hcju/isabelleps huggingface dataset:
fps_deriv_power': fixes a :: "'a fps"   and n :: "nat" shows "fps_deriv (a ^ n) = of_nat n * fps_deriv a * a ^ (n - 1)"
no_defection_def': fixes s :: "v_state"   and r_votes :: "process \<Rightarrow> val option"   and r :: "nat" shows "no_defection s r_votes r \<equiv> \<forall>r'<r. \<forall>Q\<in>Quorum. \<forall>v. votes s r' ` Q = {Some v} \<longrightarrow> r_votes ` Q \<subseteq> {None, Some v}"
paval_aval: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "vars a \<subseteq> dom ps" shows "paval a ps = aval a (\<lambda>v. case ps v of None \<Rightarrow> s v | Some r \<Rightarrow> r)"
fn2: fixes a :: "'b"   and b :: "'b"   and A :: "'b ptrm"   and B :: "'b ptrm"   and T :: "PreSimplyTyped.type" assumes "a \<noteq> b"   and "A \<approx> [a \<leftrightarrow> b] \<bullet> B"   and "a \<notin> ptrm_fvs B" shows "PFn a T A \<approx> PFn b T B"
preserves_ide: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "Adj.Cop.ide a" shows "Adj.Cop.ide (Adj.Cop.map a)"
sum_lookup_costs: fixes t :: "'a tree" assumes "bst t" shows "(\<Sum>x\<in>set_tree t. lookup_cost x t) = ipl t"
vifunion_vdoubleton: fixes i :: "V"   and j :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> (VLambda (set {i, j}) f `\<^sub>\<circ> set {i, j}) = f i \<union>\<^sub>\<circ> f j"
assms_1: shows "t \<in>\<^sub>A frontier (c_imp c loc)"
prems_2: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : t1s _> t2s"
asubGroup_def: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" shows "A +> H = b_ag A \<guillemotright> H "
pCons_2: shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
x_c_U_split: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> U = x \<cdot> U + x \<cdot> (0::'a) \<parallel> U"
filterlim_mono_eventually: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and G :: "'a filter"   and F' :: "'b filter"   and G' :: "'a filter"   and f' :: "'a \<Rightarrow> 'b" assumes "filterlim f F G"   and "F \<le> F'"   and "G' \<le> G"   and "\<forall>\<^sub>F x in G'. f x = f' x" shows "filterlim f' F' G'"
insert_diff: fixes A :: "'c set"   and b :: "'c"   and B :: "'c set" assumes "A \<subseteq> insert b B" shows "A - {b} \<subseteq> B"
nprv_sound: fixes H :: "fmla set"   and A :: "fmla"   and e :: "(name, nat) finfun" assumes "H \<turnstile> A"   and "\<forall>B\<in>H. eval_fmla e B" shows "eval_fmla e A"
lsl_lsu_ran: shows "range \<nu>\<^sup>\<natural> = range \<nu>"
no_smallest_element_skipped_empty: shows "no_smallest_element_skipped carrier {}"
vrat_uminus_closed: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "-\<^sub>\<rat> x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
set_valued_functor_axioms: shows "set_valued_functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F"
A_splay_ub: fixes t :: "'b tree"   and l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "bst t"   and "\<langle>l, x, r\<rangle> \<in> subtrees t" shows "A_splay x t \<le> 3 * (\<phi> t - \<phi> \<langle>l, x, r\<rangle>) + 1"
append_rows_index: fixes A :: "'b mat"   and B :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "dim_col A = dim_col B"   and "i < dim_row A + dim_row B"   and "j < dim_col A" shows "(A @\<^sub>r B) $$ (i, j) = (if i < dim_row A then A $$ (i, j) else B $$ (i - dim_row A, j))"
basis_gen_hull: fixes G :: "'b list set" shows "\<langle>\<BB> G\<rangle> = \<langle>G\<rangle>"
simps_3: shows "is_cnf \<bottom> = is_disj \<bottom>"
exist_degree_two: assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n"   and "finite E"   and "finite V"   and "2 \<le> card V" shows "\<exists>v\<in>V. degree v G = 2"
FrQ': shows "extractFrame Q = \<langle>A\<^sub>Q', \<Psi>\<^sub>Q\<rangle>"
hne: shows "half_segments_of_aform X \<noteq> []"
ennreal_indicator_less: fixes A :: "'c set"   and x :: "'c"   and B :: "'c set" shows "(indicator A x \<le> indicator B x) = (x \<in> A \<longrightarrow> x \<in> B)"
topological_basis: fixes B :: "'a set set" shows "topological_basis B = (\<forall>x. open x = (\<exists>B'\<subseteq>B. \<Union> B' = x))"
