gamma> \<turnstile> t : T "   and "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> \<theta><t> is \<theta>'<t> : T"
affine_parallel_assoc: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "affine_parallel A B"   and "affine_parallel B C" shows "affine_parallel A C"
fwd_bc: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>" shows "pout (\<sigma>\<^bsub>nid\<^esub>t n) = bc (\<sigma>\<^bsub>nid\<^esub>t n)"
Units_closed: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "x \<in> carrier Zp"
fresh_X: shows "(p \<bullet> bn \<alpha>) \<sharp>* X"
scaleR_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "real"   and y :: "'a"   and n :: "nat" assumes "class.real_normed_algebra_1 plus zero minus uminus scaleR times one dist norm sgn uniformity open" shows "power.power one times (scaleR x y) n = scaleR (x ^ n) (power.power one times y n)"
PiE_empty_range: fixes i :: "'d"   and I :: "'d set"   and F :: "'d \<Rightarrow> 'e set" assumes "i \<in> I"   and "F i = {}" shows "Pi\<^sub>E I F = {}"
relcompp_apply: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and c :: "'c" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
stoch_proc_filt_def: fixes M :: "'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'d measure"   and n :: "nat" shows "stoch_proc_filt M X N n = gen_subalgebra M (sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets N}))"
inverse_real_of_posnat_ge_real_FreeUltrafilterNat: fixes u :: "real" assumes "0 < u" shows "\<not> (\<forall>\<^sub>F n in \<U>. u \<le> inverse (real (Suc n)))"
nearly_healthy_wlp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy t" shows "nearly_healthy (wlp (Embed t))"
size_mset_remove1_mset_le_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(size (remove1_mset x M) < size M) = (x \<in># M)"
assms_2: fixes b :: "'a" assumes "b \<in> A" shows "b dvd a"
normalI: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "subgroup H G"   and "\<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H" shows "H \<lhd> G"
subsetI: fixes A :: "'d set"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
weaken_SPEC: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and \<Psi> :: "'a \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "m \<le> SPEC \<Psi>"
results'_gpv_catch_gpvE: fixes x :: "'d option"   and gpv :: "('d, 'e, 'f) gpv" assumes "x \<in> results'_gpv (catch_gpv gpv)" obtains x' :: "'d" where "x = Some x'" and "x' \<in> results'_gpv gpv"   | "x = None" and "\<not> colossless_gpv \<I>_full gpv"
sel_2: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "stl (s1 ||| s2) = stl s1 ||| stl s2"
listset_singletonE: fixes ys :: "'b list"   and A :: "'b set" assumes "ys \<in> listset [A]" obtains a :: "'b" where "a \<in> A" and "ys = [a]"
cfg_onD_state: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> R_G.cfg_on s" shows "state cfg = s"
ucast_0: shows "UCAST('c \<rightarrow> 'b) 0 = 0"
iii: shows "\<forall>x\<in>E - E'. \<not> (forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x E'\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>)"
sigma_algebra_def: fixes A :: "'b set set" shows "sigma_algebra A = ({} \<in> A \<and> (\<forall>a. a \<in> A \<longrightarrow> - a \<in> A) \<and> (\<forall>a. (\<forall>i. a i \<in> A) \<longrightarrow> \<Union> (range a) \<in> A))"
levelsCounter_auxIncreasesFirst: fixes a :: "('b \<times> bool) list" shows "\<forall>m n l1 l2. levelsCounter_aux a (m # l1) = n # l2 \<longrightarrow> m \<le> n"
zassenhaus_reconstruction_generic: fixes sl_impl :: "(int poly, int \<times> int poly list, 'a) subseqs_foldr_impl"   and sli :: "int \<times> int poly list \<Rightarrow> int poly list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> bool"   and fs :: "int poly list" assumes "correct_subseqs_foldr_impl (\<lambda>v. map_prod (poly_mod.mul_const (p ^ n) v) ((#) v)) sl_impl sli"   and "zassenhaus_reconstruction_generic sl_impl hs p n f = fs" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"