hcju/isabelleps huggingface dataset:
torder_on_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "torder_on A r = (porder_on A r \<and> total_onP A r)"
DBM_reset'_complete: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and cs :: "'c list"   and d :: "'d" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n" shows "[cs\<rightarrow>d]u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
2: shows "rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) A = rat_poly_times B A - rat_poly_times (A ^ 3) A"
starfun_le_mono: fixes N :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>n\<ge>N. f n \<le> g n" shows "\<forall>n\<ge>hypnat_of_nat N. (*f* f) n \<le> (*f* g) n"
A: shows "u \<in> sinks_aux I D (insert v U) xs"
invertible_def: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> G" shows "symmetric.sub.invertible u = (\<exists>v\<in>G. compose S u v = identity S \<and> compose S v u = identity S)"
strict_trans1: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<preceq> a"   and "c \<prec> b" shows "c \<prec> a"
mod_ring_minus64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (minus_p64 pp) (-)"
fundamental_property_of_static_slicing: fixes m :: "'node"   and as :: "'edge list"   and m' :: "'node"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and S :: "'node SDG_node set" assumes "m -as\<rightarrow>\<^sub>\<surd>* m'"   and "preds (kinds as) [cf]"   and "CFG_node m' \<in> S" obtains as' :: "'edge list" where "preds (slice_kinds S as') [cf]" and "\<forall>V\<in>Use m'. state_val (transfers (slice_kinds S as') [cf]) V = state_val (transfers (kinds as) [cf]) V" and "slice_edges S [] as = slice_edges S [] as'" and "transfers (kinds as) [cf] \<noteq> []" and "m -as'\<rightarrow>\<^sub>\<surd>* m'"
news_subcs: fixes d :: "'a"   and s :: "'a FOL_Fitting.term"   and z :: "('a, 'b) form list"   and c :: "'a" assumes "new_term d s"   and "news d z" shows "news d (subcs c s z)"
steps_impl_execs: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* ([], [], t)" shows "\<Gamma>\<turnstile> \<langle>cs,css,s\<rangle> \<Rightarrow> t"
top_on_opt_widen: fixes o1 :: "'a st option"   and X :: "char list set"   and o2 :: "'a st option" assumes "top_on\<^sub>o o1 X"   and "top_on\<^sub>o o2 X" shows "top_on\<^sub>o (o1 \<nabla> o2) X"
parCasesInputFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and T :: "('a, 'b, 'c) psi"   and A\<^sub>P\<^sub>Q :: "name list"   and \<Psi>\<^sub>P\<^sub>Q :: "'b"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> T"   and "extractFrame (P \<parallel> Q) = \<langle>A\<^sub>P\<^sub>Q, \<Psi>\<^sub>P\<^sub>Q\<rangle>"   and "distinct A\<^sub>P\<^sub>Q"   and "A\<^sub>P\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>P\<^sub>Q \<sharp>* P"   and "A\<^sub>P\<^sub>Q \<sharp>* Q"   and "A\<^sub>P\<^sub>Q \<sharp>* M"   and "\<And>P' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P' \<parallel> Q)"   and "\<And>Q' A\<^sub>P \<Psi>\<^sub>P A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>P; distinct A\<^sub>Q; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* M; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* M; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q; \<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q\<rbrakk> \<Longrightarrow> Prop (P \<parallel> Q')" shows "Prop T"
3: shows "coreflexive ((f \<sqinter> - q) * e\<^sup>T)"
intros_1: fixes a :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" shows "(a, []) \<in> conv ars"
nin_nodes_reachable: fixes n :: "nat" assumes "n \<notin> set (g_V G)" shows "\<not> reachable n"
delete2_def: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "local.delete2 x t = Time_Monad.val (delete2_tm x t)"
sublens_put_put: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'c"   and u :: "'a" assumes "mwb_lens X"   and "Y \<subseteq>\<^sub>L X" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>X\<^esub> \<sigma> u"
ide_MkIde: fixes A :: "'j \<Rightarrow> 'c" assumes "A \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>))" shows "J_C.ide (J_C.MkIde A)"
assms_2: shows "bal_i n (height \<langle>l, y, r\<rangle> + d)"
assms: shows "(let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) (mem s1)) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table (mem s1) 1))) = (let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2) in Let (mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) (mem s2)) (case_option None (\<lambda>lvl1_page_table. ptd_lookup va lvl1_page_table (mem s2) 1)))"