TESL_interp_unfold_stepwise_sporadicon: fixes K\<^sub>1 :: "clock"   and \<tau> :: "'a tag_const"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Union> {Y. \<exists>n. Y = \<lbrakk> K\<^sub>1 sporadic \<tau> on K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
imp_recovery0: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "disj_PL p (imp_PL p q) \<in> Cn A"
Ana_const: fixes c :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "arity c = 0" shows "Ana (Fun c T) = ([], [])"
f_fold_merge: fixes xs :: "'a list"   and x :: "'a" assumes "\<And>y. y \<in> set xs \<Longrightarrow> f y = f x" shows "f (fold merge xs x) = f x"
semiring: shows "semiring_of_sets \<Omega> {G a b |a b. P a b}"
iso_bij_betw_individual_blocks: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "bij_betw \<pi> bl (\<pi> ` bl)"
stirling_limit_aux1: fixes z :: "complex" shows "((\<lambda>y. Ln (1 + z * complex_of_real y) / complex_of_real y) \<longlongrightarrow> z) (at_right 0)"
universal: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "\<And>\<alpha>. [\<phi> \<alpha> in v]" shows "[\<^bold>\<forall>\<alpha>. \<phi> \<alpha> in v]"
special_points_on_x_axis''_3: shows "\<infinity>\<^sub>h \<in> circline_set x_axis"
B_theorem_3_main: shows "\<Theta> i * f i / \<gamma> (i + 1) = (- 1) ^ (n 1 + n i + i + 1) / f i * (\<Prod>l\<leftarrow>[3..<Suc (Suc i)]. \<alpha> l / \<beta> l)"
simps: fixes uu :: "'b"   and p :: "('a, 'c) antecedent" shows "labelsIn uu p = a_conc p"
discI: fixes tree :: "tree"   and x21 :: "tree"   and x22 :: "tree" assumes "tree = Branching x21 x22" shows "tree \<noteq> Leaf"
tail_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> arcs (with_proj G)" shows "tail (with_proj G) e \<in> verts (with_proj G)"
hom_power: fixes x :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "x ^ n \<circ>\<^sub>p p = (x \<circ>\<^sub>p p) ^ n"
generalized_fw_join_1_nomatch: fixes am :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme"   and ad :: "'c"   and l2 :: "('a simple_match \<times> 'd) list" assumes "\<not> simple_matches am p" shows "generalized_sfw (concat (map (\<lambda>(m2, b). map (\<lambda>u. (u, ad, b)) (option2list (simple_match_and am m2))) l2)) p = None"
subadditive: fixes f :: "('i \<Rightarrow> 'a) set \<Rightarrow> ennreal"   and A :: "'e \<Rightarrow> ('i \<Rightarrow> 'a) set"   and S :: "'e set" assumes "positive generator f"   and "Measure_Space.additive generator f"   and "A ` S \<subseteq> generator"   and "finite S" shows "f (\<Union> (A ` S)) \<le> (\<Sum>i\<in>S. f (A i))"
ennreal_tendsto_const_minus: fixes g :: "'b \<Rightarrow> ennreal"   and c :: "ennreal"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. g x \<le> c"   and "((\<lambda>x. c - g x) \<longlongrightarrow> 0) F" shows "(g \<longlongrightarrow> c) F"
hyps_3: fixes i :: "nat"   and t :: "('a, 'b) Term.term" assumes "i < length T"   and "T ! i = t \<cdot> \<delta>"   and "\<forall>x\<in>fv t \<union> fv (S ! i). (\<exists>y. \<delta> x = Var y) \<or> (\<exists>f. \<delta> x = Fun f [] \<and> P f = [])" shows "\<exists>u. term_variants_pred P t u \<and> S ! i = u \<cdot> \<delta>"
qle_trans: fixes qle :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "quasi_order qle"   and "qle x y"   and "qle y z" shows "qle x z"
inf_sub_right: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "z -\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
delta_pos_neg_least_1: fixes F :: "('a \<Rightarrow> 'b) admS dual \<times> ('a \<Rightarrow> 'b) admS \<Rightarrow> ('a \<Rightarrow> 'b) admS"   and \<delta> :: "('b \<rightarrow> 'b) \<rightarrow> 'b \<rightarrow> 'b"   and rm :: "('a \<Rightarrow> 'b) admS"   and rp :: "('a \<Rightarrow> 'b) admS" assumes "DomSolV F \<delta>"   and "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "DomSolV.delta_neg F \<le> dual rm"
simps: fixes l :: "rat"   and r :: "rat"   and sr :: "rat" shows "local.tighten_poly_bounds_epsilon l r sr = (if r - l \<le> x then (l, r, sr) else case local.tighten_poly_bounds l r sr of (l', r', sr') \<Rightarrow> local.tighten_poly_bounds_epsilon l' r' sr')"
simps_1: fixes a :: "('b, 'c) Combinators"   and x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators" shows "member a (x \<oplus> xs) = (member a x \<or> member a xs)"
fold_graph_fold: fixes A :: "'b set"   and z :: "'a" assumes "finite A" shows "fold_graph (pls' \<circ> g) z A (Finite_Set.fold (pls' \<circ> g) z A)"
mset_concat: fixes xss :: "'b list list" shows "mset (concat xss) = sum_list (map mset xss)"
o1: shows "tlookup fst i t = Some (fst ia, snd ia)"
extensional_set: fixes a :: "'s"   and b :: "'s" assumes "S.ide a"   and "S.ide b"   and "S.set a = S.set b" shows "a = b"
in_angle_out: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and P :: "'p" assumes "B Out A C"   and "P InAngle A B C" shows "B Out A P"
cols_nth: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_col A" shows "cols A ! i = col A i"
Cprops_1: shows "support_set p \<inter> C = {}"
that: fixes S :: "'b set" assumes "open S"   and "l \<in> S"   and "\<exists>\<^sub>F x in F. f x \<notin> S" shows "thesis"
IH: fixes s :: "'a net_state"   and t :: "'a net_state" assumes "st = SubnetS s t" shows "P s t"
exec_move_BlockSomeI: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and e :: "'a expr1"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and xcp' :: "'a option"   and V :: "nat"   and T :: "ty"   and v :: "'a val" assumes "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t {V:T=\<lfloor>v\<rfloor>; e} h (stk, loc, Suc (Suc pc), xcp) ta h' (stk', loc', Suc (Suc pc'), xcp')"
dom_trm_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 t :: "'b trm"   and C :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.dom_trm trm_ord t C = (\<exists>L u v p. L \<in> C \<and> decompose_literal L u v p \<and> (p = neg \<and> t = u \<or> (t, u) \<in> trm_ord))"
space_closed: fixes M :: "'b measure" shows "null_sets M \<subseteq> Pow (space M)"
discard_def: shows "discard \<equiv> snd (takedropWhile_queue (\<lambda>(t, X). right I < enat (nt - t)) data_in)"
h2: assumes "\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
mk_diagonal_dim_1: fixes as :: "'a list" shows "dim_row (mk_diagonal as) = length as"
AMin_k: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>k\<in>carrier K - {\<zero>}. AMin {x \<in> v ` carrier K. 0 < x} = v k"
rel_interior_translation: fixes a :: "'c"   and S :: "'c set" shows "rel_interior ((+) a ` S) = (+) a ` rel_interior S"
fmapU_btree_def: shows "fmapU \<equiv> btree_map"
submonoid_is_monoid: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "submonoid H G"   and "Group.monoid G" shows "Group.monoid (G\<lparr>carrier := H\<rparr>)"
intros_4: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
subdg_dom_simp: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = \<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr>"
rw_rat_strict_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "rw_rat_strict p q = (let u = punit.lt (snd q) \<oplus> fst p; v = punit.lt (snd p) \<oplus> fst q in u \<prec>\<^sub>t v \<or> u = v \<and> fst p \<prec>\<^sub>t fst q)"
poly_x_minus_y_def: fixes p :: "'a poly" shows "poly_x_minus_y p = poly_lift p \<circ>\<^sub>p x_y"
subcls_conv_subcls': fixes P :: "'c prog" shows "subcls P = subcls' P"
less_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "(d1 < d2) = (d1 = Low \<and> d2 = High)"
inf_sup_ord_3: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<squnion> y"
wf_exists_wls: fixes R :: "('sort \<times> 'sort) set"   and f :: "'sort \<Rightarrow> 'opSym"   and s :: "'sort" assumes "wf R"   and "\<And>s. (\<exists>xs. s = asSort xs) \<or> witOpS s (f s) R" shows "\<exists>X. wls s X"
a5_def: shows "a5 = is_a_proper_intersec Q B C' B' C"
paths_second_not_v0: fixes xs :: "'a list" assumes "xs \<in> paths" shows "hd (tl xs) \<noteq> v0"
un_B_inverse: fixes x :: "'c dual" shows "B (un_B x) = x"
enumerate_step: fixes S :: "'a set"   and n :: "nat" assumes "infinite S" shows "enum S n < enum S (Suc n)"