lub_below: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "class.cpo below"   and "po.chain below S"   and "\<And>i. below (S i) x" shows "below (po.lub below (range S)) x"
eq: shows "bs = take (maxindex p) bs @ drop (maxindex p) bs"
delete_point_design: fixes p :: "'a" assumes "\<And>bl. \<lbrakk>bl \<in># \<B>; p \<in> bl\<rbrakk> \<Longrightarrow> 2 \<le> card bl" shows "design (del_point p) (del_point_blocks p)"
Limit_vsingleton_in_VsetD: fixes a :: "V"   and \<alpha> :: "V" assumes "set {a} \<in>\<^sub>\<circ> Vset \<alpha>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
closed_segment_image_interval: fixes a :: "'b"   and b :: "'b" shows "{a--b} = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0..1}"
assms: shows "\<forall>t\<in>FP. \<forall>(a, b)\<in>TI. \<forall>s\<in>set \<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle>. s \<in> FP"
Rec_succ: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "'a"   and n :: "'c" assumes "Rec e r m n" shows "Rec e r (succ m) (r m n)"
taut_test: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'d" assumes "reduced_bdt_checkers ifex_of val" shows "taut_test ifex_of b = (\<forall>s. val b s)"
non_speculative_written_value_typeable: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'md \<Rightarrow> 'addr val list \<Rightarrow> 'x"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and status :: "status"   and w :: "nat"   and ad :: "'addr"   and al :: "addr_loc" assumes "ts_ok wfx (thr (h.start_state f P C M vs)) h.start_heap"   and "wf_syscls P"   and "E \<in> h.\<E>_start f P C M vs status"   and "w \<in> write_actions E"   and "(ad, al) \<in> action_loc P E w"   and "non_speculative P (vs_type_all P) (lmap snd (ltake (enat w) E))" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> value_written P E w (ad, al) :\<le> T"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}\<lbrakk>fa\<rbrakk> p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
Ints_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "plus a b \<in> ring_1.Ints one plus zero minus"
fv_regex_alt: fixes fv :: "'d \<Rightarrow> 'c set"   and r :: "'d regex" shows "fv_regex fv r = \<Union> (fv ` atms r)"
correct_routing_def: fixes r :: "'b routing_rule list" shows "correct_routing r \<equiv> is_longest_prefix_routing r \<and> valid_prefixes r"
trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst: fixes a :: "(('a, 'c) Term.term \<times> ('a, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (a \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
ord_iff_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "less_eq x y = (rep_nat x \<le> rep_nat y)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "zero_hom hom"   and "injective hom" shows "inj_zero_hom hom"
universe_closure: assumes "CMU" shows "\<exists>y. \<forall>x. P x y"
remove_all_rev_removes: fixes x :: "'b"   and xs :: "'b list" shows "x \<notin> set (remove_all_rev x xs)"
abs_eq: fixes xa :: "int"   and x :: "nat" shows "clearBit (Word.Word xa) x = Word.Word (unset_bit x xa)"
atMost_iff: fixes i :: "'c list"   and k :: "'c list" shows "(i \<in> prefix_order.atMost k) = prefix i k"
intros_4: fixes u :: "'d \<Rightarrow> 'c"   and c :: "'d"   and d :: "'c" assumes "- u c < d" shows "dbm_entry_val u None (Some c) (Lt d)"
prems_1: fixes i :: "'b" assumes "i \<in> insert i__ I" shows "(f i \<longlongrightarrow> a i) F"
embed_measure_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and M :: "'a measure" assumes "inj f"   and "inj g" shows "embed_measure (embed_measure M f) g = embed_measure M (g \<circ> f)"
optimize_matches_generic: fixes rs :: "'c rule list"   and P :: "'c match_expr \<Rightarrow> bool"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c match_expr \<Rightarrow> 'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and s :: "state"   and t :: "state" assumes "\<forall>r\<in>set rs. P (get_match r)"   and "\<And>m. P m \<Longrightarrow> matches \<gamma> (f m) p = matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>optimize_matches f rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
meet_left: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<sqsubseteq>\<^bsub>L\<^esub> x"
psi_notI: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct3 p q r"   and "psi p q r" shows "\<not> psi q p r"