Gram_Schmidt_upt_k_efficient: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "Gram_Schmidt_upt_k A k = foldl Gram_Schmidt_column_k_efficient A [0..<Suc k]"
set_restriction_card: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "set_restriction f"   and "finite A" shows "card (f A) = card A - card {a \<in> A. f {a} = {}}"
arr_mset_remove_last: assumes "l \<le> r" shows "arr_mset arr l r = arr_mset arr l (r + 1) - {#arr r#}"
has_integral_integrable_integral: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and s :: "'a set" shows "(f has_integral i) s = (f integrable_on s \<and> integral s f = i)"
ag_pOp_add_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "a \<plusminus> c = b \<plusminus> c"
huntington_id: fixes x :: "'a" shows "x \<squnion> - x = - x \<squnion> - (- x)"
div0_uint32_def: fixes x :: "uint32" shows "div0_uint32 x = undefined (div) x 0"
FEx0: fixes m :: "nat"   and g :: "form" assumes "infinite (deriv s)"   and "init s" shows "\<forall>n. contains f n (m, FEx g) \<longrightarrow> (\<exists>n'. contains f n' (0, FEx g))"
elims_2: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v" assumes "is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = va__" and "xa = []" and "xb = v'a__" and "va__ = v'a__ \<and> v'a__ \<in> V"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and p :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> E \<and> is_path v2 p v'"
bounded_w': fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "w i \<in> \<Sigma>"
intf_type_def: fixes c :: "'a"   and I :: "'b itself" shows "c ::\<^sub>i I \<equiv> True"
l_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier \<G>"   and "a \<in> carrier \<G>" shows "(x = x \<otimes> a) = (a = \<one>)"
chainI: fixes C :: "'f set set"   and A :: "'f set set" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x" shows "subset.chain A C"
And\<^sub>n_inj: shows "inj_on And\<^sub>n {s. finite s}"
kmp_def: fixes s :: "'b list"   and t :: "'b list" shows "kmp s t \<equiv> ASSERT (s \<noteq> []) \<bind> (\<lambda>_. let i = 0; j = 0; pos = None in WHILE\<^sub>T\<^bsup>I_outer s t\<^esup> (\<lambda>(i, j, pos). i + length s \<le> length t \<and> pos = None) (\<lambda>(i, j, pos). ASSERT (i + length s \<le> length t) \<bind> (\<lambda>_. WHILE\<^sub>T\<^bsup>I_in_na s t i\<^esup> (\<lambda>(j, pos). t ! (i + j) = s ! j \<and> pos = None) (\<lambda>(j, pos). let j = j + 1 in if j = length s then RETURN (j, Some i) else RETURN (j, None)) (j, pos) \<bind> (\<lambda>(j, pos). if pos = None then ASSERT (j < length s) \<bind> (\<lambda>_. let i = i + (j - \<ff> s j + 1); j = max 0 (\<ff> s j - 1) in RETURN (i, j, None)) else RETURN (i, j, Some i)))) (i, j, pos) \<bind> (\<lambda>(uu_, uu_, pos). RETURN pos))"
transpose_rank_mul_conjugate_transpose: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "vec_space.rank nc A\<^sup>H \<le> rank (A * A\<^sup>H)"
greedy_path_lhd: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P" shows "v = v0"
sheaf_morphisms_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" assumes "s \<in> \<O> U" shows "sheaf_spec_morphisms U U s = s"
mono_step'_top: fixes C :: "(char list \<Rightarrow> 'a) option acom"   and C' :: "(char list \<Rightarrow> 'a) option acom" assumes "C \<le> C'" shows "step' \<top> C \<le> step' \<top> C'"
hyps_2: shows "n irs-ns\<rightarrow>\<^sub>d* n''"
incidence_system_isomorphism_axioms: shows "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' \<pi>"
intro: fixes r1 :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and r2 :: "('d, 'a, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'a, 'b, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'a, 'g, 'h, 'e, 'f) \<tau>moves"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'g \<Rightarrow> bool" assumes "\<And>t. delay_bisimulation_final_base (r1 t) (r2 t) (bisim t) \<tau>move1 \<tau>move2 (\<lambda>(x1, m). final1 x1) (\<lambda>(x2, m). final2 x2)" shows "FWdelay_bisimulation_final_base_axioms final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"
vec_def: fixes x :: "'d" shows "vec x = (\<chi>i. x)"
simps_2: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) list" shows "exe_antisym ((x, y) # r) = (((y, x) \<in> set r \<longrightarrow> x = y) \<and> exe_antisym r)"
assms_2: shows "[z * x = 1] (mod int (order \<G>))"
match_IfaceFalse: shows "\<nexists>IfaceFalse. \<forall>i. \<not> match_iface IfaceFalse i"
I_All: fixes X :: "W.tvar"   and T' :: "ty"   and T :: "ty"   and S :: "tyS" assumes "X \<sharp> T'"   and "T \<prec> S" shows "T[X::=T'] \<prec> \<forall>[X].S"
cross3_simps_12: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * (a * c) = a * (b * c)"
no_roots_imp_same_roots: fixes p :: "'a list"   and q :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "p \<noteq> []"   and "q \<in> carrier (poly_ring R)"   and "roots p = {#}" shows "roots (p \<otimes>\<^bsub>poly_ring R\<^esub> q) = roots q"
P2_security: fixes D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf"   and n :: "nat"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "\<forall>a. lossless_spmf (D a)"   and "\<forall>b\<^sub>\<sigma>. local.etp.HCP_adv n \<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "OT_12.adv_P2 n m1 m2 D \<le> 2 * HCP_ad"
open_contains_box: assumes "open (ball s r)"   and "s \<in> ball s r" obtains a :: "complex" and b :: "complex" where "box a b \<subseteq> ball s r" and "s \<in> box a b" and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i"
simps_2: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and v :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and va :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.getMinTree (t # v # va) = (if BinomialHeapStruc.prio t \<le> BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree (v # va)) then t else BinomialHeapStruc.getMinTree (v # va))"
mult_le_0_iff: fixes a :: "'a"   and b :: "'a" shows "(a * b \<le> (0::'a)) = ((0::'a) \<le> a \<and> b \<le> (0::'a) \<or> a \<le> (0::'a) \<and> (0::'a) \<le> b)"
measure_pmf_bind: fixes M :: "'e pmf"   and f :: "'e \<Rightarrow> 'd pmf" shows "measure_pmf (M \<bind> f) = measure_pmf M \<bind> (\<lambda>x. measure_pmf (f x))"
measure_spmf_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(rel_spmf A ===> rel_pred A ===> (=)) (\<lambda>p. Sigma_Algebra.measure (measure_spmf p)) (\<lambda>q. Sigma_Algebra.measure (measure_spmf q))"
intro: fixes isom :: "('a, 'b, 'c) tuple_isomorphism" assumes "\<And>x. abst isom (repr isom x) = x"   and "\<And>y. repr isom (abst isom y) = y" shows "isomorphic_tuple isom"
the_inv_into_vimage: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and A :: "'c set" assumes "inj_on f X"   and "A \<subseteq> X" shows "the_inv_into X f -` A \<inter> f ` X = f ` A"
viewR_correct_nonEmpty: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t \<noteq> Empty" obtains a :: "'a \<times> 'b" and s :: "('a, 'b) FingerTreeStruc" where "FingerTreeStruc.viewR t = Some (a, s)" and "FingerTreeStruc.ft_invar s \<and> FingerTreeStruc.toList t = FingerTreeStruc.toList s @ [a]"
eq_norm: shows "norm_instr (body fd2 ! pc) = norm_instr instr"
j: fixes z :: "'a" assumes "z \<in> ball (0::'a) 1" shows "exp (j z) = (g \<circ> h) z"
weight_sample_uniform: fixes n :: "nat" shows "weight_spmf (sample_uniform n) = indicat_real (range Suc) n"
map_add_restrict_dom_left: fixes h :: "'c \<Rightarrow> 'd option"   and h' :: "'c \<Rightarrow> 'd option" assumes "h \<bottom> h'" shows "(h ++ h') |` dom h = h"
if_cancel: fixes c :: "bool"   and x :: "'a" shows "(if c then x else x) = x"
F_anal_valid: shows "\<forall>twoC\<in>two_chain_typeII. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i"
foldD_empty: fixes e :: "'e"   and D :: "'e set"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "e \<in> D" shows "foldD D f e {} = e"
wsorted_map\<alpha>: fixes s :: "'cedge list" assumes "wsorted' s" shows "wsorted (map \<alpha> s)"
gbinomial_Suc0: fixes a :: "'c" shows "a gchoose Suc 0 = a"
a_antitone': fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "ad y \<le> ad x"
not_less: fixes x :: "'c"   and y :: "'c" shows "(\<not> x < y) = (y \<le> x)"