nonempty_p_fusion: fixes ps :: "'a list"   and qs :: "'a list" assumes "ps \<noteq> []"   and "qs \<noteq> []" shows "p_fusion ps qs \<noteq> []"
hp_conv: fixes h :: "'b"   and l :: "'c" shows "hp (h, l) = h"
is_methdI: fixes G :: "prog"   and C :: "qtname"   and y :: "class"   and sig :: "sig"   and b :: "qtname \<times> methd" assumes "class G C = Some y"   and "methd G C sig = Some b" shows "is_methd G C sig"
inprod_cvec_def: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "\<langle>z,w\<rangle> = (let (z1, z2) = z; (w1, w2) = w in vec_cnj (z1, z2) *\<^sub>v\<^sub>v (w1, w2))"
maxR_def: fixes d :: "D2"   and x :: "X5"   and y :: "X5" shows "BSI.maxR d x y = (if (x, y) \<in> PX5d d then y else x)"
unitary_schur_decomposition: fixes A :: "complex mat"   and n :: "nat"   and es :: "complex list"   and B :: "complex mat"   and P :: "complex mat"   and Q :: "complex mat" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"   and "unitary_schur_decomposition A es = (B, P, Q)" shows "similar_mat_wit A B P Q \<and> upper_triangular B \<and> diag_mat B = es \<and> unitary P \<and> Q = adjoint P"
continuous_at_Inf_antimono: fixes f :: "'b \<Rightarrow> 'd"   and S :: "'b set" assumes "antimono f"   and "continuous (at_right (Inf S)) f"   and "S \<noteq> {}"   and "bdd_below S" shows "f (Inf S) = Sup (f ` S)"
frontier_halfspace_component_ge: fixes n :: "'c"   and c :: "real" assumes "n \<noteq> (0::'c)" shows "frontier {x. c \<le> x \<bullet> n} = plane n c"
l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM_DOM_Components.get_dom_component Shadow_DOM_DOM_Components.is_strongly_dom_component_safe Shadow_DOM_DOM_Components.is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs get_disconnected_nodes Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name"
Grp_def: fixes A :: "'n set"   and f :: "'n \<Rightarrow> 'o" shows "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
lcs_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b" assumes "class.ulcs_powerprod plus minus zero lcs" shows "lcs zero t = t"
bell01_index_3: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> $$ (2, 0) = complex_of_real (1 / sqrt 2)"
X_\<G>\<F>_Y_\<F>\<G>: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set" assumes "X \<subseteq> subformulas\<^sub>\<mu> \<phi>"   and "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"   and "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)"   and "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)" shows "X \<subseteq> \<G>\<F> \<phi> w \<and> Y \<subseteq> \<F>\<G> \<phi> w"
uncountable_def: fixes A :: "'a set" shows "uncountable A = (A \<noteq> {} \<and> (\<nexists>f. range f = A))"
part: shows "\<forall>i<length Xs. \<forall>j<length Xs. i \<noteq> j \<longrightarrow> Xs ! i \<inter> Xs ! j = {}"
preim_of_ideal_is_ideal: fixes J :: "'c set"   and B :: "'c set"   and addB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroB :: "'c"   and oneB :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d" assumes "ideal J B addB multB zeroB oneB"   and "ring_homomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "ideal (f \<^sup>\<inverse> A J) A addA multA zeroA oneA"
add_update_distrib: fixes ps1 :: "char list \<Rightarrow> int option"   and x1 :: "char list"   and y :: "int"   and ps2 :: "char list \<Rightarrow> int option"   and x2 :: "aexp" assumes "ps1 x1 = Some y"   and "ps1 ## ps2"   and "vars x2 \<subseteq> dom ps1" shows "ps1(x1 \<mapsto> paval x2 ps1) + ps2 = (ps1 + ps2)(x1 \<mapsto> paval x2 ps1)"
iUntil_conj_not: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1 \<and> \<not> Q t1. t1 \<U> t2 I. Q t2) = (P t1. t1 \<U> t2 I. Q t2)"
satisfies_atom'_stronger: fixes v :: "nat \<Rightarrow> 'a"   and a :: "'a atom" assumes "v \<Turnstile>\<^sub>a\<^sub>e a" shows "v \<Turnstile>\<^sub>a a"
check_in_heap_ptr_in_heap: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "(ptr |\<in>| object_ptr_kinds h) = h \<turnstile> ok check_in_heap ptr"
containsI: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" shows "Predicate_Compile.contains A x"
simps_2: fixes y :: "'a"   and ys :: "'a list" shows "selsort (y # ys) = (let xm = fold min_for ys y; xs' = remove1 xm (y # ys) in xm # selsort xs')"
LeastI: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and k :: "'e" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
join_upper1: fixes x :: "'c"   and y :: "'c" shows "x \<sqsubseteq> x \<squnion> y"
solve_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "solve A b = (if consistent A b then Some (solve_consistent_rref (fst (solve_system A b)) (snd (solve_system A b)), basis_null_space A) else None)"
geub_def: fixes lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and b :: "'a option" shows "\<unrhd>\<^sub>u\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> False | Some b' \<Rightarrow> lt b' c \<or> b' = c"
add_0: fixes a :: "'k" shows "(0::'k) + a = a"
lincomb_coordinates: fixes v :: "'a vec" assumes "v \<in> carrier_vec n" shows "lincomb (\<lambda>u. v $ (THE i. u = unit_vec n i)) (set (unit_vecs n)) = v"
special_subgroup_inj: shows "inj_on genby (Pow S)"
integral_norm_bound: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" shows "norm (integral\<^sup>L M f) \<le> LINT x|M. norm (f x)"
restrict_top_impl_spec: fixes s :: "'a"   and ni :: "'b"   and n :: "'c ifex"   and vr :: "'c"   and vl :: "bool" assumes "I s"   and "(ni, n) \<in> R s" shows "ospec (restrict_top_impl ni vr vl s) (\<lambda>(res, s'). (res, restrict_top n vr vl) \<in> R s \<and> s' = s)"
assms_2: shows "M 0 i + M i 0 = \<one>"
index_smult_vec_1: fixes i :: "nat"   and v :: "'b vec"   and a :: "'b" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $ i = a * v $ i"
o_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and x :: "'d" shows "(f \<circ> g) x = f (g x)"
set_intros_1: fixes x1 :: "'b"   and x2 :: "'b stream" shows "x1 \<in> sset (x1 ## x2)"
neg_numeral_less_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = (n < m)"
digraph_dg_10: fixes \<alpha> :: "V"   and \<aa> :: "V" assumes "\<Z> \<alpha>"   and "\<aa> \<in>\<^sub>\<circ> Vset \<alpha>" shows "digraph \<alpha> (dg_10 \<aa>)"
sig_inv_setD': fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and v :: "'t" assumes "r \<in> sig_inv_set' j"   and "v \<in> keys r" shows "component_of_term v < j"
only_touches_opening: fixes i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and v :: "nat"   and v' :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b" assumes "only_touches i branch xs"   and "(v, v') \<in> xs"   and "branch !. v = Some (ps, a)" shows "i = a"
f: shows "finite {i \<in> I. f i \<noteq> 0}"
distinct_replace: fixes fs :: "'a list"   and newFs :: "'a list"   and oldF :: "'a" assumes "distinct fs"   and "distinct newFs"   and "set fs \<inter> set newFs \<subseteq> {oldF}" shows "distinct (replace oldF newFs fs)"
grid_stepsize_nonneg: fixes j :: "nat" shows "0 \<le> stepsize j"
sums_def: fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b" shows "f sums s = (\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> s"
ide_char: fixes a :: "('o, 'a) arr" shows "ide a = (arr a \<and> local.dom a = a \<and> cod a = a)"
thn: shows "n \<noteq> 0 \<longrightarrow> v' \<in> E e2 \<rho>"
hyps: fixes P :: "complex mat"   and Q :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "is_quantum_predicate (wlp S P) \<and> wlp S P \<le>\<^sub>L wlp S Q"
uminus_scalar_prod: fixes v :: "'b vec"   and n :: "nat"   and w :: "'b vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "- (v \<bullet> w) = - v \<bullet> w"
mem_simps_1: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
fin2: shows "finite (supp (f (bs \<inter> supp y) y c))"
step: fixes f' :: "'c"   and y :: "'a"   and x :: "'b" assumes "\<And>x. U f' x = U f' x"   and "y \<in> U (F (C (inf (U f) (\<lambda>x. {y. P x y})))) x" shows "P x y"
abs_eq: fixes y :: "'h \<Rightarrow> 'i set" shows "responses_\<I> (Abs_\<I> y) = y"
extremum_unique: fixes a :: "'a multiset" shows "(a \<subseteq># \<Empt>) = (a = \<Empt>)"
singular_chain_mono: fixes p :: "nat"   and X :: "'c topology"   and T :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and S :: "'c set" assumes "singular_chain p (subtopology X T) c"   and "T \<subseteq> S" shows "singular_chain p (subtopology X S) c"
refine_IdI: fixes m :: "'e nres"   and m' :: "'e nres" assumes "m \<le> m'" shows "m \<le> \<Down> Id m'"
axioms_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "mpt M T" shows "qmpt M T"
is_free_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_free unfree (const name)"
offs_pred_ub_less: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" assumes "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "0 < offs_num (length ns) xs index key mi ma i" shows "ns ! i < ub"
lm040: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G"   and "finite G" shows "finite (pseudoAllocation a)"
null_char: shows "B.VV.null = B.VxV.null"